Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d)

Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(d) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(d)

I.

Question 1.
Find the algebraic equation whose roots are 3 times the roots of x3 + 2x2 – 4x + 1 = 0
Solution:
Given equation is f(x) = x3 + 2x2 – 4x + 1 = 0
We require an equation whose roots are 3 times the roots of f(x) = 0
i,e., Required equation is f([latex]\frac{x}{3}[/latex]) = 0
⇒ [latex]\left(\frac{x}{3}\right)^{3}+2\left(\frac{x}{3}\right)^{2}-\frac{4 x}{3}+1=0[/latex]
⇒ [latex]\frac{x^{3}}{27}+\frac{2}{9} x^{2}-\frac{4}{3} x+1=0[/latex]
Multiplying with 27, required equation is x3 + 6x2 – 36x + 27 = 0

Question 2.
Find the algebraic equation whose roots are 2 times the roots of x5 – 2x4 + 3x3 – 2x2 + 4x + 3 = 0
Solution:
Given equation is f(x) = x5 – 2x4 + 3x3 – 2x2 + 4x + 3 = 0
We require an equation whose roots are 2 times the roots of f(x) = 0
Required equation is f([latex]\frac{x}{2}[/latex]) = 0
⇒ [latex]\left(\frac{x}{2}\right)^{5}-2\left(\frac{x}{2}\right)^{4}+3\left(\frac{x}{2}\right)^{3}-2\left(\frac{x}{2}\right)^{2}+4\left(\frac{x}{2}\right)[/latex] + 3 = 0
⇒ [latex]\frac{x^{5}}{32}-2 \cdot \frac{x^{4}}{16}+3 \cdot \frac{x^{3}}{8}-2 \cdot \frac{x^{2}}{4}+4 \cdot \frac{x}{2}+3=0[/latex]
Multiplying with 32, the required equation is
⇒ x5 – 4x4 + 12x3 – 16x2 + 64x + 96 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d)

Question 3.
Find the transformed equation whose roots are the negative of the roots of x4 + 5x3 + 11x + 3 = 0
Solution:
Given f(x) = x4 + 5x3 + 11x + 3 = 0
We want an equation whose roots are -α1, -α2, -α3, -α4
Required equation f(-x) = 0
⇒ (-x)4 + 5(-x)3 + 11(-x) + 3 = 0
⇒ x4 – 5x3 – 11x + 3 = 0

Question 4.
Find the transformed equation whose roots are the negatives of the roots of x7 + 3x5 + x3 – x2 + 7x + 2 = 0
Solution:
Given f(x) = x7 + 3x5 + x3 – x2 + 7x + 2 = 0
We want an equation whose roots are -α1, -α2, ………., -αn
Required equation is f(-x) = 0
⇒ (-x)7 + 3(-x)5 + (-x)3 – (-x)2 + 7(-x) + 2 = 0
⇒ -x7 – 3x5 – x3 – x2 – 7x + 2 = 0
⇒ x7 + 3x5 + x3 + x2 + 7x – 2 = 0

Question 5.
Find the polynomial equation whose roots are the reciprocals of the roots of x4 – 3x3 + 7x2 + 5x – 2 = 0
Solution:
Given equation is f(x) = x4 – 3x3 + 7x2 + 5x – 2 = 0
Required equation is f([latex]\frac{1}{x}[/latex]) = 0
i.e., [latex]\frac{1}{x^{4}}-\frac{3}{x^{3}}+\frac{7}{x^{2}}+\frac{5}{x}-2=0[/latex]
Multiplying with x4
⇒ 1 – 3x + 7x2 + 5x3 – 2x4 = 0
⇒ 2x4 – 5x3 – 7x2 + 3x – 1 = 0

Question 6.
Find the polynomial equation whose roots are the reciprocals of the roots of x5 + 11x4 + x3 + 4x2 – 13x + 6 = 0
Solution:
Given equation is f(x) = x5 + 11x4 + x3 + 4x2 – 13x + 6 = 0
Required equation is f([latex]\frac{1}{x}[/latex]) = 0
[latex]\frac{1}{x^5}+\frac{11}{x^4}+\frac{1}{x^3}+\frac{4}{x^2}-\frac{13}{x}+6=0[/latex]
Multiplying by x5
⇒ 1 + 11x + x2 + 4x3 – 13x4 + 6x5 = 0
⇒ 6x5 – 13x4 + 4x3 + x2 + 11x + 1 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d)

II.

Question 1.
Find the polynomial equation whose roots are the squares of the roots of x4 + x3 + 2x2 + x + 1 = 0
Solution:
Given equation is f(x) = x4 + x3 + 2x2 + x + 1 = 0
Required equation f(√x) = 0
⇒ x2 + x√x + 2x + √x + 1 = 0
⇒ √x(x + 1) = -(x2 + 2x + 1)
Squaring both sides,
⇒ x(x + 1)2 = (x2 + 2x + 1)2
⇒ x(x2 + 2x + 1) = x4 + 4x2 + 1 + 4x3 + 4x + 2x2
⇒ x3 + 2x2 + x = x4 + 4x3 + 6x2 + 4x + 1
⇒ x4 + 3x3 + 4x2 + 3x + 1 = 0

Question 2.
Form the polynomial equation whose roots are the squares of the roots of x3 + 3x2 – 7x + 6 = 0
Solution:
Given equation is f(x) = x3 + 3x2 – 7x + 6 = 0
Required equation is f(√x) = 0
⇒ x√x + 3x – 7√x + 6 = 0
⇒ √x(x – 7) = -(3x + 6)
Squaring on both sides,
⇒ x(x – 7)2 = (3x + 6)2
⇒ x(x2 – 14x + 49) = 9x2 + 36 + 36x
⇒ x3 – 14x2 + 49x – 9x2 – 36x – 36 = 0
⇒ x3 – 23x2 + 13x – 36 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d)

Question 3.
Form the polynomial equation whose roots are cubes of the roots of x3 + 3x2 + 2 = 0
Solution:
Given equation is x3 + 3x2 + 2 = 0
Put y = x3 so that x = y1/3
∴ y + 3y2/3 + 2 = 0
∴ 3y2/3 = -(y + 2)
Cubing on both sides,
27y2 = -(y + 2)3 = -(y3 + 6y2 + 12y + 8)
∴ y3 + 6y2 + 12y + 8 = 0
⇒ y3 + 33y2 + 12y + 8 = 0
Required equation is x3 + 33x2 + 12x + 8 = 0

III.

Question 1.
Find the polynomial equation whose roots are the translates of those of the equation x4 – 5x3 + 7x2 – 17x + 11 = 0 by -2.
Solution:
Given equation is f(x) = x4 – 5x3 + 7x2 – 17x + 11 = 0
The required equation is f(x + 2) = 0
(x + 2)4 – 5(x + 2)3 + 7(x + 2)2 – 17(x + 2) + 11 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q1
Required equation is x4 + 3x3 + x2 – 17x – 19 = 0

Question 2.
Find the polynomial equation whose roots are the translates of those of x5 – 4x4 + 3x2 – 4x + 6 = 0 by -3.
Solution:
Given equation is f(x) = x5 – 4x4 + 3x2 – 4x + 6 = 0
Required equation is f(x + 3) = 0
(x + 3)5 – 4(x + 3)4 + 3(x + 3)2
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q2
Required equation is x5 + 11x4 + 42x3 + 57x2 – 13x – 60 = 0

Question 3.
Find the polynomial equation whose roots are the translates of the roots of the equation x4 – x3 – 10x2 + 4x + 24 = 0 by 2.
Solution:
Given f(x) = x4 – x3 – 10x2 + 4x + 24 = 0
Required equation is f(x – 2) = 0
(x – 2)4 – (x – 2)3 – 10(x – 2)2 + 4(x – 2) + 24 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q3
Required equation is x4 – 9x3 + 20x2 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d)

Question 4.
Find the polynomial equation whose roots are the translates of the equation 3x5 – 5x3 + 7 = 0 by 4.
Solution:
Given f(x) = 3x5 – 5x3 + 7 = 0
Required equation is f(x – 4) = 0
3(x – 4)5 – 5(x – 4)3 + 7 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q4
Required equation is x5 – 60x4 + 475x3 – 1860x2 + 3600x – 2745 = 0

Question 5.
Transform each of the following equations into ones in which of the coefficients of the second highest power of x is zero and also find their transformed equations.
(i) x3 – 6x2 + 10x – 3 = 0
Solution:
Given equation is x3 – 6x2 + 10x – 3 = 0
To remove the second term diminish the roots by [latex]-\frac{a_1}{n a_0}=\frac{6}{3}=2[/latex]
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q5(i)
Required equation is x3 – 2x + 1 = 0

(ii) x4 + 4x3 + 2x2 – 4x – 2 = 0
Solution:
Given equation is x4 + 4x3 + 2x2 – 4x – 2 = 0
Diminishing the roots by [latex]-\frac{a_1}{n a_0}=\frac{-4}{4}=-1[/latex]
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q5(ii)
Required equation is x4 – 4x2 + 1 = 0

(iii) x3 – 6x2 + 4x – 7 = 0
Solution:
Given equation is x3 – 6x2 + 4x – 7 = 0
Diminishing the roots by [latex]-\frac{a_1}{n a_0}=\frac{6}{3}=2[/latex]
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q5(iii)
Required equation is x3 – 8x – 15 = 0

(iv) x3 + 6x2 + 4x + 4 = 0
Solution:
Given equation is x3 + 6x2 + 4x + 4 = 0
To remove the second term diminish the roots by [latex]\frac{-a_1}{n a_0}=-\frac{6}{3}=-2[/latex]
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q5(iv)
Required equation is x3 – 8x + 12 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d)

Question 6.
Transform each of the following equations into ones in which the coefficients of the third highest power of x are zero.
Hint: To remove the rth term in an equation f(x) = 0 of degree n diminish the roots by ‘h’ such that [latex]f^{(n-r+1)}(h)=0[/latex]
(i) x4 + 2x3 – 12x2 + 2x – 1 = 0
Solution:
Let f(x) = x4 + 2x3 – 12x2 + 2x – 1
To remove the 3rd term, diminish the roots by h such that f”(h) = 0
f'(x) = 4x3 + 6x2 – 24x + 2
f”(x) = 12x2 + 12x – 24
f”(h) = 0
⇒ 12h2 + 12h – 24 = 0
⇒ h2 + h – 2 = 0
⇒ (h + 2) (h – 1) = 0
⇒ h = -2 or 1
Case (i):
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q6(i)
Transformed equation is x4 – 6x3 + 42x – 53 = 0
Case (ii):
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q6(i).1
Transformed equation is x4 – 6x3 – 12x – 8 = 0
∴ The required equation is x4 – 6x3 + 42x – 53 = 0
or x4 + 6x3 – 12x – 8 = 0

(ii) x3 + 2x2 + x + 1 = 0
Solution:
Let f(x) = x3 + 2x2 + x + 1
To remove the 3rd term, diminish the roots by h such that f'(h) = 0, f'(x) = 3x2 + 4x + 1
f'(h) = 0
⇒ 3h2 + 4h + 1 = 0
⇒ (3h + 1) (h + 1)
⇒ h = -1, [latex]-\frac{1}{3}[/latex]
Case (i):
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q6(ii)
Transformed equation is x3 – x2 + 1 = 0
Case (ii):
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q6(ii).1
Transformed equation is x3 + x2 + [latex]\frac{23}{27}[/latex] = 0
⇒ 27x3 + 27x2 + 23 = 0
∴ The required equation is x3 – x2 + 1 = 0 or 27x3 + 27x2 + 23 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d)

Question 7.
Solve the following equations.
(i) x4 – 10x3 + 26x2 – 10x + 1 = 0
Solution:
This is a standard reciprocal equation.
Dividing with x2
[latex]x^2-10 x+26-\frac{10}{x}+\frac{1}{x^2}=0[/latex]
[latex]\left(x^2+\frac{1}{x^2}\right)-10\left(x+\frac{1}{x}\right)+26=0[/latex] …….(1)
put a = x + [latex]\frac{1}{x}[/latex]
[latex]x^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2[/latex] = a2 – 2
Substituting in (1)
a2 – 2 – 10a + 26 = 0
⇒ a2 – 10a + 24 = 0
⇒ (a – 4)(a – 6) = 0
⇒ a = 4 or 6
Case (i): a = 4
x + [latex]\frac{1}{x}[/latex] = 4
⇒ x2 + 1 = 4x
⇒ x2 – 4x + 1 = 0
⇒ x = [latex]\frac{4 \pm \sqrt{16-4}}{2}=\frac{4 \pm 2 \sqrt{3}}{2}[/latex]
⇒ x = 2 ± √3
Case (ii): a = 6
x + [latex]\frac{1}{x}[/latex] = 6
⇒ x2 + 1 = 6x
⇒ x2 – 6x + 1 = 0
⇒ x = [latex]\frac{6 \pm \sqrt{36-4}}{2}[/latex]
⇒ x = [latex]\frac{2(3 \pm 2 \sqrt{2)}}{2}[/latex]
⇒ x = 3 ± 2√2
∴ The roots are 3 ± 2√2, 2 ± √3

(ii) 2x5 + x4 – 12x3 – 12x2 + x + 2 = 0
Solution:
Given f(x) = 2x5 + x4 – 12x3 – 12x2 + x + 2 = 0
This is an odd-degree reciprocal equation of the first type.
∴ -1 is a root.
Dividing f(x) with x + 1
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(d) III Q7(ii)
Dividing f(x) by (x + 1), we get
2x4 – x3 – 11x2 – x + 2 = 0
Dividing by x2
[latex]2 x^2-x-11-\frac{1}{x}+\frac{2}{x^2}=0[/latex]
[latex]2\left(x^2+\frac{1}{x^2}\right)-\left(x+\frac{1}{x}\right)-11=0[/latex] ……..(1)
Put a = x + [latex]\frac{1}{x}[/latex] so that
[latex]x^2+\frac{1}{x^2}=a^2-2[/latex]
Substituting in (1), the required equation is
⇒ 2(a2 – 2) – a – 11 = 0
⇒ 2a2 – 4 – a – 11 = 0
⇒ 2a2 – a – 15 = 0
⇒ (a – 3) (2a + 5) = 0
⇒ a = 3 or [latex]-\frac{5}{2}[/latex]
Case (i): a = 3
x + [latex]\frac{1}{x}[/latex] = 3
⇒ x2 + 1 = 3x
⇒ x2 – 3x + 1 = 0
⇒ x = [latex]\frac{3 \pm \sqrt{9-4}}{2}=\frac{3 \pm \sqrt{5}}{2}[/latex]
Case (ii): a = [latex]-\frac{5}{2}[/latex]
⇒ [latex]x+\frac{1}{x}=-\frac{5}{2}[/latex]
⇒ [latex]\frac{x^2+1}{x}=-\frac{5}{2}[/latex]
⇒ 2x2 + 2 = -5x
⇒ 2x2 + 5x + 2 = 0
⇒ (2x + 1) (x + 2) = 0
⇒ x = [latex]-\frac{1}{2}[/latex], -2
∴ The roots are -1, [latex]-\frac{1}{2}[/latex], -2, [latex]\frac{3 \pm \sqrt{5}}{2}[/latex]

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c)

Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(c) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(c)

I.

Question 1.
Form the polynomial equation whose roots are
(i) 2 + 3i, 2 – 3i, 1 + i, 1 – i
Solution:
The required equation is [x – (2 + 3i)] [x – (2 – 3i)] [x – (1 + i)][x – (1 – i)] = 0
⇒ [(x – 2) – 3i)] [(x – 2) + 3i] [(x – 1) – i] [(x – 1) + i] = 0
⇒ [(x – 2)2 – 9i2] [(x – 1)2 – i2] = 0
⇒ (x2 – 4x + 4 + 9) (x2 – 2x + 1 + 1) = 0
⇒ (x2 – 4x + 13) (x2 – 2x + 2) = 0
⇒ x4 – 4x3 + 13x2 – 2x3 + 8x2 – 26x + 2x2 – 8x + 26 = 0
⇒ x4 – 6x3 + 23x2 – 34x + 26 = 0

(ii) 3, 2, 1 + i, 1 – i
Solution:
Required equation is (x – 3) (x – 2) [x – (1 + i)] [ x – (1 – i)] = 0
⇒ (x2 – 5x + 6) [(x – 1) – i] [(x – 1) + i] = 0
⇒ (x2 – 5x + 6) [(x – 1)2 – i2] = 0
⇒ (x2 – 5x + 6) (x2 – 2x + 1 + 1) = 0
⇒ (x2 – 5x + 6) (x2 – 2x + 2) = 0
⇒ x4 – 5x3 + 6x2 – 2x3 + 10x2 – 12x + 2x2 – 10x + 12 = 0
⇒ x4 – 7x3 + 18x2 – 22x + 12 = 0

(iii) 1 + i, 1 – i, -1 + i, -1 – i
Solution:
Required equation is [x – (1 + i)] [x – (1 – i)] [x – (-1 + i)] [x – (-1 – i)] = 0
⇒ [(x – 1) – i][(x – i) + i] [(x + 1) – i] [(x + 1) + i] = 0
⇒ [(x – 1)2 – i2] [(x + 1)2 – i2] = 0
⇒ (x2 – 2x + 1 + 1) (x2 + 2x + 1 + 1) = 0
⇒ (x2 – 2x + 2) (x2 + 2x + 2) = 0
⇒ x4 – 2x3 + 2x2 + 2x3 – 4x2 + 4x + 2x2 – 4x + 4 = 0
⇒ x4 + 4 = 0

(iv) 1 + i, 1 – i, 1 + i, 1 – i
Solution:
Required equation is [x – (1 + i)] [x – (1 – i)] [x – (1 + i)] [x – (1 – i)] = 0
⇒ [(x – 1) – i]2 [(x – 1) + i]2 = 0
⇒ [(x – 1)2 – i2] = 0
⇒ (x2 – 2x + 1 + 1)2 = 0
⇒ x4 + 4x2 + 4 – 4x3 + 4x2 – 8x = 0
⇒ x4 – 4x3 + 8x2 – 8x + 4 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c)

Question 2.
Form the polynomial equation with rational coefficients whose roots are
(i) 4√3, 5 + 2i
Solution:
For the polynomial equation with rational coeffs. the roots are conjugate surds and conjugate complex numbers.
4√3, 5 + 2i
Let α = 4√3 then β = -4√3, and γ = 5 + 2i then δ = 5 – 2i
α, β, γ, δ are the roots
α + β = 0, αβ = -48
γ + δ = 10, γδ = 25 + 4 = 29
The required equation is [x2 – (α + β)x + αβ] [x2 – (γ + δ)x + γδ] = 0
⇒ (x2 – 48) (x2 – 10x + 29) = 0
⇒ x4 – 10x3 + 29x2 – 48x2 + 480x – 1932 = 0
⇒ x4 – 10x3 – 19x2 + 480x – 1932 = 0

(ii) 1 + 5i, 5 – i
Solution:
For the polynomial equation with rational coeffs. the roots are conjugate surds and conjugate complex numbers.
Let α = 1 + 5i then β = 1 – 5i,
and γ = 5 + i then δ = 5 – i
α + β = 2, αβ = 26
γ + δ = 10, γδ = 26
The required equation is [x2 – (α + β)x + αβ] [x2 – (γ + δ)x + γδ] = 0
⇒ (x2 – 2x + 26) (x2 – 10x + 26) = 0
⇒ x4 – 12x3 + 72x2 – 312x + 676 = 0

(iii) i – √5
Solution:
For the polynomial equation with rational coeffs. the roots are conjugate surds and conjugate complex numbers.
Let α = i – √5, β = i + √5, γ = -i – √5, δ = -i + √5 are the roots
α + β = 2i, αβ = -6
γ + δ = -2i, γδ = -6
The required equation is [x2 – (α + β)x + αβ][x2 – (γ + δ)x + γδ] = 0
⇒ (x2 – 2ix – 6) (x2 + 2ix – 6) = 0
⇒ [(x2 – 6) – 2ix] [(x2 – 6) + 2ix] = 0
⇒ (x2 – 6)2 + 4x2 = 0
⇒ x4 + 36 – 12x2 + 4x2 = 0
⇒ x4 – 8x2 + 36 = 0

(iv) -√3 + i√2
Solution:
Let α = -√3 + i√2, β = -√3 – i√2, γ = √3 – i√2, γ = √3 + i√2
α + β = -2√3
αβ = (-√3)2 – (i√2)2
= 3 – i2 (2)
= 5
γ + δ = 2√3
γδ = 5
The required equation is [(x2 – (α + β)x + αβ)] [(x2 – (γ + δ)x + γδ)] = 0
⇒ (x2 + 2√3x + 5) (x2 – 2√3x + 5) = 0
⇒ (x2 + 5)2 – (2√3x)2 = 0
⇒ x4 + 25 + 10x2 – 12x2 = 0
⇒ x4 – 2x2 + 25 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c)

II.

Question 1.
Solve the equation x4 + 2x3 – 5x2 + 6x + 2 = 0 given that 1 + i is one of its roots.
Solution:
Let 1 + i be one root ⇒ 1 – i be another root
The equation having roots 1 ± i is x2 – 2x + 2 = 0
∴ x2 – 2x + 2 is a factor of x4 + 2x3 – 5x2 + 6x + 2 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q1
∴ The roots of the given equation are 1 ± i, -2 + √3

Question 2.
Solve the equation 3x3 – 4x2 + x + 88 = 0 which has 2 – √-7 as a root.
Solution:
Let 2 – √-7 (i.e) 2 – √7i is one root
⇒ 2 + √7i is another root.
The equation having roots 2 ± √7i is x2 – 4x + 11 = 0
∴ x2 – 4x + 11 is a factor of the given equation.
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q2
3x + 8 = 0
⇒ x = [latex]-\frac{8}{3}[/latex]
∴ The roots of the given equation are 2 ± √7i, [latex]-\frac{8}{3}[/latex]

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c)

Question 3.
Solve x4 – 4x2 + 8x + 35 = 0, given that 2 + i√3 is a root.
Solution:
Let 2 + i√3 be one root
⇒ 2 – i√3 is another root.
The equation having roots 2 ± i√3 is x2 – 4x + 7 = 0
∴ x2 – 4x + 7 is a factor of x4 – 4x2 + 8x + 35
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q3
∴ The roots of the given equation are 2 ± i√3, -2 ± i

Question 4.
Solve the equation x4 – 6x3 + 11x2 – 10x + 2 = 0, given that 2 + √3 is a root of the equation.
Solution:
2 + √3 is one root
⇒ 2 – √3 is another root.
The equation having the roots of 2 ± √3 is x2 – 4x + 1 = 0
∴ x2 – 4x + 1 is a factor of x4 – 6x3 + 11x2 – 10x + 2 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q4
∴ The roots of the required equation are 2 ± √3, 1 ± i

Question 5.
Given that -2 + √-7 is a root of the equation x4 + 2x2 – 16x + 77 = 0, solve it completely.
Solution:
-2 – √-7 (i.e) -2 + i√7 is one root
⇒ -2 – i√7 is another root.
The equation having the roots of -2 ± i√7 is x2 + 4x + 11 = 0
∴ x2 + 4x + 11 is a factor of x4 + 2x2 – 16x + 77
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q5
∴ The roots of the required equation are -2 ± i√7, 2 ± √3i

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c)

Question 6.
Solve the equations x4 + 2x3 – 16x2 – 22x + 7 = 0, given that 2 – √3 is a root of it.
Solution:
2 + √3 is a root ⇒ 2 – √3 is also a root.
The equation having roots 2 ± √3 is
x2 – (2 + √3 + 2 – √3)x + (2 + √3) (2 – √3) = 0
⇒ x2 – 4x + 1 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q6
x2 + 6x + 7 = 0
⇒ x = [latex]\frac{-6 \pm \sqrt{36-28}}{2}[/latex]
⇒ x = -3 ± √2
∴ Roots are 2 ± √3, -3 ± √2

Question 7.
Solve the equation 3x5 – 4x4 – 42x3 + 56x2 + 27x – 36 = 0 given that √2 + √5 is one of its roots.
Solution:
√2 + √5 is a root
⇒ -√2 – √5, -√2 + √5, -√2 – √5 are also roots.
The equation haying roots √2 + √5 is
x2 – (√2 + √5 + √2 – √5) + (√2 + √5) (√2 – √5) = 0
⇒ x2 – 2√2x – 3 = 0
The equation having roots -√2 ± √5 is
x2 – (-√2 + √5 – √2 – √5) + (√2 + √5)(-√2 – √5) = 0
⇒ x2 + 2√x – 3 = 0
The equation having roots ±√2 ± √5 is (x2 + 2√2x – 3) (x2 – 2√2x – 3) = 0
⇒ (x2 – 3)2 – (2√2x)2 = 0
⇒ x4 – 6x2 + 9 – 8x2 = 0
⇒ x4 – 14x2 + 9 = 0
∴ 3x5 – 4x4 – 42x3 + 56x2 + 27x – 36 = 0
⇒ 3x(x4 – 14x2 + 9) – 4(x4 – 14x2 + 9) = 0
⇒ (x4 – 14x2 + 9) (3x – 4) = 0
⇒ x = ±√2 ± √5 or 4/3
∴ The roots are ±√2 ± √5, 4/3

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c)

Question 8.
Solve the equation x4 – 9x3 + 27x2 – 29x + 6 = 0, given that one root of it is 2 – √3.
Solution:
2 – √3 is one root ⇒ 2 + √3 is another root.
The equation having the roots of 2 ± √3 is x2 – 4x + 1 = 0
∴ x2 – 4x + 1 is a factor of the given equation.
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q8
x2 – 5x + 6 = 0
⇒ (x – 2) (x – 3) = 0
⇒ x = 2, 3
∴ The roots of the required equations are 2 ± √3, 2, 3

Question 9.
Show that the equation [latex]\frac{a^{2}}{x-a^{\prime}}+\frac{b^{2}}{x-b^{\prime}}+\frac{c^{2}}{x-c^{\prime}}+\ldots+\frac{k^{2}}{x-k^{\prime}}[/latex]= x – m Where a, b, c ….k, m, a’, b’, c’…. k’ are all real numbers, cannot have a non-real root.
Solution:
Let α + iβ be the root of the given equation.
Suppose if β ≠ 0, then α – iβ is also a root of the given equation.
Substitute α + iβ in the given equation, and we get
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(c) II Q9
[latex]\left[\frac{a^{2}}{\left(\alpha-a^{\prime}\right)^{2}+\beta^{2}}+\frac{b^{2}}{\left(\alpha-b^{\prime}\right)^{2}+\beta^{2}}+\ldots \frac{k^{2}}{\left(\alpha-k^{\prime}\right)^{2}+\beta^{2}}+1\right][/latex]
= 0
⇒ β = 0
This is a contradiction.
∴ The given equation cannot have non-real roots.

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(b) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(b)

I.

Question 1.
Solve x3 – 3x2 – 16x + 48 = 0, given that the sum of two roots is zero.
Solution:
Let α, β, γ are the roots of x3 – 3x2 – 16x + 48 = 0
α + β + γ = 3
Given α + β = 0 (∵ Sum of two roots is zero)
∴ γ = 3
i.e. x – 3 is a factor of x3 – 3x2 – 16x + 48 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) I Q1
x2 – 16 = 0
⇒ x2 = 16
⇒ x = ±4
∴ The roots are -4, 4, 3

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 2.
Find the condition that x3 – px2 + qx – r = 0 may have the sum of its roots zero.
Solution:
Let α, β, γ be the roots of x3 – px2 + qx – r = 0
α + β + γ = p ………(1)
αβ + βγ + γα = q ………..(2)
αβγ = r ………..(3)
Given α + β = 0 (∵ Sum of two roots is zero)
From (1), γ = p
∴ γ is a root of x3 – px2 + qx – r = 0
γ3 – pγ2 + qγ – r = 0
But γ = p
p3 – p(p2) + q(p) – r = 0
p3 – p2 + qp – r = 0
∴ qp = r is the required condition.

Question 3.
Given that the roots of x3 + 3px2 + 3qx + r = 0 are in
(i) A.P., show that 2p3 – 3qp + r = 0
(ii) G.P., show that p3r = q3
(iii) H.P., show that 2q3 = r(3pq – r)
Solution:
Given equation is x3 + 3px2 + 3qx + r = 0
(i) The roots are in A.P.
Suppose a – d, a, a + d are the roots
Sum = a – d + a + a + d = -3p
⇒ 3a = -3p
⇒ a = -p ……….(1)
∵ ‘a’ is a root of x3 + 3px2 + 3qx + r = 0
⇒ a3 + 3pa2 + 3qa + r = 0
But a = -p
⇒ p3 + 3p(-p)2 + 3q(-p) + r = 0
⇒ 2p3 – 3pq + r = 0 is the required condition

(ii) The roots are in G.P.
Suppose the roots be [latex]\frac{a}{R}[/latex], a, aR
Given ([latex]\frac{a}{R}[/latex]) (a) (aR) = -r
⇒ a = -r
⇒ a = (-r)1/3
∵ ‘a’ is a root of x3 + 3px2 + 3qx + r = 0
⇒ (-r1/3)3 + 3p(-r1/3)2 + 3q(-r1/3) + r = 0
⇒ -r + 3pr2/3 – 3qr1/3 + r = 0
pr2/3 = qr1/3
⇒ pr1/3 = q
⇒ p1/3r = q is the required condition

(iii) The roots of x3 + 3px2 + 3qx + r = 0 are in H.P.
x3 + 3px2 + 3qx + r = 0 ……..(1)
Let y = [latex]\frac{1}{x}[/latex] so that [latex]\frac{1}{y^{3}}+\frac{3 p}{y^{2}}+\frac{3 q}{y}+r[/latex] = 0
Roots of ry3 + 3qy2 + 3py + 1 = 0 are in A.P.
ry3 + 3qy2 + 3py + 1 = 0 ……..(2)
Suppose a – d, a, a + d be the roots of (2)
Sum = a – d + a + a + d = [latex]-\frac{3 q}{r}[/latex]
3a = [latex]-\frac{3 q}{r}[/latex]
a = [latex]-\frac{q}{r}[/latex] ………(1)
∵ ‘a’ is a root of ry3 + 3qy2 + 3py + 1 = 0
⇒ ra3 + 3qa2 + 3pa + 1 = 0
But a = [latex]-\frac{q}{r}[/latex]
⇒ [latex]r\left(-\frac{q}{r}\right)^{3}+3 q\left(-\frac{q}{r}\right)^{2}+3 p\left(-\frac{q}{r}\right)+1=0[/latex]
⇒ [latex]\frac{-q^{3}}{r^{2}}+\frac{3 q^{3}}{r^{2}}-\frac{3 p q}{r}+1=0[/latex]
⇒ -q3 + 3q3 – 3pqr + r2 = 0
⇒ 2q3 = r(3pq – r) is the required condition.

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 4.
Find the condition that x3 – px2 + qx – r = 0 may have the roots in G.P.
Solution:
Let [latex]\frac{a}{R}[/latex], a, aR be the roots of x3 – px2 + qx – r = 0
The product of the roots = [latex]\frac{a}{R}[/latex] . a . aR = a3
product of the roots = r
⇒ a = r1/3
∵ a is a root of x3 – px2 + qx – r = 0
⇒ a3 – pa2 + qa – r = 0
But a = r1/3
⇒ (r1/3)3 – p(r1/3)2 + q(r1/3) – r = 0
⇒ r – p . r2/3 + q . r1/3 – r = 0
⇒ p . r2/3 = qr1/3
By cubing on both sides
⇒ p3r2 = q3r
⇒ p3r = q3 is the required condition

II.

Question 1.
Solve 9x3 – 15x2 + 7x – 1 = 0, given that two of its roots are equal.
Solution:
Suppose α, β, γ are the roots of 9x3 – 15x3 + 7x – 1 = 0
α + β + γ = [latex]\frac{15}{9}=\frac{5}{3}[/latex]
αβ + βγ + γα = [latex]\frac{7}{9}[/latex]
αβγ = [latex]\frac{1}{9}[/latex]
Given α = β (∵ two of its roots are equal)
2α + γ = [latex]\frac{5}{3}[/latex]
⇒ γ = [latex]\frac{5}{3}[/latex] – 2α
α2 + 2αγ = [latex]\frac{7}{9}[/latex]
⇒ α2 + 2α ([latex]\frac{5}{3}[/latex] – 2α) = [latex]\frac{7}{9}[/latex]
⇒ α2 + [latex]\frac{2 \alpha(5-6 \alpha)}{3}=\frac{7}{9}[/latex]
⇒ 9α2 + 6α(5 – 6α) = 7
⇒ 9α2 + 30α – 36α2 = 7
⇒ 27α2 – 30α + 7 = 0
⇒ (3α – 1)(9α – 7) = 0
⇒ α = [latex]\frac{1}{3}[/latex] or [latex]\frac{7}{9}[/latex]
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q1
The roots are [latex]\frac{1}{3}[/latex], [latex]\frac{1}{3}[/latex], 1

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 2.
Given that one root of 2x3 + 3x2 – 8x + 3 = 0 is double of another root, find the roots of the equation.
Solution:
Suppose α, β, γ are the roots of 2x3 + 3x2 – 8x + 3 = 0
α + β + γ = [latex]-\frac{3}{2}[/latex] ……..(1)
αβ + βγ + γα = -4 ……..(2)
αβγ = [latex]-\frac{3}{2}[/latex]
Given α = 2β (∵ one root is double the other)
Substituting in (1)
3β + γ = [latex]-\frac{3}{2}[/latex]
⇒ γ = [latex]-\frac{3}{2}[/latex] – 3β …….(4)
Substituting in (2)
αβ + γ(α + β) = -4
⇒ 2β2 + 3βγ = -4
⇒ 2β2 + 3β([latex]-\frac{3}{2}[/latex] – 3β) = -4
⇒ 2β2 – [latex]\frac{3 \beta(3+6 \beta)}{2}[/latex] = -4
⇒ 4β2 – 9β – 18β2 = -8
⇒ 14β2 + 9β – 8 = 0
⇒ (2β – 1)(7β + 8) = 0
⇒ 2β – 1 = 0 or 7β + 8 = 0
⇒ β = [latex]\frac{1}{2}[/latex] or β = [latex]-\frac{8}{7}[/latex]
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q2
∴ The roots are [latex]\frac{1}{2}[/latex], 1 and -3

Question 3.
Solve x3 – 9x2 + 14x + 24 = 0, given that two of the roots are in the ratio 3 : 2.
Solution:
Suppose α, β, γ are the roots of x3 – 9x2 + 14x + 24 = 0
α + β + γ = 9 ………(1)
αβ + βγ + γα = 14 ……….(2)
αβγ = -24 ……….(3)
∵ two roots are in the ratio 3 : 2
Let α : β = 3 : 2
⇒ β = [latex]\frac{2 \alpha}{3}[/latex]
Substituting in (1)
[latex]\frac{5 \alpha}{3}[/latex] + γ = 9
⇒ γ = 9 – [latex]\frac{5 \alpha}{3}[/latex] ………(4)
Substituting in (2)
⇒ [latex]\frac{2}{3}[/latex] α2 + γ(α + β) = 14
⇒ [latex]\frac{2}{3} \alpha^{2}+\left(9-\frac{5 \alpha}{3}\right) \cdot \frac{5 \alpha}{3}[/latex] = 14
⇒ 2α2 + 5α(9 – [latex]\frac{5 \alpha}{3}[/latex]) = 42
⇒ 2α2 + 5α [latex]\frac{(27-5 \alpha)}{3}[/latex] = 42
⇒ 6α2 + 135α – 25α2 = 126
⇒ 19α2 – 135α + 126 = 0
⇒ 19α2 – 114α – 21α + 126 = 0
⇒ 19α(α – 6) – 21(α – 6) = 0
⇒ (19α – 21)(α – 6) = 0
⇒ 19α – 21 = 0 or α – 6 = 0
⇒ α = [latex]\frac{21}{19}[/latex] or α = 6
Case (i): α = 6
β = [latex]\frac{2 \alpha}{3}[/latex]
= [latex]\frac{2}{3}[/latex] × 6
= 4
γ = 9 – [latex]\frac{5 \alpha}{3}[/latex]
= 9 – [latex]\frac{5}{3}[/latex] × 6
= 9 – 10
= -1
α = 6, β = 4, γ = -1 satisfy αβγ = -24
∴ The roots are 6, 4, -1

Case (ii): α = [latex]\frac{21}{19}[/latex]
β = [latex]\frac{2}{3} \times \frac{21}{19}=\frac{14}{19}[/latex]
γ = 9 – [latex]\frac{5 \alpha}{3}[/latex]
= 9 – [latex]\frac{5}{3} \cdot \frac{21}{19}[/latex]
= [latex]\frac{136}{19}[/latex]
These values do not satisfy αβγ= -24
∴ The roots are 6, 4, -1.

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 4.
Solve the following equations, given that the roots of each are in A.P.
(i) 8x3 – 36x2 – 18x + 81 = 0
Solution:
Given the roots of 8x3 – 36x2 – 18x + 81 = 0 are in A.P.
Let the roots be a – d, a, a + d
Sum of the roots = [latex]\frac{36}{8}[/latex]
⇒ a – d + a + a + d = [latex]\frac{9}{2}[/latex]
⇒ 3a = [latex]\frac{9}{2}[/latex]
⇒ a = [latex]\frac{3}{2}[/latex]
∴ (x – [latex]\frac{3}{2}[/latex]) is a factor of 8x3 – 36x2 – 18x + 81 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q4(i)
⇒ 8x2 – 24x – 54 = 0
⇒ 4x2 – 12x – 27 = 0
⇒ 4x2 – 18x + 6x – 27 = 0
⇒ 2x(2x – 9) + 3(2x – 9) = 0
⇒ (2x + 3) (2x – 9) = 0
⇒ x = [latex]-\frac{3}{2}, \frac{9}{2}[/latex]
∴ The roots are [latex]-\frac{3}{2}, \frac{3}{2}, \frac{9}{2}[/latex]

(ii) x3 – 3x2 – 6x + 8 = 0
Solution:
The roots of x3 – 3x2 – 6x + 8 = 0 are in A.P
Suppose a – d, a, a + d be the roots
Sum = a – d + a + a + d = 3
⇒ 3a = 3
⇒ a = 1
∴ (x – 1) is a factor of x3 – 3x2 – 6x + 8 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q4(ii)
⇒ x2 – 2x – 8 = 0
⇒ x2 – 4x + 2x – 8 = 0
⇒ x(x – 4)+ 2(x – 4) = 0
⇒ (x – 4) (x + 2) = 0
⇒ x = 4, -2
∴ The roots are -2, 1, 4

Question 5.
Solve the following equations, given that the roots of each are in G.P.
(i) 3x3 – 26x2 + 52x – 24 = 0
Solution:
Given equation is 3x3 – 26x2 + 52x – 24 = 0
The roots are in G.P.
Suppose [latex]\frac{a}{r}[/latex], a, ar are the roots.
Product = [latex]\frac{a}{r}[/latex] . a . ar = [latex]-\left(-\frac{24}{3}\right)[/latex]
⇒ a3 = 8
⇒ a = 2
∴ (x – 2) is a factor of 3x3 – 26x2 + 52x – 24
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q5(i)
Hint: 3 × 12 = 3 ×6 × 2 = (-18)(-2)
⇒ 3x2 – 20x + 12 = 0
⇒ 3x2 – 18x – 2x + 12 = 0
⇒ 3x(x – 6) – 2(x – 6) = 0
⇒ (3x – 2) (x – 6) = 0
⇒ x = [latex]\frac{2}{3}[/latex], 6
∴ The roots are [latex]\frac{2}{3}[/latex], 2, 6.

(ii) 54x3 – 39x2 – 26x + 16 = 0
Solution:
Given equation is 54x3 – 39x2 – 26x + 16 = 0
The roots are in G.P.
Suppose [latex]\frac{a}{r}[/latex], a, ar be the roots.
Product = [latex]\frac{a}{r}[/latex] . a . ar = [latex]-\frac{16}{54}[/latex]
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q5(ii)
Hint: 18 × 8 = 9 × 2 × 8 = (-9) (-16)
⇒ 54x2 – 75x + 24 = 0
⇒ 18x2 – 25x + 8 = 0
⇒ 18x2 – 9x – 16x + 8 = 0
⇒ 9x(2x – 1) – 8(2x- 1) = 0
⇒ (9x – 8) (2x – 1) = 0
⇒ x = [latex]\frac{8}{9}, \frac{1}{2}[/latex]
∴ The roots are [latex]\frac{8}{9},-\frac{2}{3}, \frac{1}{2}[/latex]

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 6.
Solve the following equations, given that the roots of each are in H.P.
(i) 6x3 – 11x2 + 6x – 1 = 0
Solution:
Given equation is 6x3 – 11x2 + 6x – 1 = 0 …….(1)
Put y = [latex]\frac{1}{x}[/latex] so that [latex]\frac{6}{\mathrm{y}^{3}}-\frac{11}{\mathrm{y}^{2}}+\frac{6}{\mathrm{y}}-1[/latex] = 0
⇒ 6 – 11y + 6y2 – y3 = 0
⇒ y3 – 6y2 + 11y – 6 = 0 ………(2)
Roots of (1) are in H.P.
⇒ Roots of (2) are in A.P.
Let a – d, a, a + d be the roots of (2)
Sum = a – d + a + a + d = 6
⇒ 3a = 6
⇒ a = 2
Product = a(a2 – d2) = 6
⇒ 2(4 – d2) = 6
⇒ 4 – d2 = 3
⇒ d2 = 1
⇒ d = 1
a – d = 2 – 1 = 1,
a = 2
a + d = 2 + 1 = 3
The roots of (2) are 1, 2, 3
The roots of (1) are 1, [latex]\frac{1}{2}[/latex], [latex]\frac{1}{3}[/latex]

(ii) 15x3 – 23x2 + 9x – 1 = 0
Solution:
Given equation is 15x3 – 23x2 + 9x – 1 = 0 …….(1)
Put y = [latex]\frac{1}{x}[/latex] so that [latex]\frac{15}{y^{3}}-\frac{23}{y^{2}}+\frac{9}{y}-1[/latex] = 0
⇒ 15 – 23y + 9y2 – y3 = 0
⇒ y3 – 9y2 + 23y – 15 = 0 ………(2)
Roots of (1) are in H.P. So that roots of (2) are in A.P.
Let a – d, a, a + d be the roots of (2)
Sum = a – d + a + a + d = 9
⇒ 3a = 9
⇒ a = 3
Product = a(a2 – d2) = 15
⇒ 3(9 – d2) = 15
⇒ 9 – d2 = 5
⇒ d2 = 4
⇒ d = 2
a – d = 3 – 2 = 1
a = 3
a + d = 3 + 2 = 5
Roots of (2) are 1, 3, 5
Hence roots of (1) are 1, [latex]\frac{1}{3}[/latex], [latex]\frac{1}{5}[/latex]

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 7.
Solve the following equations, given that they have multiple roots.
(i) x4 – 6x3 + 13x2 – 24x + 36 = 0
Solution:
(i) Let f(x) = x4 – 6x3 + 13x2 – 24x + 36
⇒ f'(x) = 4x3 – 18x2 + 26x – 24
⇒ f'(3) = 4(27) – 18(9) + 26(3) – 24
⇒ f'(3) = 108 – 162 + 78 – 24
⇒ f'(3) = 0
f(3) = 81 – 162 + 117 – 72 + 36 = 0
Hint: Choose the value of x from the factors of the G.C.D of constant terms in f(x) and f'(x).
∴ x – 3 is a factor of f'(x) and f(x)
∴ 3 is the repeated foot of f(x)
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q7(i)
x2 + 4 = 0
⇒ x = ±2i
∴ The roots of the given equation are 3, 3, ±2i

(ii) 3x4 + 16x3 + 24x2 – 16 = 0
Solution:
Let f(x) = 3x4 + 16x3 + 24x2 – 16
f(x) = 12x3 + 48x2 + 48x
= 12x(x2 + 4x + 4)
= 12x (x + 2)2
f'(-2) = 0
f(-2) = 3(16) + 16(-8) + 24(4) – 16 = 0
∴ x + 2 is a factor of f'(x) and f(x)
∴ -2 is a multiple root of f(x) = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) II Q7(ii)
3x2 + 4x – 4 = 0
⇒ 3x2 + 6x – 2x – 4 = 0
⇒ 3x(x + 2) – 2(x + 2) = 0
⇒ (x + 2) (3x – 2) = 0
⇒ x = -2, [latex]\frac{2}{3}[/latex]
∴ The roots of the given equation are -2, -2, -2, [latex]\frac{2}{3}[/latex]

III.

Question 1.
Solve x4 + x3 – 16x2 – 4x + 48 = 0, given that the product of two of the roots is 6.
Solution:
Suppose α, β, γ, δ are the roots of
x4 + x3 – 16x2 – 4x + 48 = 0 ………..(1)
Sum of the roots = -1
⇒ α + β + γ + δ = -1
and Product of the roots = αβγδ = 48
∵ Product of two roots is 6
Let αβ = 6
From (1), γδ = [latex]\frac{48}{\alpha \beta}=\frac{48}{6}[/latex] = 8
Let α + β = p and γ + δ = q
The equation having roots α, β is x2 – (α + β) x + αβ = 0
⇒ x2 – px + 6 = 0 ………..(2)
The equation having the roots γ, δ is x2 – (γ + δ) x + γδ = 0
⇒ x2 – qx + 8 = 0 ……….(3)
∴ From (1), (2) and (3)
x4 + x3 – 16x2 – 4x + 48 = (x2 – px + 6) (x2 – qx + 8)
= x4 – (p + q) x3 + (pq + 14) x2 – (8p + 86q) x + 48
Comparing the like terms,
p + q = -1
8p + 6q = 4 ⇒ 4p + 3q = 2
Solving, q = -6
∴ p = -1 + 6 = 5
Substitute the value of p in eq. (2),
x2 – 5x + 6 = 0 ⇒ x = 2, 3
Substitute/the value of q in eq. (3),
x2 + 6x + 8 = 0 ⇒ x = -2, – 4
∴ The roots of the given equation are -4, -2, 2, 3

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 2.
Solve 8x4 – 2x3 – 27x2 + 6x + 9 = 0 given that two roots have the same absolute value, but are opposite in signs.
Solution:
Suppose α, β, γ, δ are the roots of the equation
8x4 – 2x3 – 27x2 + 6x + 9 = 0
⇒ [latex]x^{4}-\frac{1}{4} x^{3}-\frac{27}{8} x^{2}+\frac{3}{4} x+\frac{9}{8}=0[/latex] …………(1)
Sum of the roots = α + β + γ + δ = [latex]\frac{1}{4}[/latex] and
Product of the roots = αβγδ = [latex]\frac{9}{8}[/latex]
Given β = -α
⇒ α + β = 0
∴ 0 + γ + δ = [latex]\frac{1}{4}[/latex]
⇒ γ + δ = [latex]\frac{1}{4}[/latex]
Let αβ = p, γδ = q, so that pq = [latex]\frac{9}{8}[/latex]
The equation having the roots α, β is x2 – (α + β)x + αβ = 0
⇒ x2 + p = 0 ……….(2)
The equation having the roots γ, δ is x2 – (γ + δ)x + γδ = 0
⇒ x2 – [latex]\frac{1}{4}[/latex] x + q = 0 ……..(3)
From (1), (2) and (3)
[latex]x^{4}-\frac{1}{4} x^{3}-\frac{27}{8} x^{2}+\frac{3}{4} x+\frac{9}{8}[/latex] = (x2 + p) [latex]\left(x^{2}-\frac{1}{4} x+q\right)[/latex]
= [latex]x^{4}-\frac{1}{4} x^{3}+(p+q) x^{2}-\frac{p}{4} x+p q[/latex]
Comparing the coefficients of x and constants
[latex]\frac{-p}{4}=\frac{3}{4}[/latex] ⇒ p = -3
pq = [latex]\frac{9}{8}[/latex]
⇒ q = [latex]\frac{9}{8} \times \frac{-1}{3}=\frac{-3}{8}[/latex]
Substitute the value of p in eq. (2),
x2 – 3 = 0 ⇒ x = ±√3
Substitute the value of q in eq. (3),
[latex]x^{2}-\frac{1}{4} x-\frac{3}{8}=0[/latex]
⇒ 8x2 – 2x – 3 = 0
⇒ (2x + 1) (4x – 3) = 6
⇒ x = [latex]-\frac{1}{2}, \frac{3}{4}[/latex]
∴ The roots of the given equation are -√3, [latex]-\frac{1}{2}, \frac{3}{4}[/latex], √3

Question 3.
Solve 18x3 + 81x2 + 121x + 60 = 0 given that one root is equal to half the sum of the remaining roots.
Solution:
Suppose α, β, γ are the roots of 18x3 + 81x2 + 121x + 60 = 0
sum = α + β + γ = [latex]\frac{-81}{18}=\frac{-9}{2}[/latex] ……….(1)
αβ + βγ + γα = [latex]\frac{121}{18}[/latex] …………(2)
αβγ = [latex]\frac{-60}{18}=\frac{-10}{3}[/latex] ………(3)
∵ One root is equal to half of the sum of the remaining two,
Let α = [latex]\frac{1}{2}[/latex] (β + γ)
Substitute in (1)
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) III Q3

Question 4.
Find the condition in order that the equation ax4 + 4bx3 + 6cx2 + 4dx + e = 0 may have a pair of equal roots.
Solution:
Let α, α, β, β are the roots of the equation.
ax4 + 4bx3 + 6cx2 + 4dx + e = 0
⇒ [latex]x^{4}+\frac{4 b}{a} x^{2}+\frac{6 c}{a} x^{2}+\frac{4 d}{a} x+\frac{e}{a}=0[/latex]
Sum of the roots, 2(α + β) = [latex]-\frac{4 b}{a}[/latex]
⇒ α + β = [latex]-\frac{2 b}{a}[/latex]
⇒ αβ = k (say)
Equation having roots α, β is x2 – (α + β) x + αβ = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) III Q4

Question 5.
(i) Show that x5 – 5x3 + 5x2 – 1 = 0 has three equal roots and find this root.
Solution:
Let f(x) = x5 – 5x3 + 5x2 – 1
f'(x) = 5x4 – 15x2 + 10x = 5x(x3 – 3x + 2)
f'(1) = 5(1) (1 – 3 + 2) = 0
f(1) = 1 – 5 + 5 – 1 = 0
x – 1 is a factor of f'(x) and f(x)
∴ 1 is a repeated root of f(x).
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) III Q5(i)
x3 + 2x2 – 2x – 1 = 0
⇒ 1 is a root of the above equation (∵ sum of the coefficients is zero)
∴ 1 is the required root.

(ii) Find the repeated roots of x5 – 3x4 – 5x3 + 27x2 – 32x + 12 = 0
Solution:
Let f(x) = x5 – 3x4 – 5x3 + 27x2 – 32x + 12
f'(x) = 5x4 – 12x3 – 15x2 + 54x – 32
f'(2) = 5(2)4 – 12(2)3 – 15(2)2 + 54(2) – 32
= 80 – 96 – 60 + 108 – 32
= 0
f(2) = (2)5 – 3(2)4 – 5(2)3 + 27(2)2 – 32(2) + 12
= 32 – 48 – 40 + 108 – 64 + 12
= 152 – 152
= 0
∴ x – 2 is a common factor of f'(x) and f(x)
2 is a multiple root of f(x) = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) III Q5(ii)
Let g(x) = x3 + x2 – 5x + 3
g'(x) = 3x2 + 2x – 5 = (3x + 5) (x – 1)
g(1) = 1 + 1 – 5 + 3 = 0
∴ x – 1 is a common factor of g'(x) and g(x)
∴ 1 is a multiple root of g(x) = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) III Q5(ii).1
x + 3 = 0 ⇒ x = -3
∴ The roots are 2, 2, 1, 1, -3

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b)

Question 6.
Solve the equation 8x3 – 20x2 + 6x + 9 = 0 given that the equation has multiple roots.
Solution:
Let f(x) = 8x3 – 20x2 + 6x + 9
f'(x) = 24x2 – 40x + 6
= 2 (12x2 – 20x + 3)
= 2[12x2 – 18x – 2x + 3]
= 2[6x(2x – 3) – 1(2x – 3)]
= 2(2x – 3) (6x – 1)
f'(x) = 0
⇒ x = [latex]\frac{3}{2}[/latex], x = [latex]\frac{1}{6}[/latex]
[latex]f\left(\frac{3}{2}\right)=8\left(\frac{3}{2}\right)^{3}-20\left(\frac{3}{2}\right)^{2}+6\left(\frac{3}{2}\right)+9[/latex]
= 27 – 45 + 9 + 9
= 0
Hence x – [latex]\frac{3}{2}[/latex] is a factor of f(x) and f'(x)
∴ [latex]\frac{3}{2}[/latex] is a multiple root of f(x) = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(b) III Q6
8x – 4 = 0
⇒ x = [latex]\frac{4}{8}=\frac{1}{2}[/latex]
∴ The roots of the equation f(x) = 0 are [latex]\frac{3}{2}, \frac{3}{2}, \frac{1}{2}[/latex]

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(a) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2A Theory of Equations Solutions Exercise 4(a)

I.

Question 1.
Form polynomial equations of the lowest degree, with roots as given below.
(i) 1, -1, 3
Solution:
Equation having roots α, β, γ is (x – α) (x – β) (x – γ) = 0
Sol. Required equation is (x – 1) (x + 1) (x – 3) = 0
⇒ (x2 – 1) (x – 3) = 0
⇒ x3 – 3x2 – x + 3 = 0

(ii) 1 ± 2i, 4, 2
Solution:
In an equation, imaginary roots occur in conjugate pairs.
Equation having roots α, β, γ, δ is (x – α) (x – β) (x – γ) (x – δ) = 0
Required equation is [x – (1 + 2i)] [x – (1 – 2i)] (x – 4) (x – 2) = 0
(x – (1 + 2i)] [x – (1 – 2i)] = [(x – 1) – 2i] [(x – 1) + 2i]
= (x – 1)2 – 4i2
= (x – 1)2 + 4
= x2 – 2x + 1 + 4
= x2 – 2x + 5
(x – 4) (x – 2) = x2 – 4x – 2x + 8 = x2 – 6x + 8
Required equation (x2 – 2x + 5) (x2 – 6x + 8) = 0
⇒ x4 – 2x3 + 5x2 – 6x3 + 12x2 – 30x + 8x2 – 16x + 40 = 0
⇒ x4 – 8x3 + 25x2 – 46x + 40 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

(iii) 2 ± √3, 1 ± 2i
Solution:
Required equation is [x – (2 + √3)] [x – (2 – √3)] [x – (1 + 2i)] [ x – (1 – 2i)] = 0
[x – (2 + √3)] [x – (2 – √3)]
= [(x – 2) – √3] [(x – 2) + √3]
= (x – 2)2 – 3
= x2 – 4x + 4 – 3
= x2 – 4x + 1
[x – (1 + 2i)] [x – (1 – 2i)] = [(x – 1) – 2i] [(x – 1) + 2i]
= (x – 1)2 – 4i2
= x2 – 2x + 1 + 4
= x2 – 2x + 5
Substituting in (1), the required equation is
(x2 – 4x + 1) (x2 – 2x + 5) = 0
⇒ x4 – 4x3 + x2 – 2x3 + 8x2 – 2x + 5x2 – 20x + 5 = 0
⇒ x4 – 6x3 + 14x2 – 22x + 5 = 0

(iv) 0, 0, 2, 2, -2, -2
Solution:
Required equation is (x – 0) (x – 0) (x – 2) (x – 2) (x + 2) (x + 2) = 0
⇒ x2 (x – 2)2 (x + 2)2 = 0
⇒ x2 (x2 – 4)2 = 0
⇒ x2 (x4 – 8x2 + 16) = 0
⇒ x6 – 8x4 + 16x2 = 0

(v) 1 ± √3, 2, 5
Solution:
Required equation is [x – (1 + √3)] [x – (1 – √3)][(x – 2) (x – 5)] = 0 ………(1)
[x – (1 + √3)] [x – (1 – √3)] = [(x – 1) – √3] [(x – 1) + √3]
= (x – 1)2 – 3
= x2 – 2x + 1 – 3
= x2 – 2x – 2
(x – 2) (x – 5) = x2 – 2x – 5x + 10 = x2 – 7x + 10
Substituting in (1), the required equation is
(x2 – 2x – 2) (x2 – 7x + 10) = 0
⇒ x4 – 2x3 – 2x2 – 7x3 + 14x2 + 14x + 10x2 – 20x – 20 = 0
⇒ x4 – 9x3 + 22x2 – 6x – 20 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

(vi) 0, 1, [latex]-\frac{3}{2}[/latex], [latex]-\frac{5}{2}[/latex]
Solution:
Required equation is
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a) I Q1(vi)

Question 2.
If α, β, γ are the roots of 4x3 – 6x2 + 7x + 3 = 0, then find the value of αβ + βγ + γα.
Solution:
α, β, γ are the roots of 4x3 – 6x2 + 7x + 3 = 0
α + β + γ = [latex]-\frac{a_{1}}{a_{0}}=\frac{6}{4}[/latex]
αβ + βγ + γα = [latex]\frac{a_{2}}{a_{0}}=\frac{7}{4}[/latex]
αβγ = [latex]-\frac{a_{3}}{a_{0}}=-\frac{3}{4}[/latex]
∴ αβ + βγ + γα = [latex]\frac{7}{4}[/latex]

Question 3.
If 1, 1, α are the roots of x3 – 6x2 + 9x – 4 = 0, then find α.
Solution:
1, 1, α are roots of x3 – 6x2 + 9x – 4 = 0
Sum = 1 + 1 + α = 6
⇒ α = 6 – 2 = 4

Question 4.
If -1, 2 and α are the roots of 2x3 + x2 – 7x – 6 = 0, then find α.
Solution:
-1, 2, α are roots of 2x3 + x2 – 7x – 6 = 0
Sum = -1 + 2 + α = [latex]-\frac{1}{2}[/latex]
⇒ α = [latex]-\frac{1}{2}[/latex] – 1 = [latex]-\frac{3}{2}[/latex]

Question 5.
If 1, -2 and 3 are roots of x3 – 2x2 + ax + 6 = 0, then find a.
Solution:
1, -2 and 3 are roots of x3 – 2x2 + ax + 6 = 0
⇒ 1(-2) + (-2)3 + 3 . 1 = a
⇒ a = -2 – 6 + 3 = -5

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

Question 6.
If the product of the roots of 4x3 + 16x2 – 9x – a = 0 is 9, then find a.
Solution:
α, β, γ are the roots of 4x3 + 16x2 – 9x – a = 0
αβγ = 9
⇒ [latex]\frac{a}{4}[/latex] = 9
⇒ a = 36

Question 7.
Find the values of s1, s2, s3, and s4 for each of the following equations.
(i) x4 – 16x3 + 86x2 – 176x + 105 = 0
(ii) 8x4 – 2x3 – 27x2 + 6x + 9 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a) I Q7
Solution:
(i) Given equation is x4 – 16x3 + 86x2 – 176x + 105 = 0
We know that
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a) I Q7.1
(ii) Equation is 8x4 – 2x3 – 27x2 + 6x + 9 = 0
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a) I Q7.2

II.

Question 1.
If α, β and 1 are the roots of x3 – 2x2 – 5x + 6 = 0, then find α and β.
Solution:
α, β and 1 are the roots of x3 – 2x2 – 5x + 6 = 0
Sum = α + β + 1 = 2
⇒ α + β = 1
product = αβ = -6
(α – β)2 = (α + β)2 – 4αβ
= 1 + 24
= 25
α – β = 5, α + β = 1
Adding
2α = 6
⇒ α = 3
α + β = 1
⇒ β = 1 – α
= 1 – 3
= -2
∴ α = 3 and β = -2

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

Question 2.
If α, β and γ are the roots of x3 – 2x2 + 3x – 4 = 0, then find
(i) Σα2β2
(ii) Σαβ(α + β)
Solution:
Since α, β, γ are the roots of x3 – 2x2 + 3x – 4 = 0 then
α + β + γ = 2
αβ + βγ + γα = 3
αβγ = 4
(i) Σα2β2 = α2β2 + β2γ2 + γ2α2
= (αβ + βγ + γα)2 – 2αβγ(α + β + γ)
= 9 – 2 . 2 . 4
= 9 – 16
= -7

(ii) Σαβ(α + β) = α2β + β2γ + γ2α + αβ2 + βγ2 + γα2
= (αβ + βγ + γα) (α + β + γ) – 3αβγ
= 2 . 3 – 3 . 4
= 6 – 12
= -6

Question 3.
If α, β and γ are the roots of x3 + px2 + qx + r = 0, then find the following.
(i) [latex]\sum \frac{1}{\alpha^{2} \beta^{2}}[/latex]
(ii) [latex]\frac{\beta^{2}+\gamma^{2}}{\beta \gamma}+\frac{\gamma^{2}+\alpha^{2}}{\gamma \alpha}+\frac{\alpha^{2}+\beta^{2}}{\alpha \beta}[/latex] or [latex]\Sigma \frac{\beta^{2}+\gamma^{2}}{\beta \gamma}[/latex]
(iii) (β + γ – 3α) (γ + α – 3β) (α + β – 3γ)
(iv) Σα3β3
Solution:
α, β and γ are the roots of x3 + px2 + qx + r = 0,
α + β + γ = -p
αβ + βγ + γα = q
αβγ = -r
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a) II Q3

(iii) (β + γ – 3α) (γ + α – 3β) (α + β – 3γ) = (α + β + γ – 4α) (α + β + γ – 4β) (α + β + γ – 4γ)
= (-p – 4α) (-p – 4β) (-p – 4γ)
= -(p + 4α) (p + 4β) (p + 4γ)
= -(p3 + 4p2 (α + β + γ) + 16p (αβ + βγ + γα) + (64αβγ)
= -(p3 – 4p3 + 16pq – 64r)
= 3p3 – 16pq + 64r

(iv) Σα3β3 = α3β3 + β3γ3 + γ3α3
(αβ + βγ + γα)2 = α2β2 + β2γ2 + γ2α2 + 2αβγ (α + β + γ)
⇒ q2 = α2β2 + β2γ2 + γ2α2 + 2pr
⇒ α2β2 + β2γ2 + γ2α2 = q2 – 2pr
∴ α3β3 + β3γ3 + γ3α3 = (α2β2 + β2γ2 + γ2α2) (αβ + βγ + γα) – αβγ Σα2β
= (q2 – 2pr) . q + r[(αβ + βγ + γα) (α + β + γ) – 3αβγ]
= q3 – 2pqr + r(-pq + 3r)
= q3 – 2pqr – pqr + 3r2
= q3 – 3pqr + 3r2

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

III.

Question 1.
If α, β, γ are the roots of x3 – 6x2 + 11x – 6 = 0, then find the equation whose roots are α2 + β2, β2 + γ2, γ2 + α2.
Solution:
1st Method:
Let α, β, γ are the roots of the equation x3 – 6x2 + 11x – 6 = 0
∴ α + β + γ = 6, αβ + βγ + γα = 11
Let y = α2 + β2 = α2 + β2 + γ2 – γ2
⇒ y = (α + β + γ)2 – 2(αβ + βγ + γα) – x2
⇒ y = 36 – 22 – x2
⇒ x2 = 14 – y
⇒ x = [latex]\sqrt{14-y}[/latex]
Substitute x = [latex]\sqrt{14-y}[/latex] in x3 – 6x2 + 11x – 6 = 0
⇒ ([latex]\sqrt{14-y}[/latex])3 – 6([latex]\sqrt{14-y}[/latex])2 + 11([latex]\sqrt{14-y}[/latex]) – 6 = 0
⇒ (14 – y) [latex]\sqrt{14-y}[/latex] – 6(14 – y) + 11 [latex]\sqrt{14-y}[/latex] – 6 = 0
⇒ -6(14 – y + 1) = [latex]\sqrt{14-y}[/latex] [-11 – 14 + y]
⇒ -6(15 – y) = ([latex]\sqrt{14-y}[/latex]) (y – 25)
Squaring on both sides
i.e., [-6(15 – y)]2 = [[latex]\sqrt{14-y}[/latex](y – 25)]2
⇒ 36(225 – 30y + y2) = (14 – y)(y2 – 50y + 625)
⇒ 8100 – 1080y + 36y2 = 14y2 – 700y + 8750 – y3 + 50y2 – 625y
⇒ 8100 – 1080y + 36y2 = -y3 + 64y2 – 1325y + 8750
⇒ y3 – 28y2 + 245y – 650 = 0
∴ The required equation is x3 – 28x2 + 245x – 650 = 0
2nd Method:
Let α, β, γ are the roots of x3 – 6x2 + 11x – 6 = 0
It is an odd-degree reciprocal equation of class two.
∴ x – 1 is a factor of x3 – 6x2 + 11x – 6
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a) III Q1
∴ x3 – 6x2 + 11x – 6 = (x – 1) (x2 – 5x + 6) = (x – 1) ( x – 2) (x – 3)
∴ The roots of x3 – 6x2 + 11x – 6 = 0 are α = 1, β = 2, γ = 3
Now α2 + β2 = 12 + 22 = 5
β2 + γ2 = 22 + 32 = 13
γ2 + α2 = 32 + 12 = 10
Therefore the cubic equation with roots α2 + β2, β2 + γ2, γ2 + α2 is (x – 5) (x – 13) (x – 10) = 0
⇒ x3 – (5 + 13 + 10) x2 + (65 + 130 + 50)x – 650 = 0
⇒ x3 – 28x2 + 245x – 650 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

Question 2.
If α, β, γ are the roots of x3 – 7x + 6 = 0, then find the equation whose roots are (α – β)2, (β – γ)2, (γ – α)2
Solution:
1st Method:
Let α, β, γ are the roots of the equation x3 – 7x + 6 = 0 …….(1)
α + β + γ = 0, αβγ = -6
Let y = (α – β)2 = (α + β)2 – 4αβ
⇒ y = (-γ)2 – 4([latex]\frac{6}{\gamma}[/latex])
⇒ y = γ2 + [latex]\frac{24}{\gamma}[/latex]
⇒ y = x2 + [latex]\frac{24}{x}[/latex]
⇒ xy = x3 + 24
⇒ xy = 7x – 6 + 24 [from (1)]
⇒ x(y – 7) = 18
⇒ x = [latex]\frac{18}{y-7}[/latex]
Substituting x = [latex]\frac{18}{y-7}[/latex] in x3 – 7x + 6 = 0
([latex]\frac{18}{y-7}[/latex])3 – 6([latex]\frac{18}{y-7}[/latex]) + 6 = 0
⇒ (18)3 – 7(18) (y – 7)2 + 6(y – 7)3 = 0
⇒ 5832 – 126(y2 – 14y + 49) + 6(y3 – 21y2 + 147y – 343) = 0
⇒ 972 – 21(y2 – 14y + 49) + (y3 – 21y2 + 147y – 343) = 0
⇒ y3 – 42y2 + 441y – 400 = 0
∴ The equation with roots (α – β)2, (β – γ)2, (γ – α)2 is x3 – 42x2 + 441x – 400 = 0
2nd Method:
α, β, γ are the roots of x3 – 7x + 6 = 0
By trial and error method x = 1 satisfies this equation.
∴ x – 1 is a factor of x3 – 7x + 6
Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a) III Q2
∴ x3 – 7x + 6 = (x – 1) (x2 + x – 6) = (x – 1)(x + 3)(x – 2)
∵ α, β, γ are the roots of x3 – 7x + 6 = 0
α = 1, β = -3, γ = 2,
Now (α – β)2 = [1 – (-3)]2 = (4)2 = 16
(β – γ)2 = [-3 – 2]2 = 25
(γ – α)2 = [2 – 1]2 = 1
∴ The cubic equation whose roots are (α – β)2, (β – γ)2, (γ – α)2 is (x – 16) (x – 25) (x – 1) = 0
⇒ x3 – (16 + 25 + 1) x2 + (400 + 25 + 16)x – 400 = 0
⇒ x3 – 42x2 + 441x – 400 = 0

Inter 2nd Year Maths 2A Theory of Equations Solutions Ex 4(a)

Question 3.
If α, β, γ are the roots of x3 – 3ax + b = 0, prove that Σ(α – β) (α – γ) = 9a.
Solution:
α, β, γ are the roots of x3 – 3ax + b = 0
∴ α + β + γ = 0, αβ + βγ + γα = -3a, αβγ = -b
Σ(α – β) (α – γ) = Σ[α2 – α(β + γ) + βγ]
= Σ[α2 + α2 + βγl
= 2(α2 + β2 + γ2) + (βγ + γα + αβ)
= 2(α + β + γ)2 – 4(αβ + βγ + γα) + (αβ + βγ + γα)
= 0 – 4(-3a) + (-3a)
= 9a

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(c)

Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Quadratic Expressions Solutions Exercise 3(c) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2A Quadratic Expressions Solutions Exercise 3(c)

I.

Question 1.
Solve the following inequations by the algebraic method.
(i) 15x2 + 4x – 4 ≤ 0
Solution:
15x2 + 4x – 4 ≤ 0
⇒ 15x2 – 6x + 10x – 4 ≤ 0
⇒ 3x(5x – 2) + 2(5x – 2) ≤ 0
⇒ (3x + 2) (5x – 2) ≤ 0
Co-efficient of x2 = 15 > 0,
Given Expression is ≤ 0
⇒ x lies between [latex]\frac{-2}{3}[/latex] and [latex]\frac{2}{5}[/latex]
i.e., [latex]\frac{-2}{3} \leq x \leq \frac{2}{5}[/latex]

(ii) x2 – 2x + 1 < 0
Solution:
x2 – 2x + 1 < 0
⇒ (x – 1)2 < 0
There is no real value of ‘x’ satisfying this inequality
Solution set = Φ (or) Solution does not exist.

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(c)

(iii) 2 – 3x – 2x2 ≥ 0
Solution:
-(2x2 + 3x – 2) ≥ 0
⇒ -(2x2 + 4x – x – 2) ≥ 0
⇒ -[2x(x + 2) – 1(x + 2)] ≥ 0
⇒ -(2x – 1) (x + 2) ≥ 0
Co-efficient of x2 = -2 < 0,
The given expression is ≥ 0
⇒ x lies between -2 and [latex]\frac{1}{2}[/latex]
i.e., -2 ≤ x ≤ [latex]\frac{1}{2}[/latex]

(iv) x2 – 4x – 21 ≥ 0
Solution:
x2 – 4x – 21 ≥ 0
⇒ x2 – 7x + 3x – 21 ≥ 0
⇒ x(x – 7) + 3(x – 7) ≥ 0
⇒ (x + 3) (x – 7) ≥ 0
Co-efficient of x2 = 1 > 0,
The given expression is ≥ 0
x does not lie between -3 and 7
i.e., {x/x ∈ (-∞, -3] ∪ [7, ∞)}

II.

Question 1.
Solve the following inequations by graphical method.
(i) x2 – 7x + 6 > 0
Solution:
f(x) = x2 – 7x + 6
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(c) II Q1(i)
f(x) > 0 ⇒ y > 0
Solutions are given by x < 1 and x > 6

(ii) 4 – x2 > 0
Solution:
Let f(x) = 4 – x2
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(c) II Q1(ii)
f(x) > 0 ⇒ y > 0
Solution set = {x/-2 < x < 2}

(iii) 15x2 + 4x – 4 < 0
Solution:
Let f(x) = 15x2 + 4x – 4
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(c) II Q1(iii)
f(x) ≤ 0 ⇒ y ≤ 0
Solution set = [latex]\left\{x / \frac{-2}{3} \leq x \leq \frac{2}{5}\right\}[/latex]

(iv) x2 – 4x – 21 ≥ 0
Solution:
Let f(x) = x2 – 4x – 21
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(c) II Q1(iv)
f(x) ≥ 0 ⇒ y ≥ 0
Solution set = {x/x ∈ (-∞, -3] ∪ [7, ∞)}

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(c)

Question 2.
Solve the following inequations.
(i) [latex]\sqrt{3 x-8}[/latex] < -2
Solution:
L.H.S. is positive and R.H.S. is negative.
∴ The given inequality holds for no real x.
Solution set = Φ (or) Solution does not exist.

(ii) [latex]\sqrt{-x^{2}+6 x-5}[/latex] > 8 – 2x
Solution:
[latex]\sqrt{-x^{2}+6 x-5}[/latex] > 8 – 2x
⇔ -x2 + 6x – 5 > 0
and (i) 8 – 2x < 0 (or) (ii) 8 – 2x ≥ 0
We have -x2 + 6x – 5 = -(x2 – 6x + 5) = -(x – 1) (x – 5)
Hence -x2 + 6x – 5 ≥ 0 ⇔ x ∈ [1, 5]
(i) -x2 + 6x – 5 ≥ 0 and 8 – 2x < 0
⇔ x ∈ [1, 5] and x > 4
⇔ x ∈ [4, 5] ………(1)
(ii) -x2 + 6x – 5 ≥ 0 and 8 – 2x ≥ 0
∵ [latex]\sqrt{\left(-x^{2}+6 x-5\right)}[/latex] > 8 – 2x
⇔ -x2 + 6x – 5 > (8 – 2x)2 and 8 – 2x ≥ 0
⇔ -x2 + 6x – 5 > 64 + 4x2 – 32x and x ≤ 4
⇔ -5x2 + 38x – 69 > 0 and x ≤ 4
⇔ 5x2 – 38x + 69 < 0 and x ≤ 4
⇔ 5x2 – 15x – 23x + 69 < 0 and x ≤ 4
⇔ (5x – 23)(x – 3) < 0 and x ≤ 4
⇔ x ∈ (3, [latex]\frac{23}{5}[/latex]) and x ≤ 4
⇔ x ∈ (3, [latex]\frac{23}{5}[/latex]) ∩ (-∞, 4)
⇔ x ∈ (3, 4)
Hence the solution set of the given equation is x ∈ (4, 5) ∪ (3, 4)
⇒ x ∈ (3, 5) (or) 3 < x ≤ 5.

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(b)

Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Quadratic Expressions Solutions Exercise 3(b) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2A Quadratic Expressions Solutions Exercise 3(b)

I.

Question 1.
If the quadratic equations ax2 + 2bx + c = 0 and ax2 + 2cx + b = 0, (b ≠ c) have a common root, then show that a + 4b + 4c = 0
Solution:
Let α be the common roots of the equations ax2 + 2bx + c = 0 and ax2 + 2cx + b = 0
2 + 2bα + c = 0
2 + 2cα + b = 0
on Subtracting,
2α(b – c) + c – b = 0
2α(b – c) = b – c
2α = 1 (b ≠ c)
α = [latex]\frac{1}{2}[/latex]
Substitute α = [latex]\frac{1}{2}[/latex] in ax2 + 2bx + c = 0 is
[latex]a\left(\frac{1}{4}\right)+2 b \frac{1}{2}+c=0[/latex]
⇒ a + 4b + 4c = 0
∴ a + 4b + 4c = 0

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(b)

Question 2.
If x2 – 6x + 5 = 0 and x2 – 12x + p = 0 have a common root, then find p.
Solution:
Given x2 – 6x + 5 = 0, x2 – 12x + p = 0 have a common root.
If α is the common root then
α2 – 6α + 5 = 0, α2 – 12α + p = 0
α2 – 6α + 5 = 0
⇒ (α – 1) (α – 5) = 0
⇒ α = 1 or 5
If α = 1 then α2 – 12α + p = 0
⇒ 1 – 12 + p = 0
⇒ p = 11
If α = 5 then α2 – 12α + p = 0
⇒ 25 – 60 + p = 0
⇒ p = 35
∴ p = 11 or 35

Question 3.
If x2 – 6x + 5 = 0 and x2 – 3ax + 35 = 0 have a common root, then find a.
Solution:
The roots of the equation x2 – 6x + 5 = 0 are
(x – 1) (x – 5) = 0
⇒ x = 1, x = 5
Case (i): x = 1 is a common root then it is also root for the equation x2 – 3ax + 35 = 0
⇒ 1 – 3a(1) + 35 = 0
⇒ a = 12
Case (ii): x = 5 is a common root then
(5)2 – 3a(5) + 35 = 0
⇒ 60 – 15a = 0
⇒ a = 4
∴ a = 12 (or) a = 4

Question 4.
If the equation x2 + ax + b = 0 and x2 + cx + d = 0 have a common root and the first equation has equal roots, then prove that 2(b + d) = ac.
Solution:
Let α be the common root.
∴ x2 + ax + b = 0 has equal roots.
Its roots are α, α
α + α = -a
⇒ α = [latex]-\frac{a}{2}[/latex]
α . α = b
⇒ α2 = b
∴ α is a root of x2 + cx + d = 0
⇒ α2 + cα + d = 0
⇒ b + c([latex]-\frac{a}{2}[/latex]) + d = 0
⇒ 2(b + d) = ac

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(b)

Question 5.
Discuss the signs of the following quadratic expressions when x is real.
(i) x2 – 5x + 4
Solution:
x2 – 5x + 4 = (x – 1) (x – 4)
a = 1 > 0
The expression x2 – 5x + 2 is positive if x < 1 or x > 4 and is negative if 1 < x < 4

(ii) x2 – x + 3
Solution:
∆ = b2 – 4ac
= (-1)2 – 4 (1) (3)
= 1 – 12
= -11 < 0
a = 1 > 0, ∆ < 0
⇒ The given expression is positive for all real x.

Question 6.
For what values of x, the following expressions are positive?
(i) x2 – 5x + 6
Solution:
x2 – 5x + 6 = (x – 2) (x – 3)
Roots of x2 – 5x + 6 = 0 are 2, 3 which are real.
The expression x2 – 5x + 6 is positive if x < 2 or x > 3
∴ a = 1 > 0

(ii) 3x2 + 4x + 4
Solution:
Here a = 3, b = 4, c = 4
∆ = b2 – 4ac
= 16 – 48
= -32 < 0
∴ 3x2 + 4x + 4 is positive ∀ x ∈ R
∴ a = 3 > 0 and ∆ < 0
ax2 + bx + c and ‘a’ have same sign ∀ x ∈ R, if ∆ < 0

(iii) 4x – 5x2 + 2
Solution:
Roots of 4x – 5x2 + 2 = 0 are [latex]\frac{-4 \pm \sqrt{16+40}}{-10}[/latex]
i.e., [latex]\frac{2 \pm \sqrt{14}}{5}[/latex] which is real
∴ The expression 4x – 5x2 + 2 is positive when
[latex]\frac{2-\sqrt{14}}{5}<x<\frac{2+\sqrt{14}}{5}[/latex] [∵ a = -5 < 0]

(iv) x2 – 5x + 14
Solution:
Here a = 1, b = -5, c = 14
∆ = b2 – 4ac
= 25 – 56
= -31 < 0
∴ ∆ < 0 ∵ a = 1 > 0 and ∆ < 0
⇒ x2 – 5x + 14 is positive ∀ x ∈ R.

Question 7.
For what values of x, the following expressions are negative?
(i) x2 – 7x + 10
Solution:
x2 – 7x + 10 = (x – 2)(x – 5)
Roots of x2 – 7x + 10 = 0 are 2, 5 which are real.
∴ The expression x2 – 7x + 10 is negative if 2 < x < 5, ∵ a = 1 > 0

(ii) 15 + 4x – 3x2
Solution:
The roots of 15 + 4x – 3x2 = 0 are [latex]\frac{-4 \pm \sqrt{16+180}}{-6}[/latex]
i.e., [latex]\frac{-5}{3}[/latex], 3
∴ The expression 15 + 4x – 3x2 is negative if
-5x < [latex]\frac{-5}{3}[/latex] or x > 3, ∵ a = -3 < 0

(iii) 2x2 + 5x – 3
Solution:
The roots of 2x2 + 5x – 3 = 0 are [latex]\frac{-5 \pm \sqrt{25+24}}{4}[/latex]
i.e., -3, [latex]\frac{1}{2}[/latex]
∴ The expression 2x2 + 5x – 3 is negative if -3 < x < [latex]\frac{1}{2}[/latex], ∵ a = 2 > 0

(iv) x2 – 5x – 6
Solution:
x2 – 5x – 6 = (x – 6) (x + 1)
Roots of x2 – 5x – 6 = 0 are -1, 6 which are real.
∴ The expression x2 – 5x – 6 is negative if -1 < x < 6, ∵ a = 1 > 0

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(b)

Question 8.
Find the changes in the sign of the following expressions and find their extreme values.
Hint: Let α, β are the roots of ax2 + bx + c = 0 and α < β
(1) If x < α or x > β, ax2 + bx + c and ‘a’ have same sign.
(2) If α < x < β, ax2 + bx + c and ‘a’ have opposite sign.

(i) x2 – 5x + 6
Solution:
x2 – 5x + 6 = (x – 2) (x – 3)
(1) If 2 < x < 3, the sign of x2 – 5x + 6 is negative, ∵ a = 1 > 0
(2) If x < 2 or x > 3, the sign of x2 – 5x + 6 is positive, ∵ a = 1 > 0
Since a > 0, the minimum value of x2 – 5x + 6 is [latex]\frac{4 a c-b^{2}}{4 a}[/latex]
= [latex]\frac{4(1)(6)-(-5)^{2}}{4(1)}[/latex]
= [latex]\frac{24-25}{4}[/latex]
= [latex]-\frac{1}{4}[/latex]
Hence the extreme value of the expression x2 – 5x + 6 is [latex]-\frac{1}{4}[/latex]

(ii) 15 + 4x – 3x2
Solution:
15 + 4x – 3x2 = 15 + 9x – 5x – 3x2
= 3(5 + 3x) – x(5 + 3x)
= (3 – x) (5 + 3x)
(1) If [latex]-\frac{5}{3}[/latex] < x < 3 the sign of 15 + 4x – 3x2 is positive, ∵ a = -3 < 0
(2) If x < [latex]-\frac{5}{3}[/latex] or x > 3, the sign of 15 + 4x – 3x2 is negative, ∵ a = -3 < 0
Since a < 0, the maximum value of 15 + 4x – 3x2 is [latex]\frac{4 a c-b^{2}}{4 a}[/latex]
= [latex]\frac{4(-3)(15)-16}{4(-3)}[/latex]
= [latex]\frac{49}{3}[/latex]
Hence the extreme value of the expression 15 + 4x – 3x2 is [latex]\frac{49}{3}[/latex]

Question 9.
Find the maximum or minimum of the following expressions as x varies over R.
(i) x2 – x + 7
Solution:
a = 1 > 0,
minimum value = [latex]\frac{4 a c-b^{2}}{4 a}[/latex]
= [latex]\frac{28-1}{4}[/latex]
= [latex]\frac{27}{4}[/latex]

(ii) 12x – x2 – 32
Solution:
a = -1 < 0,
maximum value = [latex]\frac{4 a c-b^{2}}{4 a}[/latex]
= [latex]\frac{128-144}{-4}[/latex]
= 4

(iii) 2x + 5 – 3x2
Solution:
a = -3 < 0,
maximum value = [latex]\frac{4 a c-b^{2}}{4 a}[/latex]
= [latex]\frac{(4)(-3)(5)-(2)^{2}}{4 \times-3}[/latex]
= [latex]\frac{16}{3}[/latex]

(iv) ax2 + bx + a
Solution:
If a < 0, then maximum value = [latex]\frac{4 a \cdot a-b^{2}}{4 a}[/latex] = [latex]\frac{4 a^{2}-b^{2}}{4 a}[/latex]
If a > 0, then minimum value = [latex]\frac{4 a \cdot a-b^{2}}{4 a}[/latex] = [latex]\frac{4 a^{2}-b^{2}}{4 a}[/latex]

II.

Question 1.
Determine the range of the following expressions.
(i) [latex]\frac{x^{2}+x+1}{x^{2}-x+1}[/latex]
Solution:
Let y = [latex]\frac{x^{2}+x+1}{x^{2}-x+1}[/latex]
⇒ x2y – xy + y = x2 + x + 1
⇒ x2y – xy + y – x2 – x – 1 = 0
⇒ x2 (y – 1) – x(y + 1) + (y – 1) = 0
∴ x is real ⇒ b2 – 4ac ≥ 0
⇒ (y + 1 )2 – 4(y – 1 )2 ≥ 0
⇒ (y + 1)2 – (2y – 2)2 ≥ 0
⇒ (y + 1 + 2y – 2) (y + 1 – 2y + 2) ≥ 0
⇒ (3y – 1) (-y + 3) ≥ 0
⇒ -(3y – 1) (y – 3) ≥ 0
a = Coeff of y2 = -3 < 0
But The expression ≥ 0
⇒ y lies between [latex]\frac{1}{3}[/latex] and 3
∴ The range of [latex]\frac{x^{2}+x+1}{x^{2}-x+1}[/latex] is [[latex]\frac{1}{3}[/latex], 0]

(ii) [latex]\frac{x+2}{2 x^{2}+3 x+6}[/latex]
Solution:
Let y = [latex]\frac{x+2}{2 x^{2}+3 x+6}[/latex]
Then 2yx2 + 3yx + 6y = x + 2
⇒ 2yx2 + (3y – 1)x + (6y – 2) = 0
∴ x is real ⇒ discriminant ≥ 0
⇒ (3y – 1)2 – 4(2y)(6y – 2) ≥ 0
⇒ 9y2 + 1 – 6y – 48y2 + 16y ≥ 0
⇒ -39y2 + 10y + 1 ≥ 0
⇒ 39y2 – 10y – 1 < 0
⇒ 39y2 – 13y + 3y – 1 < 0
⇒ 13y(3y – 1) + 1(3y – 1) ≤ 0
⇒ (3y – 1) (13y + 1) ≤ 0
∴ a = Coeff of y2 = 39 > 0 and the exp ≤ 0
⇒ y lies between [latex]\frac{-1}{13}[/latex] and [latex]\frac{1}{3}[/latex]
∴ Range of [latex]\frac{x+2}{2 x^{2}+3 x+6}[/latex] is [latex]\left[-\frac{1}{13}, \frac{1}{3}\right][/latex]

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(b)

(iii) [latex]\frac{(x-1)(x+2)}{x+3}[/latex]
Solution:
Let y = [latex]\frac{(x-1)(x+2)}{x+3}[/latex]
⇒ yx + 3y = x2 + x – 2
⇒ x2 + (1 – y)x – 3y – 2 = 0
x ∈ R ⇒ (1 – y2) – 4(-3y – 2) ≥ 0
⇒ 1 + y2 – 2y + 12y + 8 ≥ 0
⇒ y2 + 10y + 9 ≥ 0
y2 + 10y + 9 = 0
⇒ (y + 1) (y + 9) = 0
⇒ y = -1, -9
y2 + 10y + 9 ≥ 0
∴ a = Coeff of y2 = 1 > 0 and exp ≥ 0
⇒ y ≤ -9 or y ≥ -1
∴ Range = (-∞, -9] ∪ [-1, ∞)

(iv) [latex]\frac{2 x^{2}-6 x+5}{x^{2}-3 x+2}[/latex]
Solution:
Let y = [latex]\frac{2 x^{2}-6 x+5}{x^{2}-3 x+2}[/latex]
⇒ yx2 – 3yx + 2y = 2x2 – 6x + 5
⇒ (y – 2)x2 + (6 – 3y)x + (2y – 5) = 0
x ∈ R ⇒ (6-3y)2 – 4(y – 2) (2y – 5) ≥ 0
⇒ 36 + 9y2 – 36y – 4(2y2 – 9y + 10) ≥ 0
⇒ 36 + 9y2 – 36y – 8y2 + 36y – 40 ≥ 0
⇒ y2 – 4 ≥ 0
y2 – 4 = 0
⇒ y2 = 4
⇒ y = ±2
y2 – 4 ≥ 0
⇒ y ≤ -2 or y ≥ 2
⇒ y does not lie between -2, 2,
∵ y2 Coeff is > 0 and exp is also ≥ 0
∴ Range of [latex]\frac{2 x^{2}-6 x+5}{x^{2}-3 x+2}[/latex] is (-∞, -2] ∪ [2, ∞)

Question 2.
Prove that [latex]\frac{1}{3 x+1}+\frac{1}{x+1}-\frac{1}{(3 x+1)(x+1)}[/latex] does not lie between 1 and 4, if x is real.
Solution:
Let y = [latex]\frac{1}{3 x+1}+\frac{1}{x+1}-\frac{1}{(3 x+1)(x+1)}[/latex]
⇒ y = [latex]\frac{x+1+3 x+1-1}{(3 x+1)(x+1)}[/latex]
⇒ y = [latex]\frac{4 x+1}{3 x^{2}+4 x+1}[/latex]
⇒ 3yx2 + 4yx + y = 4x + 1
⇒ 3yx2 + (4y – 4)x + (y – 1) = 0
x ∈ R ⇒ (4y – 4)2 – 4(3y)(y – 1) ≥ 0
⇒ 16y2 + 16 – 32y – 12y2 + 12y ≥ 0
⇒ 4y2 – 20y + 16 ≥ 0
4y2 – 20y + 16 = 0
⇒ y2 – 5y + 4 = 0
⇒ (y – 1)(y – 4) = 0
⇒ y = 1, 4
4y2 – 20y + 16 ≥ 0
⇒ y ≤ 1 or y ≥ 4
⇒ y does not lie between 1 and 4
Since y2 Coeff is the and exp ≥ 0.

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(b)

Question 3.
If x is real, prove that [latex]\frac{x}{x^{2}-5 x+9}[/latex] lies between 1 and [latex]\frac{-1}{11}[/latex].
Solution:
Let y = [latex]\frac{x}{x^{2}-5 x+9}[/latex]
⇒ yx2 + (-5y – 1)x + 9y = 0
x ∈ R ⇒ (5y – 1)2 – 4y(9y) ≥ 0
⇒ 25y2 + 1 + 10y – 36y2 ≥ 0
⇒ -11y2 + 10y + 1 ≥ 0 ……….(1)
-11y2 + 10y + 1 = 0
⇒ -11y2 + 11y – y + 1 = 0
⇒ 11y(-y + 1) + 1(-y + 1) = 0
⇒ (-y + 1) (11y + 1) = 0
⇒ y = 1, [latex]\frac{-1}{11}[/latex]
-11y2 + 10y + 1 ≥ 0
∴ y2 Coeff is -ve, but the exp is ≥ 0 from (1)
⇒ [latex]\frac{-1}{11}[/latex] ≤ y ≤ 1
⇒ y lies between 1 and [latex]\frac{-1}{11}[/latex]

Question 4.
If the expression [latex]\frac{x-p}{x^{2}-3 x+2}[/latex] takes all real value for x ∈ R, then find the bounts for p.
Solution:
Let y = [latex]\frac{x-p}{x^{2}-3 x+2}[/latex], given y is real
Then yx2 – 3yx + 2y = x – p
⇒ yx2 + (-3y – 1)x + (2y + p) = 0
∵ x is real ⇒ (-3y – 1)2 – 4y(2y + p) ≥ 0
⇒ 9y2 + 6y + 1 – 8y2 – 4py ≥ 0
⇒ y2 + (6 – 4p)y + 1 ≥ 0
∵ y is real ⇒ y2 + (6 – 4p)y + 1 ≥ 0
⇒ The roots are imaginary or real and equal
⇒ ∆ ≤ 0
⇒ (6 – 4p)2 – 4 ≤ 0
⇒ 4(3 – 2p)2 – 4 ≤ 0
⇒ (3 – 2p)2 – 1 ≤ 0
⇒ 4p2 – 12p + 8 ≤ 0
⇒ p2 – 3p + 2 ≤ 0
⇒ (p – 1)(p – 2) ≤ 0
If p = 1 or p = 2 then [latex]\frac{x-p}{x^{2}-3 x+2}[/latex] is not defined.
∴ 1 < p < 2

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(b)

Question 5.
If c2 ≠ ab and the roots of (c2 – ab)x2 – 2(a2 – bc)x + (b2 – ac) = 0 are equal, then show that a3 + b3 + c3 = 3abc or a = 0.
Solution:
Given equation is (c2 – ab)x2 – 2(a2 – bc)x + (b2 – ac) = 0
Discriminant = 4(a2 – bc)2 – 4(c2 – ab) (b2 – ac)
= 4[(a2 – bc)2 – (c2 – ab) (b2 – ac)]
= 4(a4 + b2c2 – 2a2bc – b2c2 + ac3 + ab3 – a2bc)
= 4(a4 + ab3 + ac3 – 3a2bc)
= 4a(a3 + b3 + c3 – 3abc)
The roots are equal ⇒ discriminant = 0
4a(a3 + b3 + c3 – 3abc) = 0
a = 0 or a3 + b3 + c3 – 3abc = 0
i.e., a = 0 or a3 + b3 + c3 = 3abc

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Quadratic Expressions Solutions Exercise 3(a) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2A Quadratic Expressions Solutions Exercise 3(a)

I.

Question 1.
Find the roots of the following equations.
(i) x2 – 7x + 12 = 0
Solution:
a = 1, b = -7, c = 12
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q1(i)
∴ The roots are 4, 3

(ii) -x2 + x + 2 = 0
Solution:
a = -1, b = 1, c = 2
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q1(ii)
∴ The roots are 2, -1

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

(iii) 2x2 + 3x + 2 = 0
Solution:
a = 2, b = 3, c = 2
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q1(iii)

(iv) √3x2 + 10x – 8√3 = 0
Solution:
a = √3 , b = 10, c = -8√3
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q1(iv)
∴ The roots are [latex]\frac{2}{\sqrt{3}}[/latex], -4√3

(v) 6√5x2 – 9x – 3√5 = 0
Solution:
a = 6√5 , b = -9, c = -3√5
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q1(v)
∴ The roots are [latex]\frac{\sqrt{5}}{2},-\frac{1}{\sqrt{5}}[/latex]

Question 2.
Form a quadratic equation whose roots are:
(i) 2, 5
Solution:
α + β = 2 + 5 = 7
αβ = 2 × 5 = 10
The required equation is x2 – (α + β)x + αβ = 0
⇒ x2 – 7x + 10 = 0

(ii) [latex]\frac{m}{n},-\frac{n}{m}[/latex], (m ≠ 0, n ≠ 0)
Solution:
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q2(ii)

(iii) [latex]\frac{p-q}{p+q}, \frac{-p+q}{p-q}[/latex], (p ≠ ±q)
Solution:
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q2(iii)

(iv) 7 ± 2√5
Solution:
α + β = 7 + 2√5 + 7 – 2√5 = 14
αβ = (7 + 2√5) (7 – 2√5)
= 49 – 20
= 29
The required equation is x2 – (α + β)x + αβ = 0
⇒ x2 – 14x + 29 = 0

(v) -3 ± 5i
Solution:
α + β = -3 + 5i – 3 – 5i = -6
αβ = (-3 + 5i) (-3 – 5i)
= 9 + 25
= 34
The required equation is x2 – (α + β)x + αβ = 0
⇒ x2 + 6x + 34 = 0

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

Question 3.
Find the nature of the roots of the following equation, without finding the roots.
(i) 2x2 – 8x + 3 = 0
Solution:
a = 2, b = -8, c = 3
b2 – 4ac = 64 – 24 = 40 > 0
∴ The roots are real and distinct.

(ii) 9x2 – 30x + 25 = 0
Solution:
a = 9, b = -30, c = 25
b2 – 4ac = 900 – 900 = 0
∴ The roots are rational and equal.

(iii) x2 – 12x + 32 = 0
Solution:
a = 1, b = -12, c = 32
b2 – 4ac = 144 – 128
= 16
= (4)2
= perfect square
∴ The roots are rational and distinct.

(iv) 2x2 – 7x + 10 = 0
Solution:
a = 2, b = -7, c = 10
b2 – 4ac = 49 – 80 = -31 < 0
∴ The roots are complex conjugate numbers.

Question 4.
If α, β are the roots of the equation ax2 + bx + c = 0, find the values of the following expressions in terms of a, b, c.
(i) [latex]\frac{1}{\alpha}+\frac{1}{\beta}[/latex]
(ii) [latex]\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}[/latex]
(iii) α4β7 + α7β4
(iv) [latex]\left(\frac{\alpha}{\beta}-\frac{\beta}{\alpha}\right)^{2}[/latex], if c ≠ 0
(v) [latex]\frac{\alpha^{2}+\beta^{2}}{\alpha^{-2}+\beta^{-2}}[/latex]
Solution:
α, β are the roots of the equation ax2 + bx + c = 0
α + β = [latex]\frac{-b}{a}[/latex], αβ = [latex]\frac{c}{a}[/latex]
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q4
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q4.1
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q4.2
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q4.3

Question 5.
Find the values of m for which the following equations have equal roots.
(i) x2 – 15 – m(2x – 8) = 0
Solution:
Given equation is x2 – 15 – m(2x – 8) = 0
⇒ x2 – 2mx + 8m – 15 = 0
a = 1, b = -2m, c = 8m – 15
b2 – 4ac = (-2m)2 – 4(1) (8m – 15)
= 4m2 – 32m + 60
= 4(m2 – 8m + 15)
= 4(m – 3) (m – 5)
If the equation ax2 + bx + c = 0 has equal roots then its discriminant is zero.
∴ The roots are equal
⇒ b2 – 4ac = 0
⇒ 4(m – 3) (m – 5) = 0
⇒ m – 3 = 0 or m – 5 = 0
⇒ m = 3 or 5

(ii) (m + 1)x2 + 2(m + 3)x + (m + 8) = 0
Solution:
Given equation is (m + 1 )x2 + 2(m + 3)x + (m + 8) = 0
a = m + 1, b = 2(m + 3), c = m + 8
b2 – 4ac = [2(m + 3)]2 – 4(m + 1) (m + 8)
= 4(m2 + 6m + 9) – 4(m2 + 8m + m + 8)
= 4m2 + 24m + 36 – 4m2 – 36m – 32
= -12m + 4
= -4(3m – 1)
∴ The roots are equal
⇒ b2 – 4ac = 0
⇒ -4(3m – 1) = 0
⇒ 3m – 1 = 0
⇒ 3m = 1
⇒ m = [latex]\frac{1}{3}[/latex]

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

(iii) x2 + (m + 3)x + m + 6 = 0
Solution:
Given equation is x2 + (m + 3)x + m + 6 = 0
a = 1, b = m + 3, c = m + 6
∴ The roots are equal
⇒ b2 – 4ac = 0
⇒ (m + 3)2 – 4(1) (m + 6) = 0
⇒ m2 + 6m + 9 – 4m – 24 = 0
⇒ m2 + 2m – 15 – 0
⇒ m2 + 5m – 3m – 15 = 0
⇒ m(m + 5) – 3(m + 5) = 0
⇒ (m + 5) (m – 3) = 0
⇒ m = -5, 3

(iv) (3m + 1)x2 + 2(m + 1)x + m = 0
Solution:
Given equation is (3m + 1)x2 + 2(m + 1)x + m = 0
a = 3m + 1, b = 2(m + 1), c = m
b2 – 4ac = 4(m + 1)2 – 4m(3m + 1)
= 4[(m + 1)2 – m(3m + 1)]
= 4(m2 + 2m + 1 – 3m2 – m)
= 4(-2m2 + m + 1)
= -4(2m2 – m – 1)
= -4(m – 1) (2m + 1)
∴ The roots are equal
⇒ discriminant = 0
⇒ b2 – 4ac = 0
⇒ -4(m – 1) (2m + 1) = 0
⇒ m – 1 = 0 or 2m + 1 = 0
⇒ m = 1 or m = [latex]\frac{-1}{2}[/latex]

(v) (2m + 1)x2 + 2(m + 3)x + (m + 5) = 0
Solution:
Given equation is (2m + 1)x2 + 2(m + 3)x + m + 5 = 0
a = 2m + 1, b = 2(m + 3), c = m + 5
∴ The roots are equal
⇒ b2 – 4ac = 0
⇒ 4(m + 3)2 – 4(2m + 1) (m + 5) = 0
⇒ 4(m2 + 6m + 9 – 2m2 – 10m – m – 5) = 0
⇒ -m2 – 5m + 4 = 0
⇒ m2 + 5m – 4 = 0
⇒ m = [latex]\frac{-5 \pm \sqrt{25+16}}{2}[/latex]
⇒ m = [latex]\frac{-5 \pm \sqrt{41}}{2}[/latex]

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

Question 6.
If α and β are the roots of x2 + px + q = 0, form a quadratic equation whose roots are (α – β)2 and (α + β)2.
Solution:
∵ α, β are the roots of the equation x2 + px + q = 0
α + β = -p, αβ = q
(α – β)2 + (α + β)2 = 2(α2 + β2)
= 2[(α + β)2 – 2αβ]
= 2[p2 – 2q]
(α – β)2 (α + β)2 = [(α + β)2 – 4αβ)] (α + β)2
= (p2 – 4q) (p2)
∴ The required equation is x2 – (sum of the roots) x + product of the roots = 0
∴ x2 – 2(p2 – 2q)x + p2(p2 – 4q) = 0

Question 7.
If x2 + bx + c = 0, x2 + cx + b = 0 (b ≠ c) have a common root, then show that b + c + 1 = 0.
Solution:
If α is a common root of x2 + bx + c = 0, x2 + cx + b = 0 then
α2 + bα + c = 0 ………..(1)
α2 + cα + b = 0 ……….(2)
(1) – (2) ⇒ (b – c)α + (c – b) = 0
⇒ α – 1 = 0
⇒ α = 1
From (1), 1 + b + c = 0
Hence b + c + 1 = 0

Question 8.
Prove that the roots of (x – a) (x – b) = h2 are always real.
Solution:
Given equation is (x – a) (x – b) = h2
x2 – (a + b)x + (ab – h2) = 0
Discriminant = (a + b)2 – 4(ab – h2)
= (a + b)2 – 4ab + 4h2
= (a – b)2 + (2h)2 > 0
∴ The roots are real.

Question 9.
Find the condition that one root of the quadratic equation ax2 + bx + c = 0 shall be n times the other, where n is a positive integer.
Solution:
Let the roots of the equation ax2 + bx + c = 0 be α, nα
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) I Q9

Question 10.
Find two consecutive positive even integers, the sum of whose squares is 340.
Solution:
Let the two consecutive positive even integers be 2λ, 2λ + 2
Sum of squares = 340
⇒ (2λ)2 + (2λ + 2)2 = 340
⇒ λ2 + (λ + 1)2 = 85
⇒ λ2 + λ2 + 2λ + 1 – 85 = 0
⇒ 2λ2 + 2λ – 84 = 0
⇒ λ2 + λ – 42 = 0
⇒ (λ + 7) (λ – 6) = 0
⇒ λ = 6, λ = -7
∵ Given numbers are positive λ = 6
∴ The two consecutive positive even integers are
2λ = 2(6) = 12 and 2λ + 2 = 12 + 2 = 14

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

II.

Question 1.
If x1, x2 are the roots of the quadratic equation ax2 + bx + c = 0 and c ≠ 0, find the value of (ax1 + b)-2 + (ax2 + b)-2 terms of a, b, c.
Solution:
x1, x2 are the roots of the equation ax2 + bx + c = 0
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) II Q1
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) II Q1.1

Question 2.
If α, β are the roots of the quadratic equation ax2 + bx + c = 0, form a quadratic equation whose roots are α2 + β2 and α-2 + β-2.
Solution:
α, β are the roots of ax2 + bx + c = 0, then
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) II Q2
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) II Q2.1

Question 3.
Solve the following equation:
2x4 + x3 – 11x2 + x + 2 = 0
Solution:
2x4 + x3 – 11x2 + x + 2 = 0
Dividing by x2
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) II Q3
Substituting in (1)
2(a2 – 2) + a – 11 = 0
⇒ 2a2 – 4 + a – 11 = 0
⇒ 2a2 + a – 15 = 0
⇒ (a + 3) (2a – 5) = 0
⇒ a + 3 = 0 or 2a – 5 = 0
⇒ a = -3 or 2a = 5
⇒ a = -3 or a = [latex]\frac{5}{2}[/latex]
Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a) II Q3.1
⇒ 2x2 + 2 = 5x
⇒ 2x2 – 5x + 2 = 0
⇒ (2x – 1) (x – 2) = 0
⇒ 2x – 1 = 0 or x – 2 = 0
⇒ x = [latex]\frac{1}{2}[/latex], 2
∴ The roots are [latex]\frac{1}{2}[/latex], 2, [latex]\frac{-3 \pm \sqrt{5}}{2}[/latex]

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

Question 4.
Solve 31+x + 31-x = 10
Solution:
31+x + 31-x = 10
[latex]\text { 3. } 3^{x}+\frac{3}{3^{x}}=10[/latex]
Put a = 3x so that 3a + [latex]\frac{3}{a}[/latex] = 10
⇒ 3a2 + 3 = 10a
⇒ 3a2 – 10a + 3 = 0
⇒ (a – 3) (3a – 1) = 0
⇒ a – 3 = 0 or 3a – 1 = 0
⇒ a = 3 or a = [latex]\frac{1}{3}[/latex]
Case (i): a = 3
⇒ 3x = 31
⇒ x = 1
Case (ii): a = [latex]\frac{1}{3}[/latex]
⇒ 3x = 3-1
⇒ x = -1
∴ The roots are 1, -1

Question 5.
Solve 4x-1 – 3 . 2x-1 + 2 = 0
Solution:
4x-1 – 3 . 2x-1 + 2 = 0
Put a = 2x-1 so that a2 = (2x-1)2 = 4x-1
∴ a2 – 3a + 2 = 0
⇒ (a – 2) (a – 1) = 0
⇒ a – 2 = 0 or a – 1 = 0
⇒ a = 2 or 1
Case (i): a = 2
2x-1 = 21
⇒ x – 1 = 1
⇒ x = 2
Case (ii): a = 1
2x-1 = 20
⇒ x – 1 = 0
⇒ x = 1
∴ The roots are 1, 2

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

Question 6.
[latex]\sqrt{\frac{x}{x-3}}+\sqrt{\frac{x-3}{x}}=\frac{5}{2}[/latex], when x ≠ 0 and x ≠ 3
Solution:
[latex]\sqrt{\frac{x}{x-3}}+\sqrt{\frac{x-3}{x}}=\frac{5}{2}[/latex]
Put a = [latex]\sqrt{\frac{x}{x-3}}[/latex]
[latex]a+\frac{1}{a}=\frac{5}{2}[/latex]
⇒ [latex]\frac{a^{2}+1}{a}=\frac{5}{2}[/latex]
⇒ 2a2 + 2 = 5a
⇒ 2a2 – 5a + 2 = 0
⇒ (2a – 1) (a – 2) = 0
⇒ 2a – 1 = 0 or a – 2 = 0
⇒ a = [latex]\frac{1}{2}[/latex] or 2
Case (i): a = 2
[latex]\sqrt{\frac{x}{x-3}}[/latex] = 2
⇒ [latex]\frac{x}{x-3}[/latex] = 4
⇒ x = 4x – 12
⇒ 3x = 12
⇒ x = 4
Case (ii): a = [latex]\frac{1}{2}[/latex]
[latex]\sqrt{\frac{x}{x-3}}=\frac{1}{2}[/latex]
⇒ [latex]\frac{x}{x-3}=\frac{1}{4}[/latex]
⇒ 4x = x – 3
⇒ 3x = -3
⇒ x = -1
∴ The roots are -1, 4.

Question 7.
[latex]\sqrt{\frac{3 x}{x+1}}+\sqrt{\frac{x+1}{3 x}}=2[/latex], when x ≠ 0 and x ≠ -1
Solution:
Put a = [latex]\sqrt{\frac{3 x}{x+1}}[/latex]
[latex]a+\frac{1}{a}=2[/latex]
⇒ [latex]\frac{a^{2}+1}{a}[/latex] = 2
⇒ a2 + 1 = 2a
⇒ a2 – 2a + 1 = 0
⇒ (a – 1)2 = 0
⇒ a – 1 = 0
⇒ a = 1, 1
∴ [latex]\sqrt{\frac{3 x}{x+1}}[/latex] = 1
⇒ [latex]\frac{3 x}{x+1}[/latex]
⇒ 3x = x + 1
⇒ 2x = 1
⇒ x = [latex]\frac{1}{2}[/latex]
∴ The root is [latex]\frac{1}{2}[/latex]

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

Question 8.
Solve [latex]2\left(x+\frac{1}{x}\right)^{2}-7\left(x+\frac{1}{x}\right)+5=0[/latex], when x ≠ 0.
Solution:
[latex]2\left(x+\frac{1}{x}\right)^{2}-7\left(x+\frac{1}{x}\right)+5=0[/latex]
Put a = x + [latex]\frac{1}{x}[/latex]
⇒ 2a2 – 7a + 5 = 0
⇒ (2a – 5)(a – 1) = 0
⇒ 2a – 5 = 0 or a -1 = 0
⇒ a = [latex]\frac{5}{2}[/latex] or 1
Case (i): a = [latex]\frac{5}{2}[/latex]
x + [latex]\frac{1}{x}[/latex] = [latex]\frac{5}{2}[/latex]
⇒ [latex]\frac{x^{2}+1}{x}=\frac{5}{2}[/latex]
⇒ 2x2 + 2 = 5x
⇒ 2x2 – 5x + 2 = 0
⇒ (2x – 1) (x – 2) = 0
⇒ 2x – 1 = 0 or x – 2 = 0
⇒ x = [latex]\frac{1}{2}[/latex] or 2
Case (ii): a = 1
[latex]x+\frac{1}{x}=1[/latex]
⇒ [latex]\frac{x^{2}+1}{x}[/latex]
⇒ x2 + 1 = x
⇒ x2 – x + 1 = 0
⇒ x = [latex]\frac{1 \pm \sqrt{1-4}}{2}=\frac{1 \pm \sqrt{3 i}}{2}[/latex]
∴ The roots are [latex]\frac{1 \pm \sqrt{3 i}}{2}, \frac{1}{2}[/latex], 2

Question 9.
Solve [latex]\left(x^{2}+\frac{1}{x^{2}}\right)-5\left(x+\frac{1}{x}\right)+6=0[/latex], when x ≠ 0
Solution:
Put a = x + [latex]\frac{1}{x}[/latex]
⇒ a2 = [latex]\left(x+\frac{1}{x}\right)^{2}[/latex]
⇒ a2 = [latex]x^{2}+\frac{1}{x^{2}}+2[/latex]
⇒ [latex]x^{2}+\frac{1}{x^{2}}[/latex] = a2 – 2
∴ a2 – 2 – 5a + 6 = 0
⇒ a2 – 5a + 4 = 0
⇒ (a – 1) (a – 4) = 0
⇒ a = 1 or 4
Case (i): a = 1
x + [latex]\frac{1}{x}[/latex] = 1
⇒ [latex]\frac{x^{2}+1}{x}[/latex] = 1
⇒ x2 + 1 = x
⇒ x2 – x + 1 = 0
⇒ x = [latex]\frac{1 \pm \sqrt{1-4}}{2}=\frac{1 \pm \sqrt{3 i}}{2}[/latex]
Case (ii): a = 4
x + [latex]\frac{1}{x}[/latex] = 4
⇒ [latex]\frac{x^{2}+1}{x}[/latex] = 4
⇒ x2 + 1 = 4x
⇒ x2 – 4x + 1 = 0
⇒ x = [latex]\frac{4 \pm \sqrt{16-4}}{2}[/latex]
⇒ x = [latex]\frac{4 \pm 2 \sqrt{3}}{2}[/latex]
⇒ x = 2 ± √3
∴ The roots are 2 ± √3, [latex]\frac{1 \pm \sqrt{3 i}}{2}[/latex]

Inter 2nd Year Maths 2A Quadratic Expressions Solutions Ex 3(a)

Question 10.
Find a quadratic equation for which the sum of the roots is 7 and the sum of the squares of the roots is 25.
Solution:
Let α, β be the roots of quadratic equation
α + β = 7, α2 + β2 = 25
⇒ (α + β)2 – 2αβ = 25
⇒ 49 – 25 = 2αβ
⇒ αβ = 12
The required equation is x2 – (α + β)x + αβ = 0
⇒ x2 – 7x + 12 = 0

AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Andhra Pradesh BIEAP AP Inter 1st Year Chemistry Study Material 13th Lesson Organic Chemistry-Some Basic Principles and Techniques Textbook Questions and Answers.

AP Inter 1st Year Chemistry Study Material 13th Lesson Organic Chemistry-Some Basic Principles and Techniques

Very Short Answer Questions

Question 1.
Write the reagents required for the conversion of benzene to methyl benzene.
Answer:
Benzene reacts with methyl chloride in presence of anhydrous AlCl3 to form methyl benzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 1
This reaction is called as Friedal craft’s alkylation.

Question 2.
How is nitrobenzene prepared.
Answer:
Benzene undergoes nitration with nitration mixture (cone. HNO3 + Cone H2SO4) at less than 60° C to form nitrobenzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 2

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 3.
Write the conformations of ethane.
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 3

Question 4.
How do you prepare ethyl chloride from ethylene?
Answer:
Ethylene reacts with hydrogen chloride to form Ethyl chloride in presence of anhydrous AlCl3 catalyst.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 4

Question 5.
Write the IUPAC names of :
a) CH3 – CH2 – CH2 – CH = CH3
b) AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 5
c) AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 6
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 7

Question 6.
Write the structures of Trichloroethanoic acid, Neopentane, P – nitro benzaldehyde. [Mar. ’13]
Answer:
Trichloro ethanoic acid – CCl3 – COOH
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 8

Question 7.
Discuss Lassaigne’s test.
Answer:
Lassaigne’s Test: –

  • A small dry sodium piece is taken in a fusion tube and heated gently till the sodium melts.
  • To this molten sodium a small amount of organic compound is added and heated until it changes red hot.
  • This red hot tube is plunged in a china dish containing water and this content is boiled, cooled and filtered.
  • The obtained filtrate is known as Lassaigne’s extract.
  • This test is used to detect the elements Nitrogen, Sulphur, halogens etc.,

Detection of Nitrogen : –
The Lassaigne’s extract is made alkaline by adding few drops of dil. NaOH. to this freshly prepared FeSO4 solution is added and warmed then few drops of FeCl3 are added followed by acidification with Cone. HQ (or) H2SO4. A bluish green colouration indicates the nitrogen.
Na + C + N → NaCN
2NaCN + FeSO4 → Na2SO4 + Fe (CN)2
Fe(CN)2 + 4NaCN → Na4 [Fe(CN)6]
3Na4 [Fe(CN)6] 4- 4FeCl3 → Fe4 [Fe(CN)6]3 + 12 NaCl (Prussian blue colouration)

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 8.
Explain the principle of chromatography.
Answer:
Chromatography has been developed as a method of separation components of a mixture generally between the stationary phase and a mobile phase.
Chromatography involves the three steps given below :
a) Adsorption and retention of a mixture of substances on the stationary phase and separation of adsorbed substances by the mobile phase to different distances on the stationary phase.
b) Recovery of the substances separated by a continuous flow of the mobile phase (known as elution) and
c) Qualitative and quantitative analysis of the eluted substances.

Question 9.
Explain why an organic liquid vaporizes at a temperature below it’s boiling point in it’s steam distillation.
Answer:
In steam distillation, the liquid boils when the sum of vapour pressures due to the organic liquid (P1) and that due to water (P2) becomes equal to the atmospheric pressure (P) i.e., P = P1 + P2.
∴ P1 is lower than P the organic liquid vaporises at lower temperature than it’s boiling point.

Question 10.
Explain the following :
a) Crystallisation
b) Distillation
Answer:
a) Crystallisation:

  • This technique is used for the purification of solid organic compounds.
  • This method is based on the difference in the solubilities of the compound and the impurities in a suitable solvent.
  • The impure compound is dissolved in a solvent in which it is partially soluble at room temperature and  appreciably soluble at higher temperature.
  • The solution is concentrated to get a nearly saturated solution on cooling the solution pure compound crystallises.
  • On repeating this process finally gets the very pure compound.

b) Distillation :
This method is used to separate

    1. Volatile liquids from non volatile impurities and
    2. The liquids having enough difference in their boiling points.
  1. Liquids having different boiling points vapourises at different temperatures.
  2. These vapours are cooled and the liquids formed are collected separately.
    Eg : CHCl3 (b.pt. 334 K) and C6H5NH2 (b.pt. 457 K) are easily separated by distillation technique.

Short Answer Questions

Question 1.
Complete the following reaction and name the products A, B and C. [T.S. Mar. ’15] [A.P. Mar. 16]
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 9
Answer:
CaC2 + 2H2O → Ca(OH)2 + C2H2
Acetylene (= A)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 10
A = C2H2 (Acetylene)
B = C6H6 (Benzene)
C = C6H5CH3 (Methyl benzene)

Question 2.
Name the products A, B and C formed in the following reactions. Give the equations for the reactions.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 11
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 12
A = 1, 2- dibromo ethane
B = Acetylene
C = 1, 1, 2, 2 – tetra bromo ethane

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 3.
How does acetylene react with : a) Bromine b) Hydrogen ? Write the balanced equations for the above reactions. Name the products.
Answer:
a) Addition of Br2: Acetylene reacts with bromine to form finally 1, 1, 2, 2 – tetra bromo ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 13
b) Addition of hydrogen : Acetylene undergoes addition reaction with hydrogen in the presence of Ni catalyst and gives ethylene and ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 14

Question 4.
What is substitution reaction ? Explain any two substitutuin reactions of benzene.
Answer:
Substitution : When an atom or a group in molecule is substituted by another atom or group then the reaction is called electrophillic substitution reaction.
e.g. : 1) Friedal craft’s alkylation : C6H6 reacts with chlorine in the presence of AlCl3 and form chloro benzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 15
e.g. : 2) Nitration : C6H6 when heated with a mixture of cone. HNO3 and cone. H2SO4, below 60° C, to give nitrobenzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 16

Question 5.
What is dehydrohalogenation ? Write the equation for the formation of alkene from alkyl halide.
Answer:
Removal of hydrogen and halogen from the adjacent carbon atoms is called dehydrohalogenation.
Alkene from alkyl halide : When ethyl chloride is heated with alcoholic KOH, ethylene is formed due to dehydrohalogenation
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 17

Question 6.
Which type of compounds react with Ozone ? Explain with one example.
Answer:
1) Unsaturated hydrocarbons react with ozone and form addition compound called ozonide. On hydrolysis the ozonide gives carbonyl compound. This is called ozonolysis.

2) Unsaturated hydrocarbons usually react with ozone. Ozonolysis is used for the location of the double bond in unsaturated compounds like alkene, alkyne and benzene.
Example : Ethylene undergoes addition reaction with ozone and form ozonide. If, on hydrolysis, in presence of Zn dust, gives formaldehyde and H2O2.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 18

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 7.
Give two examples each for position and functional isomerism. [A.P. Mar. ’16] [Mar. ’13]
Answer:
Position isomerism : This type of isomerism arises due to the difference in the position of substituent group or in the position of multiple bond.
e.g.:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 19

e.g.: 2. CH3 – CH2 – C = CH 1 – butyne
CH3 – C = C – CH3 2 – butyne

Functional group isomerism : This type of isomerism arises in carbon compounds having the same molecular formula but with different functional groups.
e.g.: 1. CH3 – CH2 – CH2 – OH 1 – proparal (C3H3O)
CH3 – CH2 – O – CH3 methoxy ethane (C3H8O)
e.g.: 2. CH3 – CH2 – OH Ethanol
CH3 – O – CH3 methoxy methane

Question 8.
Explain the mechanism of halogenations of methane.
Answer:
Methane undergoes halogenation (with Cl2) to form finally tetrachloro methane. This is a substitution reaction and takes place in presence of light.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 20
Mechanism:
Halogenation process involves the free radical chain mechanism
This mechanism involves 3 steps
i) Initiation
ii) Propagation
iii) Termination

i) Initiation : The reaction initiated by the cleavage of chlorine molecule in presence of light. Cl – Cl bond is weaker than the C – C and C – H bonds
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 21
ii) Propagation : Chlorine free radical attacks the CH4 molecule and generates methyl free radical.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 22
The methyl radical formed attacks the 2nd molecule of chlorine to form another chlorine free radical.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 23
Propagation takes place in several steps as follows.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 24
iii) Termination : The reaction terminated after sometime due to consumption of reactants. The following are the possible chain terminating steps
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 25

Question 9.
How is ethylene prepared from ethyl alcohol ? Write the equation.
Answer:
Ethylalcohol reacts with Conc.H2SO4 at 170°C to form Ethylene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 26
The above reaction is dehydration reaction. (H2O removal takes place)

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 10.
Explain the reactions of acetylene with : a) Na in NH3 b) chromic acid. Write the equations and name the products.
Answer:
a) With Na in NH3 : Acetylene reacts with Na in liquid NH3 to give monosodium acetylide and disodium acetylide.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 27
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 28
b) Chromic acid : Acetylene is oxidised to acetic acid by chromic acid.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 29

Question 11.
Explain crystallisation and sublimation phenomenon which are used in the purification of organic compounds.
Answer:
Crystallisation: The principle used here is that the impurities are insoluble or soluble in the given solvent at any temperature, but the substance to be purified is sparingly soluble at room temperature but highly soluble at highest temp. (i.e.,) almost near the boiling temp. of the solvent.
This method is useful to purify solid compounds.

Sublimation : Some solid substances when heated pass directly into vapour state without melting. Those vapour on cooling form directly solid with out condensing to liquid. This phenomenon is called sublimation.

Impure compound is taken in a beaker covered with a watch glass and heated on an electric plate. The compound sublimates and solidifies on the lower surface of the watch glass. Impurities settle in the beaker. Pure compounds in separated by scratching the watch glass.
This is also a purification method of solids.

Question 12.
Describe solvent extraction method to purify a compound.
Answer:
Solvent extractiân : Suppose an organic compound ‘A’ is more soluble in an organic solvent that in water but present in aq. solution. The aq. solution is shoken with the organic solvent. ‘A’ goes into the organic solvent which is immiscible with water. The organic layer is separated and distilled to remove the liquid solvent. The compound remains with distillation flask.

Question 13.
Explain the estimation of phosphorus and sulphur in the given organic compounds.
Answer:
Estimation of phosphorus : Known mass of organic compound (‘a’ gm) is heated with fuming HNO3 in a carius tube. ‘P’ is oxidised to H3PO4 acid. This acid is precipitated as ammonium phosphomolybdate (‘b’ gm) by adding ammonia and ammonium molybdate solutions.
% g of phosphorus = [latex]\frac{100}{a}[/latex] × [latex]\frac{b}{1877}[/latex] × 31 gm
Molecular mass of (NH4)3 PO4. 12 MO O3 (ammonium phosphomolybdate) = 1877
Estimation of sulphur : Known mass of organic compound (‘a’ gm) is heated with sodium peroxide in a carius tube. If sulphur is present it is oxidised to sulphuric acid. The acid is precipitated as barium sulphate (big) by adding excess of aq. BaCl2 solution. The ppt. is filtered, washed, dried and weighed.
% g of sulphur = [latex]\frac{100}{a}[/latex] × [latex]\frac{b \times 32}{233}[/latex] g
Molecular mass of BaSO4 = 233

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 14.
Explain addition of HBr to propene with the ionic mechanism.
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 30
Electrophilic addition mechanism : (Ionic Mechanism)
Step (i) ; Formation of carbonium ion
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 31
Step (ii) : Stable carbonium ion is attacked by Br
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 32

Question 15.
What is the product formed when sodium propionoate is heated with soda lime, (imp)
Answer:
Sodium propionate is heated with sodalime to form ethane. This reaction is decarboxylation reaction. NaOH + CaO is called as Decarboxylating agent.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 33

Long Answer Questions

Question 1.
Explain the classification of hydrocarbons.
Answer:
Classification of hydrocarbons : Hydrocarbons are classified as aliphatic hydrocarbons and aromatic hydrocarbons. Aliphatic hydrocarbons are again classified as open chain hydrocarbons and closed chain hydrocarbons i.e., cyclohydrocarbons. Both open chain and closed chain hydrocarbons are again classified as hydrocarbons containing C – C single bonds. > C = C < compounds, – C = C – hydrocarbons. Aromatic compounds are called benzenoids as they are related to benzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 34

Question 2.
Write IUPAC names of the following compounds :
a) AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 35
b) CH2 = CH – C C – CH3
c) CH3 – CH = C (CH3)2
d) AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 36 CH2 – CH2 – CH = CH2
e)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 37
Answer:
a) 1, 3 – Buta diene
b) CH2 = CH – C ≡ C – CH3
Pent, 1 -ene 3 – yne
c) CH3 – CH = C (CH3)2
2 – Methyl 2 – Butene
d) AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 38
e)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 37
4 – ethyl Dec 1, 5, 8 – Triene

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 3.
Describe two methods of preparation of ethane. Give any three reactions of ethane.
Answer:
Methods of preparation of ethane:
1. Decarboxylation:
Ethane is prepared by heating sodium propionate with sodalime.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 39

2. Kolbe’s electrolysis:
Ethane is obtained by the electrolysis of potassium acetate solution.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 40
H2 gas evolves at cathode.
Ethane and CO2 are formed at anode.

Chemical properties of ethane :
1. Halogenation:
Ethane reacts with chlorine in the presence of sunlight. Hydrogen atoms are substituted by halogen atoms successively. The final product is Hexachloro ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 41

2. Nitration:
Ethane reacts with nitric acid at 400° C and gives nitro ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 42

3. Pyrolysis:
When ethane is heated in the absence of oxygen it decomposes giving ethylene and hydrogen.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 43

Question 4.
Write the structural formulas and IUPAC names for all possible isomers having the number of double (or) triple bond as indicated :
a) C4H8 (one double bond)
b) C5H8 (one triple bond)
c) C5H12 (No multiple bonds).
Answer:
a) The possible Isomers of C4H8 (one double bond)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 44
b) The possible Isomers of C5H8 (one triple bond)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 45
c) The possible Isomers of C5H12 (No multiple bonds)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 46

Question 5.
Write chemical equations for combustion reaction of the following hydrocarbons.
a) Butane
b) Pentene
c) Hexyne.
Answer:
a) Combustion of Butane :
C4H10 + [latex]\frac{13}{2}[/latex] O2 → 4CO2 + 5H2O + Energy

b) Combustion of Pentene:
C5H10 + [latex]\frac{15}{2}[/latex] O2 → 5CO2 + 5H2O + Energy

c) Combustion of Hexyne:
C6H10 + [latex]\frac{17}{2}[/latex] O2 → 6CO2 + 5H2O + Energy

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 6.
Addition HBr to propene yields 2 – bromopropane, while in the presence of benzoyl peroxide, the same reaction yields 1 – bromo propane. Explain and give mechanism.
Answer:

  • Addition of HBr to propene yields 2 – Bromopropene. Here the reaction mechanism is electrophillic. addition mechanism.
  • Addition of HBr to propene in presence of benzoyl peroxide the reaction proceeds through free radical mechanism.
    AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 47

Electrophilic addition mechanism : (Ionic Mechanism)
Step (i) : Formation of carbonium ion
HBr → H+ + Br
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 48

Question 7.
Describe two methods of preparation of ethylene. Give equation for the reactions of ethylene with the following,
a) Ozone
b) Hypohalous acid
c) Cold and dil. alkKMno4
d) Heated with O2 at high pressure
Answer:
Preparation of ethylene:
1. Dehydrohalogenation of alkyl halide: When ethylbromide is heated with alcoholic KOH, ethylene is formed.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 49

2. Dehydration of ethanol: When ethanol is heated with cone. H2SO4 at 170°C, ethylene is obtained.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 50

Chemical properties :
a) With O3: Ethylene undergoes addition reaction with ozone and gives a cyclic compound called ozonide. It undergoes hydrolysis in the presence of zinc dust to give formaldehyde.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 51

b) With HOCl : Ethylene reacts with hypochlorous acid and gives ethylene chlorohydrin.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 52

c) Cold and dil. alk. KMnO4: Pink coloured cold and dil. alkaline KMn04 solution is Bayer’s reagent. Ethylene decolourises Bayers reagent to give ethylene glycol. This is test for unsaturation.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 53

d) Polymerisation : When Ethylene is heated in the presence of 02 at a temperature of 200° C and high pressure of 1500 – 2000 atmospheres gives polyethene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 54

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 8.
How does ethylene react with the following reagents ? Give the chemical equations and names of the products formed in the reactions,
a) Hydrogen halide
b) Hydrogen
c) Bromine
d) Water
e) Oxygen in presence of Ag at 200° C.
Answer:
(a) Reaction with hydrogen halide : Ethylene reacts with Hydrogen halides to give ethyl halides.
H2C = CH2 + HX → CH3 – CH2 X
e.g.: H2C – CH2 + HCl → C2H5Cl Ethyl chloride

(b) With Hydrogen : Ethylene reacts with H2 in presence of Ni, Pt or Pd to give ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 55

(c) With Bromine : Ethylene decolourises red coloured Br2 in CCl4 to give 1,2- dibromo ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 56

(d) With water : Ethylene reacts with dil. H2SO4 to give ethyl alcohol.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 57

(e)With Sulphur monochloride : Ethylene reacts with S2Cl2 to give musturd gas.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 58

(f) With O2 at presence of Ag: Air oxidises ethylene to ethylene oxide known as epoxide in presence of Ag catalyst at 200 – 400°C.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 59

Question 9.
An alkene ‘A’ on ozonolysis gives a mixture of ethanal and pentan 3-one. Write the reaction, structure of the products and alkene – A. Give the IUPAC name of alkene – A.
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 60

Question 10.
An alkene ‘A1 contains three C – C, eight C – H bonds and one C = C bond. ‘A’ on ozonolysis gives two moles of an aldehyde of molar mass 44o . Write IUPAC name of ‘A’.
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 61
IUPAC name of ‘A’ is 2 – Butene

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 11.
Give two methods of preparation of acetylene. How does it react with water and Ozone ?
Answer:
Preparation of acetylene :
1. Dehydrohalogenation : Acetylene is obtained when dibromo ethane is heated with alcoholic KOH.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 62

2. Acetylene is obtained by heating iodoform with silver powder.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 63

3. Addition of water : Acetylene undergoes addition reaction with water in the presence of mercuric sulphate and sulphuric acid. Vinyl alcohol formed in the reaction undergoes rearrangement and gives acetaldehyde . [Mar. 06, June 04]
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 64

4. Reaction with ozone : Acetylene reacts with ozone to form an ozonide which on hydrolysis in presence of Zn forms glyoxal. [June 04]
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 65

Question 12.
How does acetylene react with the following reagents ? Give the corresponding equations and name the products formed in the reactions
a) Acetic acid
b) Water
c) Hydrogen
d) Halogens
e) Hydrogen halide
f) Ammonical AgNO3 and Cl2Cl2 [T.S. Mar. 16]
Answer:
(a) Reaction with Acetic acid : Acetylene on treatment with acetic acid gives vinyl acetate in the 1st step and then gives ethylidene acetate. Hg2+ ions act as catalyst.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 66

(b) Addition of water: Acetylene undergoes addition reaction with water in the presence of mercuric sulphate and sulphuric acid. Vinyl alcohol formed in the reaction undergoes rearrangement gives acetaldehyde.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 67

(c) Addition of hydrogen : Acetylene undergoes addition reaction with hydrogen in the presence of Ni catalyst and gives ethylene and ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 68

(d) Addition of halogens : Acetylene undergoes addition reaction with halogens to give 1, 1, 2, 2 tetra haloethane. Dihalo alkene is formed in the first step which then adds on another molecule of halogen to give tetra halo ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 69

(e) With Hydrogen halide : Acetylene on addition with HCl gives Vinyl chloride and finally 1, 1 – dichloroethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 70

(f) With Ammonical AgNO3 solution : Acetylene gas is passed through ammonical AgNO3 solution to form a white precipitate of silver acetylide. .
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 71
With Ammonical Cu2Cl2 solution : Acetylene gas is passed through Ammonical cuprous chloride solution to give a red precipitate of cuprous acetylide.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 72

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 13.
Describe any two methods of preparation of benzene with corresponding equations. Benzene does not behave like an alkenes why ? How do we get methyl benzene from benzene ?
Answer:
Preparation of Benzene:
1. When sodium benzoate distilled with sodalime. Benzene is formed.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 73

2. Polymerisation of Acetylene : When Acetylene gas is passed through red hot Cu or Fe tubes, it polymerises and gives Benzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 74
Constitution of Benzene : Benzene molecule cannot be represented by a single structure. It has the following Resonance structures.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 75
The resonance energy of Benzene is 150.48 KJ/mole. Thus, it has more stability. Hence does not undergo addition reactions like alkenes. Benzene is an aromatic molecule.
Benzene to toluene (Friedal Craft’s Alkylation) :
Benzene reacts with methyl chloride in presence of AlCl3 and forms methyl benzene (Toluene).
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 76

Question 14.
How do we get benzene from acetylene ? Give the corresponding equation. Explain the halogenations, alkylation, acylation, nitration and Sulphonation of benzene.
Answer:
Preparation of Benzene from acetylene: On passing acetylene gas through red hot iron tubes, it trimerises to give benzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 77
Chemical Properties:
(1) Halogenation : Benzene reacts with chlorine in the presence of FeCl3 or AlCl3 to give chioro-benzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 78

(2) Friedel – Craft’s Alkylation : Benzene reacts with alkyl halides in the presence of AlCl3 to give alkyl benzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 79

(3) Friedel – Craft’s Acylation : Benzene reacts with acetyl chloride in the presence of AlCl3 to give acetophenone.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 80

(4) Nitration : Benzene when heated with a mixture of cone. H2SO4 and cone. HNO3 below 60°C to give Nitrobenzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 81

(5) Sulphonation : With fuming H2SO4, benzene, reacts to form benzene sulphonic acid.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 82

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 15.
Explain the differences between structural isomers and stereo isomers.
Answer:
Structural isomerism : When the isomerism is due to difference in the arrangement of atoms within the molecule, without any reference to space, the phenomenon is known as Structural isomerism. In this type of isomerism, the isomers possess the same molecular formula, but different structural formula.
Structural isomerism is further classified into different types :
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 83
Stereoisomerism : The stereoisomers have the same molecular and structural formula, but differ in the arrangement of atoms or groups in space. Thus the phenomenon exhibited by two or more compounds with the same molecular and structural formulae, but different spatial arrangement of atoms or groups. The spatial arrangement of atoms or groups is also referred to as configuration of the molecule. It is further classified into three types.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 84

Question 16.
What is the differences between conformation and configuration in open chain molecules.
Answer:
Configurational Isomerism : (Optical and geometrical isomerism): These are the stereo – isomers which are discrete, stable and isolable substances. They cannot be inter-converted into one another without making and breaking of new bonds. These isomers cannot be superimposed on each other.
These isomers are further classified as enantiomers, diastereomers and geometrical isomers.

Conformational Isomers : These are the stereo – isomers that easily convert into one another by the rotation around “C – C” bonds. These are in dynamic equilibrium with one another and cannot be separated under ordinary conditions.
This type of isomerism is in alkanes such as n – butane.

Question 17.
What do you understand about geometrical isomerism ? Explain the geometrical isomers of 2 – butene.
Answer:
Geometrical isomerism (Cis – trans isomerism): The isomers which possess the same structural formula but differ in spatial arrangement of the groups around the double bond are called geometrical isomers and the phenomenon is known as geometrical isomerism.
When the same groups lie on the same side of double bond, it is Cis – isomer, while when the same groups lie on opposite side of double bond, the isomer is trans.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 85

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 18.
Explain the method of writing E – Z configurations for geometrical isomers taking CHCl = CFBr as your example.
Answer:
E – Z configurations for geometrical isomers : The following procedure is followed in specifying the configurations of compounds.
i) Arrange the atoms / groups attached to each doubly bonded carbon in the order of their atomic numbers.

ii) Choose the atom/group higher priority on each doubly bonded carbon. If the atoms/groups of higher priority on each carbon are on the same side of the molecule, the letter “Z” is used to denote the configuration of such isomer. When the atoms/groups of higher priority on each carbon are on the opposite sides of the molecule, the letter ‘E’ is prefixed before the name to indicate configuration.
Example : CHCl = CFBr
Among H and Cl, 17Cl gets more priority than 1H
Among H and Br, 35Br gets more priority than 9F
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 86

Question 19.
If an alkene contains on carbons at double bond Cl. Br and – CH2 – CH2 – OH – CH (CH3)2, Write the E and Z configurations of it.
Answer:
From the given data, the structural formulae of the molecule is
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 87

Question 20.
Write a note on :
a) Distillation
b) Fractional distillation
c) Distillation under reduced pressure
d) Steam distillation.
Answer:
a) Distillation : This process is useful for the purification of liquids contaminated with nonvolatile impurities. The impure liquid is boiled in a distillation flask and the vapours are condensed and collected in a receiver. This method can also be used to separate liquids if only their boiling points differ by above 40°C. However, in the case of liquids that have boiling point difference less than 40°C fractional distillation method is used.

b) Fractional distillation : Fractional distillation for liquids that have B.Pt difference less than 40°C. The technique here is that vapours of liquid mixture. When pass through long fractionating column vapours of the liquid with high B.Pt condense and those with low B.Pt pass over through the condenser get condensed and collected in the receiver.
Here, long tubes having different shapes and designs to fit for particular requirements are used.

These are called fractionating columns. The liquid mixture is taken in a distillation flask i.e., fitted with a fractionating column at its mouth. At the upper end of the column, there is a provision to connect it to the water condenser.

c) Distillation under reduced pressure : This method is useful to purify liquids that have very high boiling points and those which decompose at or below their boiling points. If the external pressure is reduced the liquid boils at lower temperature than its normal boiling point without decomposition.

d) Steam distillation : Here liquids which are immiscrible in water passes high boiling point and steam volatile are purified. In this method steam is passed into the hot liquid that is to be purified. The mixture of steam and vapour of volatile organic compound come out. This is because of the sum of the vapour pressures of steam and the liquid to be purified become equal to the atmospheric pressure. They are passed through the condenser and condensed and finally collected in the receiver. The water layer and the organic liquid layer are separated using a separating funnel.

Question 21.
Write a brief note on chromatography.
Explain the following :
a) Column chromatography
b) Thin layer chromatography
c) Partion chromatography.
Answer:
Chromatography has been developed as a method of separation components of a mixture generally between the stationary phase and a mobile phase.
Chromatography involves the three steps given below :
a) Adsorption and retention of a mixture of substances on the stationary phase and separation of adsorbed substances by the mobile phase to different distances on the stationary phase.

b) Recovery of the substances separated by a continuous flow of the mobile phase (known as elution) and

c) Qualitative and quantitative analysis of the eluted substances.

Classification :
Two general chromatography techniques are discussed below. They are :

  1. adsorption chromatography
  2. partition chromatography.

Adsorption chromatography is based on the adsorption of different compounds on an adsorbent to different degrees. Generally used adsorbents are silica gel and alumina. A mobile phase is allowed to move over a stationary phase, the adsorbent. The components of the mixture move to different distances over the stationary phase.

Differential adsorption principle is used in
a) column chromatography and
b) thin layer chromatography.

Column chromatography : In the column chromatography the components of a mixture are separated by a column of adsorbent packed in a glass tube. The column is fitted with a stopcock at its lower end. The mixture to be adsorbed on the adsorbent is placed at the top of the stationary phase. A suitable eluant, either a single solvent or a mixture of solvents is allowed to flow down the column slowly. Depending on the degree to which the compounds are adsorbed the components are separated. The most readily absorbed substances are retained near the top and other come down accordingly to various distances.

Thin layer chromatography (TLC): This also involves adsorption differences. Here the adsorbent, say silica gel or alumina is coated over a glass plate of suitable size in thin layer. The plate is called TLC plate or chromoplate. The solution of the mixture to be separated is applied as a small spot at about 2 cm from the bottom of the plate. The plate is then kept in a closed jar containing the eluant. As the eluant rises up the plate, the components of the mixture move up along with the eluant to various distances depending on their degree of adsorption.

The relative adsorption of a component of the mixture is expressed in terms of its RETARDATION FACTOR (Rf) value.
Rf = [latex]\frac{\text { Distance moved by the subs tance from base line }(\mathrm{X})}{\text { Dis tance moved by the solvent from base line }(\mathrm{Y})}[/latex]
Partition chromatography : This is based on continuous differential partitioning of components of a mixture between the Stationary phase and the mobile phase. In paper chromatography, for example, a special paper called chromatography paper contains water trapped in it which acts as the stationary phase. The chromatography paper spotted with the solution of the mixture at the base is suspended in a suitable solvent or a mixture of solvents.

This solvent acts as the mobile phase. The solvent rises up the paper by capillary action and moves over the spot. The paper selectively retains different components as per their differing partition in mobile and stationary phase. The paper strip so developed is known as chromatogram. The spots of the separated coloured compounds are detected and for colourless compounds other methods like spraying suitable reagent, are used.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 22.
Explain the estimation of nitrogen of an organic compound by
a) Duma’s method
b) Kjeldahl’s method.
(or)
Describe Duma’s and Kjeldahl’s method for the estimation of Nitrogen.
Answer:
Estimation of nitrogen : There are two methods to estimate nitrogen in the given organic compound known as
i) Duma’s method and
ii) Kjeldahed’s method.

i) Duma’s method : In this method a known weight of organic compound is heated strongly with coarse cupric oxide. Carbon and hydrogen get oxidised to carbon dioxide and water vapour respectively. Nitrogen if present is converted to nitrogen gas. Even if some nitrogen is converted to its oxides. They are reduced by hot copper gauze to nitrogen. The product gases are collected over a solution of potassium hydroxide. CO2 is absorbed by KOH solution. Nitrogen is collected over potassium hydroxide solution and its volume is found out.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 88
Suppose that ‘a’ g of organic compound gives V1 ml of N2 at room temperature TK. If atmosphere pressure is P and aqueous tension at TK is P, then the pressure of nitrogen gas at TK is (P – p). Let (P – p) = P1 Reduce the volume of nitrogen to standard temperature 273 K and standard pressure 760 mm.
Volume of nitrogen at STP is v = [latex]\frac{\mathrm{P}_1 \mathrm{~V}_1}{\mathrm{~T}_1} \times \frac{273}{760}[/latex]
28 g of nitrogen at STP occupies 22400 ml
? g. of nitrogen at STP occupies Vml. of nitrogen.
28 × [latex]\frac{V}{22400}[/latex]g
‘a’ g. of organic compound [latex]\frac{28 \times V}{22400}[/latex] g of nitrogen.
100 g organic compound has ? of nitrogen = [latex]\frac{100}{a} \times \frac{28 \times V}{22400}[/latex] g

ii) Kjeldah’s method : This is another method to estimate nitrogen. In this method, the compound is heated with concentrated sulphuric acid in the presence of small amount of CuSO4.
Nitrogen is quantitatively converted into ammonium sulphate. The contents of the flask are transferred to another flask and heated with excess of sodium hydroxide solution to liberate ammonia gas. Ammonia gas so liberated is passed and absorbed in a known volume of known concentrated sulphuric acid that is relatively more in amount than that is required to neutralise NH3 gas. Now, the excess of acid remained after the neutralisation by NH3 is titrated against a standard solution of alkali. From the above, the amount of H2SO4 used to neutralise NH3 is calculated. From this, the mass of ammonia formed is calculated and from that percentage of nitrogen is calculated.
Organic compound + H2SO4 →(NH4)2 SO4
(NH4)2SO4 + 2 NaOH → Na2SO4 + 2 H2O + 2 NH3
2 NH3 + H2SO4 → (NH4)2SO4
Calculation :
Let the mass of organic compound taken be ‘a’ g.
Let the volume of H2SO4 initially taken be ‘Vml‘ and its molarity M.
After passing the NH3 gas into the above acid, if the remaining acid is titrated with M molar NaOH and it consumes V1 ml. of NaOH for complete neutralisation, then from the formula
[latex]\frac{\mathrm{MV}_1}{\mathrm{n}_1}[/latex] (NaOH) = [latex]\frac{\mathrm{MV}_2}{\mathrm{n}_2}[/latex] (H2SO4)
From the stoichiometric equation
2 NaOH + H2SO4 → Na2SO4 + 2H2O
n1 = number of moles of NaOH = 2; n2 = number of moles of H2SO4 = 1
[latex]\frac{\mathrm{MV}_1}{2}[/latex] = [latex]\frac{\mathrm{MV}_2}{1}[/latex] or V2 = [latex]\frac{\mathrm{V}_1}{2}[/latex] ml
Therefore, the volume of H2SO4 neutralised by NH3 is [V – [latex]\frac{\mathrm{V}_1}{2}[/latex]] ml
(or) it is equal to 2 [V – [latex]\frac{\mathrm{V}_1}{2}[/latex]] ml. of M molar NH3 solution.
1000 ml. of 1M NH3 solution contains 17g. of NH3 or 14g. of N2
2[V – [latex]\frac{\mathrm{V}_1}{2}[/latex]] ml ml. of ‘M’ NH3 solution contains
[latex]\frac{14 \times M \times 2\left[V-\frac{V_1}{2}\right]}{1000}[/latex] g. of nitrogen.
Percentage of nitrogen = [latex]\frac{14 \times M \times 2\left[V-\frac{V_1}{2}\right]}{1000} \times \frac{100}{a}[/latex]

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 23.
Explain inductive effect with a suitable example.
Answer:
Inductive effect: The electron donating or electron with drawing effect of a groups an atom that is transmitted by the polarisation of electrons in σ bonds is called inductive effect.
Illustration : Consider the molecule CH3 – CH2 – CH2 – Cl. There is a ‘σ’ covalent bond between carbon atom and chlorine atom. The electron pair between them is not equally shared. The more electronegative chlorine atom tends to attract the shared pair more towards itself. Due to this, the electron density tends to be greater nearer chlorine atom than carbon atom. It is generally represented
as AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 89 But, carbon atom bonded to chlorine atom is itself attached to other carbon atoms. Therefore, the effect can be transitted further AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 90
“Inductive effect is defined as the polarisation of a bond caused by the polarization of adjacent σ bond”.
(-I) effect i.e., electron with drawing effect is in the order
+NH3 > NO2 > CN > SO3H > CHO > CO > COOH > COCl > COOR > CONH2 > F > G > Br > I > OH > OR > NH2 > C6H5 > H.
+ I effect is – [latex]\bar{N} R[/latex] > – [latex]\overline{O}[/latex] ; – Se > S > -O ; – C(CH3)3 > – CH(CH3)2 > – CH2CH3 > – CH3.

Question 24.
Write a note on mesomeric effect.
Answer:
Mesomeric effect:
“The electron pair displacement caused by an atom or group along a chain by a conjugative mechanism is called the mesomeric effect of that atom or group”.
Salient features of the mesomeric effect:
i) Permanent effect operating in the ground state of the molecule.
ii) Lone pairs and n electrons are involved and operate through conjugative mechanism of electron displacement.
iii) It influences the physical properties, reaction rates etc.

Groups which tend to increase the electron density of the rest of the molecule are said to have (+M) effect. Such groups tend to posses lone pairs of electrons.
e.g. : AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 91
Groups that decrease the electron density of the rest of the molecule are said to have (-M) effect. Unsaturated groups having polar character have – M effect.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 92. Here C = O group decreases the electron density of the remaining molecule. It has – M effect.
+ M effect is – F > – Cl > – Br > – I;
-NR2 > OR > F;
– NH2 > – OH > – F;
– OR > – SR > SeR
– O > – OR
– M effect is = O > = NR > = CR2
= NR2< =NR ≡ N > ≡ CR

Question 25.
Describe resonance effect with one example.
Answer:
Resonance effect: It is the polarity produced in a molecule by the interactions of two n bonds or between a π bond and a lone pair of electrons present on adjacent atoms. This effect is transmitted through the chain.
If the transfer of electron is away from the atoms or substituent group attached to the conjugated system, then the molecule gets some of its positions high electron density as in aniline and it is given (+ R). If the shift of electrons are towards the atom or substituent group it is (- R) as in nitrobenzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 93
Groups showing (+ R) are X, – OH, – OR, – COOR, – NH2, – NHR, – NR2, – NHCOR etc.
(- R) effect are – COOH, CHO, > C = O, – CN, – NO2.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 26.
Explain how many types of organic reactions are possible.
Answer:
Organic reactions are mainly classified into four types known as
i) Addition rections.
ii) Substitution reactions
iii) Elimination reactions
iv) Molecular rearrangements.

i) Addition reactions : In these reactions the reagent and the substract combine together to give a single product.
e.g.: AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 94
Depending on the reagent added in the slow rate determining step, addition reactions are again a) Electrophilic classified addition reactions, b) Nucleophilic addition reactions c) Free radical addition reactions.

ii) Substitution reactions : In these reactions on atom or a group of the substrate species is replaced by another atom or group. These are again classified as a) Electrophilic substitution b) Nucleophilic substitution and c) Free radical substitution reactions on the basis of the reagent involved in the rate determining step.
e.g.-: [latex]\overline{\mathrm{O}}[/latex]H(aq) + R – X → HO – R + X(aq)

iii) Elimination reactions : In these reactions two or more atoms or groups of an organic substrate are removed to form multiple bonds.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 95

iv) Molecular rearrangements : Here one organic species (generally less stable) rearranges to other species (generally more stable). For example. Fries rearrangement.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 96

Question 27.
Write the possible conformations of ethane and explain which is more stable.
Answer:
Conformation of ethane : The conformational isomers of a given alkane are obtained by rotation about C – C bond and they are represented by Newman projections or line – Wedge or Sawhorse projections. Newman projections and energy diagrams for the various conformations of ethane are given below.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 97
Staggered (S) occurs at Dihedral angle 60°, 180°, 300°
Rotation about the carbon – carbon bond in ethane is though very rapid, not completely free. Two conformations of ethane known as staggered conformation (S) and the eclipsed conformation (E) are very important though infinite number of confomation are possible.

The C – H bonds in the staggered conformation are arranged so that each one bisects the angle defined by two C – H bonds on adjacent carbon. In the eclipsed conformation each C – H bond is aligned with a C – H bond on adjacent carbon.

In the staggered one the distance between the hydrogen nuclei is 2.55 A° but in eclipsed 2.29 A°. The staggered and elipsed conformations are interconvertible by rotation of one carbon with respect to the other around the a bond that connects them. Different conformations of the same molecule are also called conformers or rotamers.
Stability :
Staggered (S) form is more stable than eclipsed. Because in case of eclipsed form electronic repulsions are high and in case of staggered form electronic repulsions are minimum.

Question 28.
Explain aromatic electrophillic substitution reactions of benzene.
Answer:
1) Halogenation : Benzene reacts with bromine or chlorine in the presence of Lewis acids like FeCl3, AlCl3 etc., to give corresponding halobenzene.
e.g. :
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 98
Similarly with bromine, bromobenzene is formed.

2) Nitration : Benzene undergoes nitration when heated with a mixture of 1 : 1 (by volume) concentrated nitric acid and concentrated sulphuric acid (nitration mixture) at a temperature below 60° C.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 99

3) Benzene reacts with fuming sulphuric acid (oleum) and gives benzene sulphonic acid.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 100

4) Friedel – Craft’s alkylation and acylation : Benzene reacts with alkyl halides and acylh in the presence of Lewis acids (AlCl3, FeCl3) and gives alkyl benzenes and acyl benzenes.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 101
Benzene reacts with acetyl chloride in the presence of anhydrous aluminium, chloride and acetophenone.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 102

General Mechanisms : Electrophilic substitution reaction (SE) proceeds in two steps as
I. Generation of electrophile
II. a) Formation of carbocation intermediate
b) Removal of proton from carbocation intermediate.

I. Generation of electrophile E+: In the reactions halogenation, alkylation and acylatio benzene, anhydrous AlCl3, the Lewis acid produces electrophile X+ say Cl+, R+ and RCO+ by reacting the reagent chlorine, alkyl halide and acylhalide respectively.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 103
II. a) Formation of carbocation : Electrophile generated above attacks one of the benz carbons to change it to sp3 hybridised. The carbocation (Arrhenium ion) is stabilised through resonal.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 104
Sigma complex loses aromatic character due to delocalisation of electrons stopping at sp3 carb

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

b) Losing of proton : To regain aromatic character the [latex]\stackrel{+}{\mathrm{C}}[/latex] loses one proton to sp3 carbon on attack
of (AlCl4) in case of halogenation, alkylation and acylation and HSO4 in case of nitration.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 105

Question 29.
Explain electrophilic addition reactions of ethylene with mechanism.
Answer:
Electrophilic addition reactions of ethylene :
1. Addition of Hydrogen : Ethylene react with hydrozen in the presence of Pt, Pd or Ni catalyst to form ethane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 106
2. Addition of halogens : Halogens (Cl2 or Br2) react with ethylene in the presence of an inert solvent like CCl4 to form dihalo derivative
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 107
Mechanism :
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 108
3. Addition of hydrogen halides : Ethylene reacts with HBr to form ethyl bromide
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 109
4. Addition of water : In presence of few drops of cone. H2SO4, water adds to ethylene to form alcohol.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 110

Question 30.
With the help of mechanism explain free radical halogenations of alkanes.
Answer:
Methane undergoes halogenation (with Cl2) to form finally tetrachloro methane. This is a substitution reaction and takes place in presence of light.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 111
Mechanism :

  • Halogenation process involves the free radical chain mechanism
  • This mechanism involves 3 steps
    i) Initiation
    ii) Propagation
    iii) Termination

i) Initiation : The reaction initiated by the cleavage of chlorine molecule in presence of light. Cl – Cl bond is weaker than the C – C and C – H bonds
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 112

ii) Propagation : Chlorine free radical attacks the CH4 molecule and generates methyl free radical.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 113
The methyl radical formed attacks the 2nd molecule of chlorine to form another chlorine free radical.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 114

iii) Termination : The reaction terminated after sometime due to consumption of reactants.
The following are the possible chain terminating steps
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 115

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 31.
Discuss Markownikov’s rule and Kharash effect.
Answer:
Statement of Markownikoff’s rule : The rule states that when an unsymmetrical reagent adds to a double bond, the positive part of the adding reagent attaches itself to a carbon of the double bond so as to yield the more stable carbocation as an intermediate.

Mechanism : A pair of electrons from the double bond attacks the electrophilic HX to produce an achiral trigonal planar carbocation intermediate. Thehalideion X then adds to either face of the positively charged carbon forming alkyl halide product.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 116
Anti Mark Kownikoffs addition or peroxide effect or Kharasch effect : In the presence of peroxide (R – O – O – R) the addition of HBr to unsymmetrical alkene like propene takes place against Markowni- koff’s rule. As per Anti Markownkoff’s rule, the addition of HBr to an unsymmetrical alkene like propene takes place in such a way that the hydrogen atom becomes attached to the carbon atom with the fewer hydrogen atoms.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 117
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 118
The 2° free radical is more stable than 1°. Therefore. 1 – bromopropane is major product.

Question 32.
How would you convert benzene in to following compounds ?
a) Chioro benzene
b) Toluene
c) p – nitrotoluene.
Answer:
a) Benzene reacts with chlorine inpresence of FeCl3 to form chlorobenzene.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 119
b) Benzene reacts with methyl chloride in presence of AlCl3 to form toluene (Friedal Craft’s alkylation)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 120
c) From benzene p – nitro Toluene is obtained as follows.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 121

Question 33.
Why is wurtz reaction not preferred for the preparation of alkanes containing odd number of carbon atoms ? Illustrate with one example.
Answer:
Wurtz reaction Alkyl halides reacts with sodium metal in presence of dry ether to form higher alkanes.
R – X + 2Na + R – X AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 122 R – R + 2NaX

  • Wurtz reaction is preferred to prepare even no.of Carbons containing alkanes but not for odd no.of Carbons containing Alkanes.
  • For getting odd no.of Carbons containing alkanes we have to consider two different alkyl halides.

The products obtained are in mixture form, So the percentage of the desired hydrocarbon is less, i.e., yield of the desired product is low.
Methane can not be prepared by this reaction.

Question 34.
Write the equations involved in the detection of Nitrogen, Halogens and sulphur in organic compounds.
Answer:
Lassaigne’s Test : –

  • A small dry sodium piece is taken in a fusion tube and heated gently till the sodium melts.
  • To this molten sodium a small amount of organic compound is added and heated until it changes red hot.
  •  This red hot tube is plunged in a china dish containing water and this content is boiled, cooled and filtered.
  • The obtained filtrate is known as Lassaigne’s extract.
  • This test is used to detect the elements Nitrogen, Sulphur, halogens etc..

Detection of Nitrogen : –
The Lassaigne’s extract is made alkaline by adding few drops of dol. NaOH. to this freshly prepared FeSO4 solution is added and warmed then few drops of FeCl3 are added followed by acidification with Cone. HCl (or) H2SO4. A bluish green colouration indicates the nitrogen.
Na + C + N →NaCN
2NaCN + FeSO4 → Na2SO4 + Fe (CN)2
Fe(CN)2 + 4NaCN → Na4 [Fe(CN)6]
3Na4 [Fe(CN)6] + 4FeCl3 → Fe4 [Fe(CN)6]3 + 12 NaCl (Prussian blue colouration)

b) Detection of Halogens : The sodium fusion extract is acidified with HNO3 and is treated with AgNO3 solution.
Ag + X → AgX

  • White PPt indicates ClΘ ion
  • Pale yellow PPt indicates BrΘ ion
  • Yellow PPt indicates IΘ ion

c) Detection of Sulphur : To the sodium extract freshly prepared sodium nitroprusside solution. A deep violet colouration takes place.
S-2 + [Fe(CN)5NO]2+ → [Fe(CN)5 NOS]4- Violet

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 35.
Explain how carbon and Hydrogen are quantitatively determined in an organic compound.
Answer:
Estimation of carbon and hydrogen : Both the elements are estimated in the same experiment simultaneously. A known weight of the organic substance is taken and completely burnt in excess of air and copper (II) oxide carbon changes to CO2 and Hydrogen to H2O.
CxHy + (x + [latex]\frac{y}{4}[/latex])O2 → x CO2 + ([latex]\frac{y}{2}[/latex]) H2O
CO2 and H2O SO obtained are passed through already weighed ‘U‘ – tubes containing anhy, CaCl2 and caustic potash respectively. The increased weigh is of these two tubes give the weigh of H2O formed and weight of CO2 formed.
Suppose ‘a’ g of organic compound on combustion gives ‘b’ grams of water vapour and. ‘C g of CO2. Then
% g of Carbon = [latex]\frac{12}{44}[/latex] × [latex]\frac{100}{a}[/latex] × C g
% g of Hydrogen = [latex]\frac{2}{18}[/latex] × [latex]\frac{100}{a}[/latex] × b g

Question 36.
How do you determine sulphur, phosphorous and Oxygen are determined quantitatively in an organic compound ?
Answer:
Estimation of phosphorus : Known mass of organic compound (‘a’ gm) is heated with fuming HNO3 in a carius tube. ‘P’ is oxidised to H3PO4 acid. This acid is precipitated as ammonium phosphomolybdate (‘b’ gm) by adding ammonia and ammonium molybdate solutions.
% g of phosphorus = [latex]\frac{100}{a}[/latex] × [latex]\frac{b}{1877}[/latex] × 31 gm
Molecular mass of (NH4)3 PO4. 12 MO O3 (ammonium phosphomoly bdate) = 1877
Estimation of sulphur : Known mass of organic compound (‘a’ gm) is heated with sodium peroxide in a carius tube. If sulphur is present it is oxidised to sulphuric acid. The acid is precipitated as barium sulphate (big) by adding excess of aq. BaCl2 solution. The ppt. is filtered, washed, dried and weighed.
%g of sulphur = [latex]\frac{100}{a}[/latex] × [latex]\frac{b \times 32}{233}[/latex] × g
Molecular mass of BaSO4 = 233
Estimation of Oxygen : The percentage of oxygen in an organic compound is usually found by difference between the total percentage composition (100) and the sum of the percentages of all other elements. However, oxygen can also be estimated directly as follows :

A definite mass of an organic compound is decomposed by heating in a stream of nitrogen gas. The mixture of gaseous products containing oxygen is passed over red-hot coke when all the oxygen is converted to carbon monoxide. This mixture is passed through warm iodine pentoxide (I2O5) when carbon monoxide is oxidised to carbon dioxide producing iodine.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 123
On making the amount of CO produced in equation (A) equal to the amount of CO used in equation (B) by multiplying the equations (A) and (B) by 5 and 2 respectively; we find that each mole of oxygen liberated from the compound will produce two moles of carbondioxide.
Thus 88 g carbon dioxide is obtained if 32 g oxygen is liberated.
Let the mass of organic compound taken be mg
Mass of carbon dioxide produced be m1 g
∴ m1 g carbon dioxide is obtained from [latex]\frac{32 \times m_1}{88}[/latex] g O2
∴ Percentage of oxygen = [latex]\frac{32 \times m_1 \times 100}{88 \times m}[/latex]%

Question 37.
Explain Carius method for the determination of Halogens quantitatively in an organic compound.
Answer:
Estimation of halogens : (Carius method) Known mass of organic compound is heated with framing nitric acid in the presence of AgNO3 in a hard glass tube called carius tube. Carbon and hydrogen of the compound are oxidised to CO2 and H2O. Halogens forms silver halide. So obtained silver halide (Agx) is filtered, washed, dried and weighed.
% g of halogen = [latex]\frac{100 \times b \times \text { atomic mass of ‘ } x \text { ‘ }}{a \times \text { molecular mass of } \mathrm{Agx}}[/latex]
Where, a = mass of organic compound, b = mass of Agx formed
x = halogen atom.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 38.
What is carcinogenicity ? Explain with two examples.
Answer:
Benzene and several polynuclear hydrocarbons like 1, 3 – benzanthracene, 3 – methyl cholanthrene, 1, 2- benzpyrene etc., are toxic and said to carcinogenic (cancer producing).
Most of these are formed due to incomplete combustion of tobacco, coal, petroleum etc.
They undergo various chemical changes in human body and finally damage DNAto cause cancer.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 124

Solved Problems

Question 1.
How many a and it bonds are present in each of the following molecules?
(a) HC ≡ CCH = CHCH3
(b) CH2 = C = CHCH3
Solution:
(a) σC-C : 4; σC-H : 6; πC=C: 1 ; π C ≡ C : 2
(b) σC-C : 3; σC-H : 6; πC=C : 2.

Question 2.
What is the type of hybridisation of each carbon in the following compounds?
(a) CH3Cl
(b) (CH3)2CO
(c) CH3CN,
(d) HCONH2
(e) CH3CH = CHCN
Solution:
(a) sp3,
(b) sp3, sp2,
(c) sp3, sp,
(d) sp2,
(e) sp3, sp2, sp2, sp

Question 3.
Write the state of hybridisation of carbon in the following compounds and shapes of each of the molecules.
(a)H2C = O,
(b) CH3F,
(c) HC ≡ N.
Solution:
(a) sp2 hybridised carbon, trigonal planar;
(b) sp3 hybridised carbon, tetrahedral;
(c) sp hvbridised carbon, linear.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 4.
Expand each of the following condensed formulas into their complete structural formulas.
(a)CH3CH2COCH2CH3
(b) CH3CH = CH(CH2)3CH3
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 125

Question 5.
For each of the following compounds, write a condensed formula and also their bond – line formula.
(a) HOCH3CH2CH2CH(CH3)CH(CH3)CH3
(b)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 126
Solution:
Condensed formula:
(a) HO(CH2)3CH(CH3)CH(CH3)2
(b) HOCH(CN)2
Bond-line formula :
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 127

Question 6.
Expand each of the following bond-line formulas to show all the atoms including carbon and hydrogen.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 128
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 129

Question 7.
Structures and IUPAC names of some hydrocarbons are given below. Explain why the names given in the parentheses are incorrect.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 130
Solution:
(a) Lowest locant number, 2, 5, 6 is lower than 3, 5, 7,
(b) substituents are in equivalent position; lower number is given to the one that comes first in the name according to alphabetical order.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 8.
Write the IUPAC names of the compounds i-iv from their given structures.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 131
Solution:

  • The functional group present is an alcohol (OH). Hence the suffix is ‘-01’.
  • The longest chain containing – OH has eight carbon atoms. Hence the corresponding saturated hydrocarbon is octane.

The – OH is on carbon atom 3. In addition, a methyl group is attached at 6th carbon. Hence, the systematic name of this compound is 6-Methyloctan-3-ol.

i)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 132
Solution:
The functional group present is ketone (>C = 0), hence suffix ‘-one’. Presence of two keto groups is indicated by ‘di’, hence suffix becomes ‘dione’. The two keto groups are at carbons 2 and 4. The longest chain contains 6 carbon atoms, hence, parent hydrocarbon is hexane. Thus, the systematic name is Hexane – 2, 4 – dione.

ii)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 133
Solution:
Here, two functional groups namely ketone and carboxylic acid are present. The principal functional group is the cat ooxyuc acid group; hence the parent chain will be suffixed with ‘oic’ acid. Numbering of the chain starts from carbon of – COOH functional group. The keto group in the chain at carbon 5 is indicated by ‘oxo’. The longest chain including the principal functional group has 6 carbon- atoms; hence the parent hydrocarbon is hexane. The compound is, therefore, named as 5 Oxo- hexanoic acid.

iii)
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 134
Solution:
The two C=C functional groups are present at carbon atoms 1 and 3, while the C = C functional group is present at carbon 5. These groups are indicated by suffixes ‘diene’ and ‘yne’ respectively. The longest chain containing the functional groups has 6 carbon atoms; hence the parent hydrocarbon is hexane. The name of compound, therefore, is Hexa-1, 3-dien-5-yne.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 9.
Derive the structure of
(i) 2-Chlorohex- ane,
(ii) Pent-4-en-2-ol,
(iii) 3- Nitro-cydohexerte,
(iv) Cyclohex-2-en-l-ol,
(v) 6-Hydroxy-heptanal.
Solution:
(i) ‘hexane’ indicates the presence of 6 carbon atoms in the chain The functional group chloro is present at carbon 2. Hence, the structure of the compound is CH3CH2CH2CH2CH(Cl)CH3.

(ii) ‘pent’ indicates that parent hydrocarbon contains 5 carbon atoms in the chain, ‘en’ and ‘ol’ correspond to the functional groups C=C and -OH at carbon atoms 4 and 2 respectively. Thus, the structure is CH2 = CHCH2CH (OH)CH3.

(iii) Six membered ring containing a carbon- carbon double bond is implied by cyclo-hexene, which is numbered as shown in (I). The prefix 3-nitro means that a nitro group is present on C-3. Thus, complete structural formula of the compound is (II). Double bond is suffixed functional group whereas -NO2 is prefixed functional group therefore double bond gets preference over -NO2 group:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 135
(iv) ‘1 -0l’ means that a -OH group is present at C-1. OH is suffixed functional, group and gets preference over C = C bond. Thus the structure is as shown in (II) :
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 136
(v) ‘heptanal’ indicates the compound to be an aldehyde containing 7 carbon atoms in the parent chain. The ‘6-hydroxy’ indicates that -OH group is present at carbon 6. Thus, the structural formula of the compound is : CH3CH(OH)CH2CH2CH2CH2CHO. Carbon atom of – CHO group is included while numbering the carbon chain.

Question 10.
Write the structural formula of :
(a) o-Ethylanisole
(b)p-Nitroaniline,
(c) 2,3 – Dibromo -1 – phenylpentane,
(d) 4-Ethyl – 1 – fluoro-2-nitrobenzene.
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 137
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 138

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 11.
Using curved-arrow notation, show the formation of reactive intermediates when the following covalent bonds undergo heterolytic cleavage.
(a) CH3 – SCH3,
(b) CH3 – CN,
(c) CH3 – Cu
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 139

Question 12.
Giving justification, categorise the fol-lowing molecules/ions as nucleophile or electrophile :
HS, BF3, C2H5O, (CH3)3 N :,
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 140
Solution:
Nucleophiles: HS, C2H5O, (CH3)3 N:, H2N These species have unshared pair of electrons, which can be donated and shared with an electrophile.
Electrophiles: AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 141
Reactive sites have only six valence electrons; can accept electron pair from a nucleophile.

Question 13.
Identify electrophilic centre in the fol-lowing: CH3CH = O, CH3CN, CH3I.
Solution:
Among AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 142, the starred carbon atoms are electrophilic centers as they will have partial positive charge due to polarity of the bond.

Question 14.
Which bond is more polar in the following pairs of molecules:
(a) H3C – H, H3C – Br
(b) H3C – NH2, H3C – OH
(c) H3C – OH, H3C – SH
Solution:
(a) C – Br, since Br is more electronegative than H, (b) C – O, (c) C – O

Question 15.
In which C – C bond of CH3CH2CH2Br, the inductive effect is expected to be the least ?
Solution:
Magnitude of inductive effect diminishes as the number of intervening bonds increases. Hence, the effect is least in the bond between . carbon-3 and hydrogen.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 16.
Write resonance structures of CH3COO and show the movement of electrons by curved arrows.
Solution:
First, write the structure and put unshared pairs of valence electrons on appropriate atoms. Then draw the arrows one at a time moving the electrons to get the other structures.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 143

Question 17.
Write resonance structures of CH2 = CH – CHO. Indicate relative stability of the contributing structures.
Solution:
[I: Most stable, more number of covalent
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 144
bonds, each carbon and oxygen atom has an octet and no separation of opposite charge II: negative charge on more electronegative atom and positive charge on more electropositive atom; III: does not contribute as oxygen has positive charge and carbon has negative charge, hence least stable].

Question 18.
Explain why the following two structures, I and 11 cannot be the major contributors to the real structure of CH3COOCH3.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 145
Solution:
The two structures are less important contributors as they involve charge separation. Additionally, structure I contains a carbon atom with an incomplete octet.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 19.
Explain why AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 146 is more stable than AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 147 and AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 148 is the least stable cation.
Solution:
HyperconjugatiOn interaction in AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 146 is greater than in AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 147 as the AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 146 has nine C – H bonds. In AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 148, vacant p orbital is perpendicular to the plane in which C – H bonds lie; hence cannot overlap with it. Thus, AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 148 laoks hyper conjugative stability.

Question 20.
On complete combustion, 0.246 g of an organic compound gave 0.1989/ of carbon dioxide and 0.10149 of water. Determine the percentage composition of carbon and hydrogen in the compound.
Solution:
Percentage of carbon = [latex]\frac{12 \times 0.198 \times 100}{44 \times 0.246}[/latex]
= 21 .95%
Percentage of hydrogen = [latex]\frac{2 \times 0.1014 \times 100}{18 \times 0.246}[/latex]
= 4.58%

Question 21.
In Dumas’ method for estimation of nitrogen, 0.3g of an organic compound gave 5OmL of nitrogen collected at 300K temperature and 715mm pressure. Calculate the percentage composition of nitrogen in the compound. (Aqueous tension at 300K = 15 mm)
Solution:
Volume of nitrogen collected at 300K and 715mm pressure is 50 mL
Actual pressure = 715 – 15 = 700 mm
Volume of nitrogen at STP = [latex]\frac{273 \times 700 \times 50}{300 \times 760}[/latex]
= 41.9 ML
22,400 ml of N2 at STP weighs = 28 g
41.9 mL of nitrogen weighs = [latex]\frac{28 \times 41.9}{22400}[/latex] g
Percentage of nitrogen = [latex]\frac{28 \times 41.9 \times 100}{22400 \times 0.3}[/latex]
= 17.46%

Question 22.
During estimation of nitrogen present in an organic compound by Kjeldahl’s method, the ammonia evolved from 0.5 g of the compound in Kjeldahl’s estimation of nitrogen, neutralized 10 mL of 1 M H2SO4. Find out the percentage of nitrogen in the compound?
Solution:
1 M of 10 mL H2SO4 = 1 M of 20 mL NH3 1000 mL of 1 M ammonia contains 14 g nitrogen
20 ml of 1 M ammonia contains [latex]\frac{14 \times 20}{1000}[/latex] g nitrogen
Percentage of nitrogen = [latex]\frac{14 \times 20 \times 100}{1000 \times 0.5}[/latex]
= 56.0%

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 23.
In Canus method of estimation of halogen1 0.15 g of an organic compound gave 0.12 g of AgBr. Find out the percentage of bromine in the compound.
Solution:
Molar mass of AgBr = 108 + 80
= 188 g mol-1
188 g AgBr contains 80 g bromine
0.12 g AgBr contains [latex]\frac{80 \times 0.12}{188}[/latex] g bromine
Percentage of bromine = [latex]\frac{80 \times 0.12 \times 100}{188 \times 0.15}[/latex]
= 34.04%

Question 24.
In sulphur estimation, 0.157 g of an organic compound gave 0.4813 g of barium sulphate. What is the percentage of sulphur in the compound ?
Solution:
Molecular mass of BaSO4 = 137 + 32 + 64
= 233g
233 g BaSO4 contains 32 g sulphur
0.4813 g BaSO4 contains [latex]\frac{32 \times 0.4813}{233}[/latex] g sulphur
Percentage of sulphur = [latex]\frac{32 \times 0.4813 \times 100}{233 \times 0.157}[/latex]
= 42.10%

Question 25.
Write structures of different chain isomers of alkanes corresponding to the molecular formula C6H14. Also write their IUPAC names.
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 149

Question 26.
Write structures of different isomeric alkyl groups corresponding to the molecular formula C5H11. Write IUPAC names of alcohols obtained by attachment of -OH groups at different carbons of the chain.
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 150

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 27.
Write IUPAC names of the following compounds :

  1. (CH3)3 C CH2C(CH3)3
  2. (CH3)2 C(C2H5)2
  3. tetra – tert-butylmethane

Solution:

  1. 2, 2, 4, 4-Tetramethylpentane
  2. 3 3-Dimethylpentane
  3. 3, 3-Di-tert-butyl -2, 2, 4, 4 – tetramethylpentane

Question 28.
Write structural formulas of the following compounds:
(i) 3, 4, 4, 5-Tetramethylheptane
(ii) 2, 5-Dimethyhexane
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 151

Question 29.
Write structures for each of the following compounds. Why are the given names incorrect? Write correct IUPAC names.
(i) 2-Ethylpentane
(ii) 5-Ethyl – 3-methylheptane
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 152
Longest chain is of six carbon atoms and not that of five. Hence, correct name is 3-Methyl- hexane.
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 153
Numbering is to be started from the end which gives lower number to ethyl group. Hence, correct name is 3-ethyl-5-methyl-heptane.

Question 30.
Sodium salt of which acid will be needed for the preparation of propane? Write chemical equation for the reaction.
Solution:
Butanoic acid,
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 154

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 31.
Write IUPAC names of the following compounds:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 155
Solution:
(i) 2, 8 – Dimethyl – 3, 6 – decadiene;
(ii) 1, 3, 5, 7 Octatetraene;
(iii) 2 – n – Propylpent – 1 – ene;
(iv) 4 – Ethyl – 2, 6 – dimethyl – dec – 4 – ene;

Question 32.
Calculate number of sigma (σ) and pi (π) bonds in the above structures (i-iv).
Solution:
σ bonds : 33, π bonds : 2
σ bonds: 17, π bonds:4
σ bonds : 23, π bond : 1
σ bonds: 41, π bond : 1

Question 33.
Write structures and IUPAC names of different structural isomers of alkenes corresponding to C2H10.
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 156

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 34.
Draw cis and trans isomers of the following compounds. Also write their IUPAC names:
(i) CHCl = CHCl
(ii) C2H5CCH3 = CCH3C2H5
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 157

Question 35.
Which of the following compounds will show cis-trans isomerism ?
(i) (CH3)2C = CH – C2H5
(ii) CH2 = CBr2
(iii) C6H5CH = CH – CH3
(iv) CH3CH = CCl CH3
Solution:
(iii) and (iv). In structures (i) and (ii), two identical groups are attached to one of the doubly bonded carbon atom.

Question 36.
Write IUPAC names of the products obtained by addition reactions of HBr to hex- 1-ene
(i) in the absence of peroxide and
(ii) in the presence of peroxide.
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 158

Question 37.
Write structures of different isomers corresponding to the 5th member of alkyne series. Also write IUPAC names of all the isomers. What type of isomerism is exhibited by different pairs of isomers ?
Solution:
5th member of alkyne has the molecular formula C6H10. The possible isomers are :
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 159
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 160
Position and chain isomerism shown by different pairs.

AAP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques

Question 38.
How will you convert ethanoci acid into benzene?
Solution:
AP Inter 1st Year Chemistry Study Material Chapter 13 Organic Chemistry-Some Basic Principles and Techniques 161

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Andhra Pradesh BIEAP AP Inter 1st Year Chemistry Study Material 12th Lesson Environmental Chemistry Textbook Questions and Answers.

AP Inter 1st Year Chemistry Study Material 12th Lesson Environmental Chemistry

Very Short Answer Questions

Question 1.
Define the terms atmosphere and biosphere.
Answer:
Atmosphere: The blanket of gases present around the earth is called the atmosphere. It maintains the heat balance on earth. The atmosphere contains nitrogen and oxygen in large proportions.

Biosphere: Living organisms like plants, animals, and human beings constitute the biosphere.
The biosphere is related to other environment segments.

Question 2.
Explain the terms Lithosphere and Hydrosphere.
Answer:
Lithosphere : The outer mantle of the solid earth consists of minerals present in earth gust and soil. The earth inner surface contains minerals and deeper inner layers contain natural gas and soil. Mountains and hills these are all constitutes lithosphere.

Hydrosphere : All the natural water resources together constitute the hydrosphere. Hydrosphere include oceans, seas, rivers, lakes, streams etc.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 3.
Define the term Soil Pollution.
Answer:
The concentration (or) accumulation of natural bodies is called as soil. Soil gets polluted due to industrial wastes, urban wastes, agricultural pollutants, chemical, radio active pollutants etc.

Question 4.
What is Chemical Oxygen demand (COD).     [A.P. Mar. 16] [Mar. 14]
Answer:
The amount of oxygen required to oxidise organic substances present in polluted water is called Chemical Oxygen Demand (COD).
It is an index for amount of organic substances present in water.

Question 5.
What is Bio Chemical Oxygen Demand (BOD) ? [A.P. Mar. 16] [Mar. 14]
Answer:
The amount of oxygen used by the suitable micro organisms present in water during five days at 20° C is called as Bio Chemical Oxygen Demand (BOD).

Question 6.
What are Troposphere and Stratosphere ?
Answer:
Troposphere : The major portion of the atmosphere which contains air is called troposphere.
It is present 0-11 Km from the earth.

Stratosphere : Stratosphere present 11 – 50 Km from the earth and it mainly contains ozone layer. It absorbs the harmful UV radiations coming from sun.

Question 7.
Name the major particulate pollutants present in Troposphere.
Answer:
The major particulate pollutants present in troposphere are dust, mist, fumes, smoke, smog etc.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 8.
List out four Gaseous Pollutants present in the polluted air.
Answer:
Oxides of Sulphur,Nitrogen and Carbon, Ozone, Hydrocarbons etc., are gaseous pollutants present in polluted air.

Question 9.
Green house effect is caused by and gases.
Answer:
Green house effect is caused by gases such as CO2, CH4, O3, CFCs (Chloro Fluoro Carbons) and water vapour in the atmosphere.

Question 10.
Which oxides cause acid rain ? and What is its pH value ? [Mar. 13]
Answer

  • Oxides of Nitrogen, Sulphur and Carbon dissolved in rain water forms acid rain.
  • Acid rain has pH value lessthan 5.6.

Question 11.
Name two adverse effects caused by acid rains. [A.P. Mar. 16] [T.S. Mar. 15]
Answer:
Effects of acid rains

  • Acid rains are harmful for agriculture, trees and plants because it dissolves and washes away nutrients needed for their growth.
  • Acid rains affects the plants and animal life in aquatic ecosystem.
  • Acid rains damages the old buildings and historical monuments like Taj mahal.
  • Acid rains corrodes water pipes which lead to decrease the quality of drinking water.

Question 12.
What are smoke and mist ?
Answer:
Smoke : The solid particles (or) mixture of solid and liquid particles formed by the combustion of organic matter are called smoke particulates.
Eg : Cigarette smoke, oil smoke etc.

Mist: The particles produced by the spray liquids and by condensation of vapours in air is called mist.
Eg : H2SO4 – mist, herbicides, insecticides etc.
These miss their targets and travel through atmosphere to form mist.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 13.
What is classical smog ? and What is its Chemical Character (Oxidizing / reducing) ?
Answer:

  • The mixture of smoke, fog and sulphur dioxide is called classical smog. It exists in cool humid climate.
  • The chemical character of classical smog is reducing character. Hence it is also called as reducing smog.

Question 14.
Name the common components of Photo Chemical smog.
Answer:
The common components of Photo Chemical smog are O3, NO, acrolein, formaldehyde and Peroxy Acetyl Nitrate (PAN).

Question 15.
What is PAN ? What effect is caused by it ?
Answer:

  • Peroxy Acetyl Nitrate is called as PAN.
  • Peroxy Acetyl Nitrate is (PAN) is a powerful eye irritant.

Question 16.
How is Ozone formed in the Stratosphere ?
Answer:

  • UV radiations react with dioxygen (O2) molecules and split into free oxygen (O) atoms. These free oxygen atoms (0) combined to form ozone molecule.
  • The following are the reactions that takes place during the formation of ozone in stratosphere.
    AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 1

Question 17.
Give the Chemical equations involved in the Ozone depletion by CF2Cl2.
Answer:
CF2Cl2 which is released in the atmosphere mix with the normal atmospheric gases and reaches the stratosphere.

  • In stratosphere CF2Cl2 react with powerful UV radiations and liberates chlorine free radicals.
    AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 2
  • These chlorine radical reacts with ozone present in stratosphere to form chlorine monoxide radicals and molecular dioxygen.
    Cl* + O3 → ClO* + O2
  • ClO radical react with atomic oxygen and produce more Cl – radicals.
    ClO* + O → Cl* + O2

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 18.
What is Ozone hole ? Where was it first observed ?
Answer:
The depletion of ozone layer is commonly known as ozone hole.

  • It was first observed in Antarctica over the south pole.
  • It was reported by atmospheric scientists working in Antarctica.

Question 19.
What is the value of dissolved Oxygen in pure cooled, water ?
Answer:
The value of dissolved oxygen in pure cooled water is around 10 ppm.

Question 20.
Give the possible BOD values of clean water and the polluted water.
Answer:

  • The BOD value of clean water is less than 3 ppm.
  • The BOD value of water is greaterthan 4 ppm then it is said to be polluted.
  • Highly polluted water has BOD value more than 17 ppm.

Question 21.
Name three industrial Chemicals that pollute water.
Answer:
Detergents, paints, pesticides, dyes and pharmaceuticals etc.

Question 22.
What agrochemicals are responsible for water pollution ?
Answer:
Agrochemicals like chemical fertilisers, chemicals used for killing insects, fungi and weeds in crop etc., are responsible for water pollution.

Short Answer Questions

Question 1.
What are different segments of the earth’s environment ?
Answer:
Environment can be divided into four segments.

  1. Atmosphere
  2. Hydrosphere
  3. Lithosphere
  4. Biosphere

1) Atmosphere : The layer of air present around the Earth is called the atmosphere. The atmospheric air contains N2 and O2 in large proportions, while the rest of the gases like C02 are present only in smaller proportions. Atmosphere absorbs harmful radiations coming from the Sun. It plays an important role in maintaining the heat balance on Earth. If the proportions of the gases, especially O2 and N2 are disturbed by human activity, the equilibrium of the echo system is lost. It leads to disastrous consequences.

2) Hydrosphere : Hydrosphere includes all the surface and ground water resources i.e., oceans, rivers, lakes, polar ice caps etc., 97% of earth’s water is locked up in oceans. 3% is trapped in polar ice caps. Only small percentage of water is available for drinking, agricultural and industrial purpose. 80% of the earth’s surface is covered with water.

3) Lithosphere : One fifth of the total Earth surface is in the form of land. Inner layers of Earth contain minerals. Deeper inner layers of Earth contain Natural gas and oil. All these things, including hills and mountains come under Lithosphere. Plants, animals and human beings are occupied by it.

4) Biosphere : All living organisms like plants, animals and human beings constitute the Biosphere. Biosphere and other segments of the environment are interrelated. Biosphere is dependent on Atmosphere and Hydrosphere. Polluted atmosphere arrest the plant growth and bring health hazards among animals and human beings. Contaminated water causes many diseases and also death of aquatic animals.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 2.
Define the terms Sink, COD, BOD and TLV. [T.S. Mar. 16]
Answer:
Sink : The medium which retains and interacts with long lived pollutant is called the sink.
Eg : Oceans are important sinks for atmospheric CO2.

COD : The amount of oxygen required to oxidise organic substances present in polluted water is called Chemical Oxygen Demand (COD).
It is an index for amount of organic substances present in water.

BOD : The amount of oxygen used by the suitable micro organisms present in water during five days at 20° C is called as Bio Chemical Oxygen Demand (BOD).

TLV : (Threshold Limit Value): The permissiable level of the toxic substances (or) pollutants in the atmosphere which affects a person adversly when he is exposed to this for 7 – 8 hrs. in a day is called TLV.

Question 3.
Name the gaseous pollutants present in the air and explain their formation.
Answer:
Gaseous pollutants present in air are
a) Oxides of sulphur
b) Oxides of nitrogen
c) Oxides of carbon
d) Hydro carbons

a) Oxides of sulphur :

  • The oxides of sulphur formed by the fossil fuel containing sulphur are burnt.
    S + O2 → SO2
  • Sulphur dioxide oxidises to form sulphur trioxide in presence of catalyst.
    2SO2 + O2 ⇌ 2SO3
  • Sulphur trioxide can also be formed by the reaction of SO2 with O3 (or) H2O2.
    SO2 + O3 ⇌ SO3 + O2
    SO2 + H2O2 → H2SO4
  • SO2 is the most common oxide and causes the following adverse effects.
    a) It causes respiratory problems like asthma, bronchitis etc.
    b) It is poisonous to both animals and plants.
    c) It causes irritation to the eyes which results in tears and redness.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

b) Oxides of nitrogen :
The major gases present in air are oxygen and nitrogen.

  • These do not combine at normal temperature.
  • Dinitrogen and dioxygen combined at high altitudes in presence of light to form oxides of nitrogen.
    AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 3
  • No reacts with oxygen to form NO2.
    2NO + O2 → 2NO2
  • NO2 is quickly formed by the reaction of NO with O3.
    NO + O3 → NO2 + O2

Adverse Effects

  • NO2 damages the leaves of plants and effects the efficiency of photosynthesis.
  • NO2 causes lung problems.
  • NO2 causes respiratory problems.
  • NO2 affects the textile fibres and metals.

c) Oxides of carbon : –
i) Carbon monoxide (CO)

  • CO is one of the most harmful air pollutants.
  • CO gas is produced by the incomplete combustion of carbon (coal, firewood, petrol etc.)
    C(s) + [latex]\frac{1}{2}[/latex]O2(g) → C0(g)
  • CO is mainly released into atmosphere by automobiles.

Adverse effects :

  • It stops the transportation of oxygen to the organs and tissues.
  • When it is inhaled by a human being it forms stable complex with haemoglobin of blood. This complex is named as carboxy haemoglobin.
    CO + Hb → CO – Hb
  • This reduces the oxygen transportation in body and results into headache, eye problems, nerveous problems and heart problems.

ii) Carbondioxide :

  • CO2 is entered into atmosphere mainly by respiration process.
  • CO2 is also formed by the burning of fossil fuels.
    C + O2 → CO2 + heat energy
  • CO2 is formed by the decomposition of lime stone.
    CaCO3 AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 4 Cao + CO2

Effects :

  • Due to deforestation and burning of fossil fuels increases CO2 release and the balance O2 – CO2 in atmosphere disturbed.
  • The increased level of CO2 causes global warming which causes several problems like dengue, malaria etc.
  • Due to global warming efficiency of photosynthesis decreases.

d) Hydrocarbons:

  • These mainly constitutes hydrogen and carbon.
  • These are formed by the incomplete combustion of fuel used in auto mobiles.

Adverse Effects:

  • These cause cancer.
  • These are harmful to plants causing ageing, break down of tissues, flowers and leaves are shedded.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 4.
What is green house effect ? and how is it caused ?
Answer:
a) Global Warming : The Earth is heated by sunlight and some of the heat that is absorbed by the Earth is radiated back into space. But some gases like CO2, CFCs, O3, NO and water vapour present in the lower atmosphere do not allow the Earth to re-radiate the heat into space. A part of the heat so trapped in these atmospheric gases is re-emitted to the earth’s surface. This phenomenon is called the “Green house effect” (or) Global Warming.
The gases which are responsible for Green house effect are CO2, CFC’s, O3, NO, CH4 water vapour etc. and they are termed as Green house gases. The effect of increase in CO2 causes global warming.

b) Effects of global warming :

  1. If there is a 1° C increase in the temperature, the ice caps of the polar region melt and level of the sea water increases by 90 cm. Due to this, so many coastal countries will be submerged.
  2. Due to global warming, the rate of evaporation of water from the seas, rivers, ponds will increase. This leads to ultimately rains, cyclones and hurricanes.
  3. Agriculture sector will be badly affected due to the fast evaporation of surface water. There will be a shortage in the supply of water for agricultural purpose.
  4. Unseasonal rains.
  5. Increase the infectious diseases like dengue, malaria, yellow fever, sleeping sickness etc.

Prevention : To reduce the level of CO2 on the earth’s atmosphere, one must increase the number of sinks to absorb CO2. Plants absorb a major portion of the CO2. Therefore, more plants, trees, forests should be grown. The blue green algae present in the sea also gets extinct, due to water pollution. This should be prevented stopping the production of CFC etc.

Question 5.
Explain, with Chemical equations involved, the formation of acid rain.
Answer:
Acid rains are due to oxides of N, S and C (NO2, SO2 and CO2).

  • These oxides dissolve in rain water and formed as acids (HNO3, H2SO4 and H2CO3).
  • These come down to earth as rain and deposited on the earth’s surface.
  • Acid – rain is more in industrial areas.

Chemical equations involved
NO2 + NO3 → N2O5
N2O5 + H2O + 2HNO3
CO2 + H2O → H2CO3
SO3 + H2O → H2SO4
The pH of acid rain is less than 5.6.

Question 6.
Explain in detail the adverse effects caused by the acid rain.
Answer:
Acid rains are harmful because

  1. The life of old buildings will be considerable reduced.
  2. The pH of the soil changes affecting its fertility.
  3. Ammonium salts formed by acid rains can be seen as atmospheric haze.
  4. Ammonium salts in rain drops result in wet decomposition.
  5. Acid rains are harmful to agriculture, trees, plants etc.
  6. Due to acid rains aquatic life disturbed.
  7. Due to acid rains the historical monuments like Tajmahal are damaged.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 7.
How is Photochemical Smog formed ? What are its ill effects ?
Answer:

  • When unsaturated hydrocarbons and nitrogen oxides produced from auto mobiles, factories reacts with sunlight and forms photo chemical smog.
  • This occurs in warm, dry and hot climate.
  • This has high concentration of oxidising agents. Hence it is called as photo chemical smog.

Formation :

  • Fossil fuels are burnt, many pollutants are entered into troposphere.
  • Out of these many pollutants two pollutants are main constituents of photo chemical smog. These are hydrocarbons and nitric oxide.
  • These pollutants interacts with sun light and following reaction takes place.
    2NO(g) + O2(g) → 2NO2(g)
    NO2(g) → NO(g) + O(g)
  • This oxygen atoms formed in the above reaction combine with O2 to produce ozone.
    O(g) + O2(g) ⇌ O3(g)
    NO(g) + O3(g) → NO2(g) + O2(g)
  • O3 is poisonous gas and both NO2 and O3 are strong oxidising agents.
  • These react with unburnt hydro carbons in the polluted air to produce organic substances like HCHO, PAN etc.
  • The common components of photo chemical smog are ozone, NO, acrolein, form aldehyde and PAN.

Question 8.
How is Ozone layer depleted in the atmosphere and what are the harmful effects caused by Ozone layer depletion ? [T.S. Mar. 16] [A.P. Mar. 15]
Answer:

  • Ozone layer is present in stratosphere of the atmosphere.
  • Ozone in the stratosphere is due to the following reactions.
    AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 5
  • But due to industrialisation, certain chemical substances enter into stratosphere and destroy the ozone.
  • The following are the major substances that causes depletion of ozone layer.
    1. CFCs (Chloro fluoro carbons (or) freons)
    2. NO
    3. Cl2 (Chlorine)
  • CFCs are colourless, odourless, lighter, non flammable, non toxic organic molecules which are used in refrigeratos, air conditioners, in the manufacturing of plastic foam and for cleaning computer parts.
  • These CFCs enter in the stratosphere and deplete the ozone as follows.
    AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 6
  • Chlorine radicals (Cl*) are continuously generated and decompose O3.
  • NO released by super sonic jet planes and that formed by burning fossil fuels enters into stratosphere and decompose ozone.
  • Cl2 decompose into Cl* radicals and these Cl* deplete O3 by chain reactions.

Effects of depletion of the ozone layer (ozone hole): Due to depletion of ozone more U.V. enters into troposphere. This UV radiation leads to

  1. ageing of skin
  2. cataracts
  3. skin burns
  4. skin cancer
  5. damage to fish production
  6. killing many phytoplanktons
  7. effect plant proteins leading to the harmful mutation of cells
  8. evaporate surface water through the stomata of the leaves.
  9. decrease moisture content of the soil.
  10. damage paints, fibres, by fading them faster.
  11. effect on photo synthesis etc.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 9.
List out the industrial wastes that cause water pollution and what are the international standards fixed for drinking water ?
Answer:
Industrial wastes from food processing plants and paper and pulp mills are oxygen – demanding wastes. So they cause depletion of D.0 from the water.

Salts, trace of elements like Copper, Zinc, Arsenic etc., metals coming out of chromium plating Industry pollute water. They effect the human health and aquatic animals.

In Japan, at Minimata, water was polluted with mercury released from industry. The mercury enter the fish and the people who consumed the fish are adversely effected.

Mining and Nuclear power plant pollute the water with radioactive substances.

International standards for drinking water

Fluoride concentration:

  • The drinking water sample is tested for fluoride ion concentration. Its deficiency causes diseases like tooth decay etc.
  • The permissible level of concentration of fluoride in water is up to 1 ppm.
  • Fluoride concentration above 2 ppm causes brown mottling of teeth.
  • Fluoride concentration greater than 10 ppm causes harmful effects to bones and teeth.

Lead :

  • The prescribed upper limit of concentration limit of lead in drinking water is about 50 ppm.
  • Lead can damage kidney, liver, reproductive system etc.

Sulphate :

  • Sulphate concentration greater than 500 ppm in drinking water causes laxative effect.
    Nitrate :
  • Nitrate concentration maximum limit in drinking water is about 50 ppm.
  • Excess of nitrate in drinking causes disease such as methemoglobinemia.

Metals :

Maximum prescribed concentration of some common metals recommended in drinking water given in the following table.
Maximum prescribed concentration of Some Metals in Drinking Water.
AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 7

Question 10.
Explain in detail the strategies adopted in Green Chemistry to avoid environment pollution.
Answer:
The ways of using the knowledge and the principles of chemistry and other sciences to develop methods to reduce as far as possible the pollution of the environment are known as green chemistry.
Ex : In the dry cleaning of clothes, earlier (CCl2 = CCl2) tetrachloroethane was used. This com-pound contaminates the ground water and is a suspected carcinogen. Therefore, using this compound is replaced by a process in which liquefied carbon dioxide with a suitable detergent is used. This would not pollute ground water much. Now-a-days hydrogen peroxide is used for bleaching clothes in laundries. This gives better results and decreases the consumption of water.
Ex : Synthesis of ethanol is now commercially prepared by one step by the oxidation of ethane in presence of catalyst.
AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 8

Long Answer Questions

Question 1.
What is environmental pollution ? How many types of pollution are encountered ?
Answer:
Environmental pollution is defined as the addition of any external material like organic, Inorganic biological, radio logical (or) any change in nature which may harm (or) affect badly the living organism directly (or) indirectly, immediately (or) slowly.
The types of pollution encountered are the following :

  1. Air pollution
  2. Water pollution
  3. Soil pollution
  4. Oil pollution
  5. Noise pollution etc.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 2.
Explain the following in detail.
a) Global warming
b) Ozone depletion
c) Acid rain
d) Eutrophication.
Answer:
a) Global Warming : The Earth is heated by sunlight and some of the heat that is absorbed by the Earth is radiated back into space. But some gases like CO2, CFC’s, O3, NO and water vapour present in the lower atmosphere do not allow the Earth to re-radiate the heat into space. A part of the heat so trapped in these atmospheric gases is re-emitted to the earth’s surface. This phenomenon is called the “Green house effect” (or) Global warming.
The gases which are responsible for Green house effect are CO2, CFC’s, O3, NO, CH4 water vapour etc. and they are termed as Green house gases. The effect of increase in CO2 causes global warming.

b) Effects of depletion of the ozone layer (ozone hole) : Due to depletion of ozone more U.V. enters into troposphere. This UV radiation leads to

  1. ageing of skin
  2. cataracts
  3. skin burns
  4. skin cancer
  5. damage to fish production
  6. killing many phytoplanktons
  7. effect plant proteins leading to the harmful mutation of cells
  8. evaporate surface water through the stomata of the leaves.
  9. decrease moisture content of the soil.
  10. damage paints, fibres, by fading them faster.
  11. effect on photo synthesis etc.

c) Acid rain :

  • Acid rains are due to oxides of N, S and C.(NO2, SO2 and CO2)
  • These oxides dissolve in rain water and formed as acids. (HNO3, H2SO4 and H2CO3)
  • These come down to earth as rain and deposited on the earth’s surface.
  • Acid – rain is more in industrial areas.

Chemical equations involved:-
NO2 + NO3 → N2O5
N2O5 + H2O + 2HNO3
CO2 + H2O→ H2CO3
SO3 + H2O → H2SO4
The pH of acid rain is lessthan 5.6.

Acid rains are harmful because

  1. The life of old buildings will be considerable reduced.
  2. The pH of the soil changes affecting its fertility.
  3. Ammonium salts formed by acid rains can be seen as atmospheric haze.
  4. Ammonium salts in rain drops result in wet decomposition.
  5. Acid rains are harmful to agriculture, trees, plants etc.
  6. Due to acid rains aquatic life disturbed.
  7. Due to acid rains the historical monuments like Tajmahal are damaged.

d) Eutrophication : Water present in ponds and lakes becomes over nutritious when organic sub-stances from agriculture and industry are thrown into it. It can support the luxuriant growth of algae and thus the lakes and ponds become marshy. This phenomenon is called Eutrophication.

AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry

Question 3.
Green Chemistry is to avoid environmental pollution. Explain.
Answer:
The ways of using the knowledge and the principles of chemistry and other sciences to develop methods to reduce as far as possible the pollution of the environment are known as green chemistry.
Ex : In the dry cleaning of clothes, earlier (CCl2 = CCl2) tetrachloroethane was used. This com-pound contaminates the ground water and is a suspected carcinogen. Therefore, using this com-pound is replaced by a process in which liquefied carbon dioxide with a suitable detergent is used. This would not pollute ground water much. Now-a-days hydrogen peroxide is used for bleaching clothes in laundries. This gives better results and decreases the consumption of water.
Ex : Synthesis of ethanol is now commercially prepared by one step by the oxidation of ethane in presence of catalyst.
AP Inter 1st Year Chemistry Study Material Chapter 12 Environmental Chemistry 9
Thus green chemistry is a cost effective approach and it involves reduction in material, energy consumption and waste greneration.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements – Group 14

Andhra Pradesh BIEAP AP Inter 1st Year Chemistry Study Material 11th Lesson The p-Block Elements – Group 14 Textbook Questions and Answers.

AP Inter 1st Year Chemistry Study Material 11th Lesson The p-Block Elements – Group 14

Very Short Answer Questions

Question 1.
Discuss the variation of oxidation states in the group -14 elements.
Answer:

  • The common oxidation states exhibited by group-14 elements are +4 and +2.
  • Carbon exhibits negative oxidation states.
  • Heavier elements exhibits +2 oxidation state.
  • The tendency to show +2 oxidation state increases in the order Ge < Sn < pb.
  • pb exhibits +2 oxidation state as stable state because of inert pair effect.

Question 2.
How the following compounds behave with water
a) BCl3
b) CCl4.
Answer:
a) BCl3 reacts with water (hydrolysis) to form boric acid.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 1
b) CCl4 does not undergo hydrolysis due to lack of d-orbitals in the central atom ‘C1 and due to its highly non polar nature, CCl4 does not acts as Lewis acid.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 3.
Are BCl3 and SiCl4 electron-deficient compounds ? Explain.
Answer:

  • BCl3 and SiCl4 are electron-deficient compounds.
  • These two compounds behave as Lewis acids.
  • These compounds are electron pair acceptors.
  • The following reacts support the electron deficiency of BCl3 and SiCl4.
    AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 2

Question 4.
Give the hybridization of carbon in
a) CO3-2
b) diamond
c) graphite
d) fullerene
Answer:
a) In C03-2 carbon atom undergoes sp2 hybridisation.
b) In Diamond carbon atom undergoes sp3 hybridisation.
c) In Graphite carbon atom undergoes sp2 hybridisation.
d) In Fullerenes carbon atom undergoes sp2 hybridisation.

Question 5.
Why is’CO’poisonous ? [T.S. Mar. 16]
Answer:
‘CO’ gas is highly poisonous because it has the ability to form a stable complex with haemoglobin.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 3
Carboxy haemoglobin is 300 times more stable than oxyhaemoglobin.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 6.
What is allotropy ? Give the crystalline allotropes of carbon. [Mar. 13]
Answer:

  • The phenomenon of existence of an element in different physical forms having similar (or) same chemical properties is called allotropy.
  • Crystalline allotropes of carbon are
    a) Diamond
    b) Graphite
    c) Fullerenes.

Question 7.
Classify the following oxides as neutral, acidic, basic or amphoteric,
a) CO
b) B2O3
c) SiO2
d) CO2
e) Al2O3
f) PbO2
g) Tl2O3
Answer:
a) CO is neutral oxide.
b) B2O3 is acidic oxide.
c) SiO2 is acidic oxide.
d) CO2 is acidic oxide.
e) Al2O3 is amphoteric oxide.
f) PbO2 is amphoteric oxide.
g) Tl2O3 is basic oxide.

Question 8.
Name any two manmade silicates.
Answer:
Glass and cement are man made silicates.

Question 9.
Write the outer electron configuration of group -14 elements.
Answer:
The general outer most electronic configuration of group – 14 elements is ns2np2.
1) Carbon – [He] 2s22p2
2) Silicon – [Ne] 3s23p2
3) Germanium – [Ar] 3d104s24p2
4) Tin – [Kr] 4d10 5s2 5p2
5) Lead – [Xe] 4f4 5d10 6s2 6p2

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 10.
How does graphite function as a lubricant ?
Answer:
Graphite is soft and it has layer lattice. In this structure one of the layer slided over another due to weak Vander Waal’s force of attraction. These layers are slippery. Hence it is greasy and function as lubricant.

Question 11.
Graphite is a good conductor – explain.
Answer:
In graphite carbon undergoes sp2 hybridisation. Each carbon forms three a – bonds with three neighbouring carbon atoms. Fourth electron forms TC – bond and it is delocalised. Due to the presence of these moving (or) free electrons graphite acts as good conductor.

Question 12.
Explain the structure of silica.
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 4

  • Silica is a giant molecule with 3 – dimensional structure.
  • In Silica eight membered rings are formed with alternate Silicon and Oxygen atoms.
  • Each ‘Si’ is tetrahedrally surrounded by four Oxygen atoms.
  • Each Oxygen at the vertex of the tetrahedron is shared by two Silicon atoms.
  • In SiO2 Silicon atom undergoes sp3 hybridisation.

Question 13.
What is ‘Synthesis gas’?
Answer:

  • Water gas is also called as synthesis gas.
  • It is a mixture of CO and H2.
  • It s prepared by passing steamover hot coke.
  • It is used for the synthesis of methanol and a number of hydrocarbons. Hence it is called synthesis gas.

Question 14.
What is producer gas?
Answer:

  • Producer gas is mixture of CO and N2.
  • It is prepared by passing air over hot coke.

Question 15.
Diamond has high melting point – Explain.
Answer:

  • In Diamond each carbon undergoes sp3 hybridisation and it is surrounded by four other carbon atoms with strong a – bonds tetrahedrally.
  • The C – C bond energy in diamond is very high and it has 3 – dimensional structure.
  • Due to these reasons diamond has high melting point.
  • It has melting point 4200 K.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 16.
Give the use of CO2 in photosynthesis.
Answer:
The process of converting the atmospheric CO2 into Carbohydrates by green plants is known as ‘photosynthesis’.
In Photosynthesis CO2 changes to carbohydrates such as glucose.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 5

Question 17.
How does CO2 increase the green house effect ?
Answer:

  • Green plants absorbs CO2 gas for photosynthesis and releases O2 gas.
  • Due to deforestation, decomposition of lime stone and burning of fossil fuels CO2 concentration is increased in atmosphere.
  • The increase of CO2 level disturbs the O2 – CO2 balance in the atmosphere and it is responsible for green house effect (or) global warming.

Question 18.
What are silicones ?
Answer:

  • Silicones are the organo Silicon polymers containing R2 SiO – repeating unit.
  • These are synthetic compounds containing Si – O – Si.
  • Linkage preparation : These are formed by the hydrolysis of chlorosilanes.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 19.
Give the uses of silicones.
Answer:
Uses of silicones :

  • These are used in surgical and cosmetic plants.
  • These are used in preparation of silicone rubbers.
  • These are used as sealoant, greases etc.
  • These are used in preparing water proof clothes and papers.
  • These are used as insulators.
  • These are used in paints and enamels.

Question 20.
What is the effect of water on tin ?
Answer:

  • Tin metal reacts with steam to form tin dioxide and dihydrogen gas.
  • In this reaction steam is decomposed.
    Sn + 2H2O AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 6 SnO2 + 2H2

Question 21.
Write an account of SiCl4.
Answer:

  • Silicon tetrachloride (SiCl4) is also called as tetra chloro silico methane.
  • SiCl4 can acts as Lewis acid due to availability of 3d orbital in ‘Si’.
  • SiCl4 undergoes hydrolysis due to presence of vacant 3d-orbital. Here water molecules forms dative bonds with empty 3d-orbitals of Siratom.

Uses :

  • SiCl4 and NH3 mixture used to produce smoke screens.
  • Ultra pure Silicon is used to make transistors.
  • SiO2 prepared from SiCl4 used in epoxypaints, resis etc..

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 22.
SiO2 is a solid while CO2 is a gas – explain.
Answer:

  • Silica (SiO2) has giant molecular structure.
  • In SiO2 ‘Si’ undergoes sp3 hybridisation.
  •  It is a 3 – dimension structure in which each ‘Si’ atom is tetrahedrally surrounded by four oxygen atoms.
  • Hence it exists as solid compound.
  • CO2 has linear structure.
  • In CO2 ‘C’ undergoes sp hybridisation.
  • In between CO2 molecule weak Vander Waal’s forces are present.
  • In CO2 molecule two double bonds are present.
  •  Hence CO2 exists as a gas.

Question 23.
Write the use of ZSM – 5.
Answer:

  1. ZSM – 5 is a zeolite.
  2. It is used to convert alcohols directly into gasoline.

Question 24.
What is the use of dry ice ?
Answer:

  1. Solid CO2 is called as dry ice.
  2. It is used as refrigirent for frozen food and ice – creams.

Question 25.
How is water gas prepared ?
Answer:
Water gas is prepared by passing superheated steam over hot coke.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 7

Question 26.
How is producer gas prepared ?
Answer:
Producer gas is prepared by passing air over white not coke.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 8

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 27.
C-C bond length in graphite is shorter than C-C bond length in diamond – explain.
Answer:

  1. In graphite each carbon undergoes sp2 hybridisation and hence bond length is 1.42 A° (or) 141.5 pm.
  2. Graphite has hexagonal layer like lattice. It is a 2-dimensional structure.
  3. In diamond each carbon undergoes sp3 – hybridisation and hence bond length is 1.54 A° (or) 154 pm.
  4. Diamond has regular tetrahedral giant polymeric structure. It is a 3-dimensional structure.

Question 28.
Diamond is used as precious stone – explain.
Answer:

  • Diamonds are used as precious stones.
  • Diamonds are clear, colourless form of pure carbon.
  • These are hardest substances occurring naturally.
  • The weight of diamond expressed in carats.
    1 carat = 200 mg.

Question 29.
Carbon never shows co-ordination number greater than four while other members of carbon family show co-ordination number as high as six – explain.
Answer:
Carbon never shows co-ordination number greater than four because of absence of d-orbitals in carbon atom.

The other members of carbon family show co-ordination number as high as six because of availability of d – orbitals.

Question 30.
Producer gas is less efficient fuel than water gas – explain.
Answer:

  1. Producer gas has calorific value 5439.2 KJ/m3
  2. Water gas has calorific value 13000 KJ/m3.
  3. Due to high calorific value of watergas, it is more efficient fuel than producer gas (or) producer gas is less efficient than watergas.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 31.
SiF6-2 is known while SiCl6-2 is not. Explain. [A.P. Mar. 16]
Answer:
SiF6-2 is known while SiCl6-2 is not because

  • Si+4 has small size so it cannot be accomodate six large chloride ions.
  • The interaction between lone pairs of Cl ion and Si+4 is not very strong.

Short Answer Questions

Question 1.
Explain the difference in properties of diamond and graphite on the basis of their structure.
Answer:
Diamond
a) Each carbon is sp3 hybridised.
b) Each carbon is bonded to 4 other carbons tetrahedrally.
c) It has a 3 dimensional structure.
d) C – C bond length is 1.54 Å and bond angle is 109° 28′.
e) Carbon atoms are firmly held with strong covalent bonds.
f) Diamond is very hard.
g) Density = 3.5 g/cc.
h) Graphite is a conductor due to the presence of free electrons.
i) It is transparent to light and X-rays. It has high refractive index (2.45).

Graphite
a) Each carbon is sp2 hybridised.
b) Each carbon is bonded to 3 other carbon atoms to form hexagonal rings. It has sheet like structure.
c) It has a 2 dimensional structure.
d) C – C bond length in hexagonal rings is 1.42 A° and bond angle is 120°.
e) The distance between two adjacent layers is 3.35 A°. These layers are held by weak Vander Waal’s forces.
f) Graphite is soft.
g) Density – 2.2g/cc.
h) Diamond is an insulator due to the absence of free electrons.
i) It has layer, lattice. The layers are slippery. Hence it is greasy.

Question 2.
Explain the following.
a) PbCl2 reacts with Cl2 to give PbCl4
b) PbCl4 is unstable to heat,
c) Lead is not known to form PbI4.
Answer:
a) PbCl2 + Cl2 → PbCl2
But PbCl4 is unstable than PbCl2. Because compounds of lead in +2 oxidation state are stable than +4 oxidation state.

b) PbCl4 is unstable to heat:

  • In PbCl4 lead exhibits +4 oxidation state.
  • he compounds of lead in +2 oxidation state are stable than +4 oxidation state. Hence PbCl4 is unstable to heat.

c) Lead is not known to form PbI4:

  • Pb – I bond formed initially during the reaction does not release enough energy to unpair the 6s electrons.
  • Lead compounds in +2 state are stable than +4 state.
    Due to inert pair effect Pb exhibits stable +2 oxidation state.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 3.
Explain the following :
a) Silicon is heated with methyl chloride at high temperature in presence of copper.
b) SiO2 is treated with HF.
c) Graphite is a Lubricant
d) Diamond is an abrasive.
Answer:
a)

  • Methyl chloride reacts with silicon at high temperature in presence of copper catalyst to form various types of methyl substituted chlorosilane of formula MeSiCl3, Me2SiCl2, Me3SiCl with small amount Me4Si.
  • Hydrolysis of dimethyl dichloro silane followed by condensation polymerisation forms a straight chain polymer (silicone) (Me = CH3 – group)
    AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 9

b) SiO2 is treated with HF to form SiF4. Which on hydrolysis to form H4SiO4 and H2SiF6.
SiO2 + 4HF → SiF4 + 2H2O
SiF4 + 4H2O → H4SiO4 + 2H2SiF6

c) Graphite is soft and it has layer lattice. In this structure one of the layer slided over another due to weak Vander Waal’s force of attraction. These layers are slippery. Hence it is greasy and function as lubricant.

d) The covalent bonds in diamond are very strong and difficult to break, Hence diamond is used as an abrasive for sharpening hard tpols, in making dyes and in the manufacturing of tungsten filaments etc.

Question 4.
What do you understand by
a) Allotropy
b) Inert pair effect
c) Catenation.
Answer:
a) Allotropy :
The phenomenon of existence of an element in different physical forms having similar (or) same chemical properties is called allotropy.
Crystalline allotropes of carbon are
a) Diamond
b) Graphite.

b) Inert pair effect : The reluctance of ‘ns’ pair of electrons to take part in bond formation is known as inert pair effect.
(or)
The occurrence of oxidation states two units less than the group oxidation states is known as inert pair effect.
Eg : Lead exhibits +2 oxidation state as stable oxidation state due to inert pair effect. (Instead of +4 state).

c) Catenation : The phenomenon of self linkage of atoms among themselves to form long chains (or) rings is called as catenation.
Carbon has highest catenation tendency due to its high bond energy (348 KJ/mole).

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 5.
If the starting material for the manufacturing of Silicons is RSiCi3. Write the structure of the product formed.
Answer:
When RSiCl3 type of compound is used for the manufacturing of silicones a cross – linked silicon is formed.
Eg : When Methyl trichloro silane (CH3SiCl3) undergoes hydrolysis to give monomethyl silane triol. This undergoes polymerisation to form a very complex cross-linked polymer (Silicone).
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 10

Question 6.
Write a short note on Zeolites.
Answer:
Zeolites : The three dimensional structure contain no metal ions. If some of the Si+4 in them is replaced by Al+3 and an additional metal ion an infinite three dimensional lattice is formed. Replacements of one or two silicon atoms in [Si2O8]-2n form zeolites. Zeolites act as ion exchanges and as molecular sieves. The structures of zeolites permit the formation of cavities of different sizes. Water molecules and a variety of other molecules like NH3, CO2 and ethanol can be trapped in these cavities. Then zeolites serve as molecular sieves. They trap Ca+2 ions from hard water and replace them by Na+2 ions.
Uses of Zeolites :
Zeolites are used as catalysts in petrochemical industries for cracking of hydrocarbons and in Isomerisation.
Zeolite ZSM – 5 is used to convert alcohols directly into gasoline.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 7.
Write a short note on Silicates.
Answer:
Silicates : Many building materials are silicates.
Ex : Granites, slates, bricks and cement. Ceramics and glass are also silicates. The Si – O bonds in silicates are very strong. They do not dissolve in any of the common solvents nor do they mix with other substances readily.
The silicates can be divided into six types. They are mentioned here.

  1. Orthosilicate or Nesosilicates : Their general formula may be M211 (SiO4).
    Ex : Willemite Zn2 (SiO4).
  2. Pyrosilicates or sorosilicates or Disilicates : These contain SiO7-6 units. Pyrosilicates are rare.
    Ex: Thorveitite Ln2 (Si2O7).
  3. Chain silicates : They have the units (SiO3)n2n-
    Ex : Spodumene LiA/ (SiO3)2.
    Amphiboles are one type chain silicates. Generally double chains are formed in them.
  4. Cyclic silicates : They are silicates having ring structures. They may be formed of general formula (SiO3)n2n-. Rings containing three, four, six and eight tetrahedral units are known. But rings with three and six are the most common.
    Ex : Beryl Be3Al2 (Si6O18)
  5. Sheet silicates : When SiO4 units share three corners the structure formed is an inifinite two dimensional sheet. The empirical formula (Si2O5)n2n-. These compounds appear in layer struc-tures. They can be cleaved.
    Ex : Kaolin Al2 (OH)4 Si2O5.
  6. Frame work silicates or three dimensional silicates : Sharing all the four corners of a SiO4 tetrahedron results in three dimensional lattice of formula SiO2.
    Ex : Quartz, Tridymite; Cristobalite; Feldspar and ultramarine (Na8 [Al6 Si6 O24] S2), zeolites.

Question 8.
What are Silicones ? How are they obtained ?
Answer:

  • Silicones are the organo silicon polymers containing R2 SiO – repeating unit.
  • These are synthetic compounds containing Si – 0 – Si.
    Preparation : These are formed by the hydrolysis of chlorosilanes.
  • Methyl chloride reacts with Silicon at high temperature in presence of copper catalyst to form various types of methyl substituted chlorosilane of formula MeSiCl3, Me2SiCl2, Me3SiCl with small amount Me4Si
  • Hydrolysis of dimethyl dichloro silane followed by condensation polymerisation forms a straight chain polymer (Silicone) (Me = CH3 – group).
    AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 11

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 9.
Write a short note on Fullerene.
Answer:
Fullerenes :

  • Fullerenes are one type of crystalline allotropes of carbon.
  • These are formed by heating graphite in an electric arc in presence of inertgases such as Helium (or) Argon.
  • These have smooth structure without dangling bonds. Hence Fullerenes are the only pure forms of carbon.
  • C60 molecule is called Buck minster fullerene and it’s shave was like a soccer ball.
  • C60 contains twenty 6 – membered rings and twelve 5 – membered rings.
    AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 12
  •  In C60 6 – membered rings can combine with 5 (or) 6 – mem-bered rings while 5 – membered rings only combine with 6 – membered rings.
  • In Fullerens each carbon undergoes sp2 hybridisation.
  • Fullere has aromatic nature due to delocalisation of electrons in unhybrid p-orbitals.
  • The C – C bond length in these compound lies between single and double bond lengths.
  • Spherical fullerene are also named as bucky balls.
  • The ball shaped molecule has 60 vertices.
  • The C – C bond distances are 1,43A° and 1.38A° respectively.

Question 10.
Why SiO2 does not dissolve in water ?
Answer:
Silica (SiO2) is a non reactive compound in it’s normal state.

  1. This non reactivity is due to very high Si – O bond enthalpy.
  2. Silica is a giant molecule with 3-dimensional structure.
  3. In Silica each silicon atom is covalently bonded in a tetrahedral manner to four oxygen atoms
  4. Hence SiO2 is insoluble in water.
  5. But slightly dissolves at high pressures when heated.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 11.
Why is diamond hard ?
Answer:
In diamond, each carbon undergoes sp3 hybridization. A carbon atom is bound to four carbon atoms, arranged in a tetrahedral symmetry, with single bonds. A three dimensional arrangement of the tetrahedral structures result in giant molecule. The bond energy is very high (348 kJ mol-1). It is very difficult to break the bonds. So, diamond is hard.

Question 12.
What happens when the following are heated
a) CaCO3
b) CaCO3 and SiO2
c) CaCO3 and excess of coke.
Answer:
CaCO3 up on heating gives Quick lime.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 13
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 14
Quick lime (CO) with silica gives Calcium silicate.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 15
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 16
Coke reacts with Quick lime and form Carbides.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 17

Question 13.
Why does Na2CO3 solution turn into a suspension, when saturated with CO2 gas.
Answer:
An aq. solution of Na2CO3 when saturated with CO2, gives Sodium bicarbonate (NaHCO3).
Na2CO3 + H2O + CO2 → 2NaHCO3
NaHCO3 is less soluble compared to Sodium carbonate, hence suspension is formed.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 14.
What happens when
a) CO2 is passed through slaked lime
b) CaC2 is heated with N2.
Answer:
a) Slaked lime, Ca(OH)2 is turned milky on passing CO2 with the formation of insoluble calcium
carbonate Ca(OH)2 + CO2 → CaCO3 + H2O on passing more ‘CO2‘, CaCO3 is converted into Calcium bicarbonate.
CaCO3 + H2O + CO2 → Ca(HCO3)2
b) CaC2 on heating with N2 gives calcium cyanamide.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 18

Question 15.
Write a note on the anomalous behaviour of carbon in the group -14.
Answer:
Carbon shows ariamalous behaviour in group – 14 elements. The following facts support that anamalous behaviour.
Except carbon all other elements of group -14 has available d-orbitals and can expand octet in valency shell.
Carbon occurs in free state but not the other elements of this group.

  • Maximum covalency of carbon is four but for silicon is six.
  • C – C bond energy is very high (348 kJ/Mde).
  • Carbon can form multiple bonds with C, O, S, etc.
  • Hydrocarbons are more stable thermally than silanes.

Long Answer Questions

Question 1.
What are Silicones ? How are they prepared ? Give one example. What are their uses?
Answer:

  • Silicones are the organo silicon polymers containing R2 SiO – repeating unit.
  • These are synthetic compounds containing Si – 0 – Si.
    Preparation : These are formed by the hydrolysis of chlorosilanes.
  • Methyl chloride reacts with Silicon at high temperature in presence of copper catalyst to form various types of methyl substituted chlorosilane of formula MeSiCl3, Me2SiCl2, Me3SiCl with small amount Me4Si
  • Hydrolysis of dimethyl dichloro silane followed by condensation polymerisation forms a straight chain polymer (Silicone) (Me = CH3 – group).
    AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 11

Uses of Silicones:

  • These are used in surgical and cosmetic plants.
  • These are used in preparation of Silicons rubbers.
  • These are used as sealoant, greases etc.
  • These are used in preparing water proof clothes and papers.
  • These are used as insulators.
  • These are used in paints and enamels.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 2.
Explain the structure of Silica. How does it react with
a) NaOH and
b) HF.
Answer:

  • Silica is a giant molecule with 3 – dimensional structure.
  • In Silica eight membered rings are formed with alternate Silicon and Oxygen atoms.
  • Each ‘Si’ is tetrahedrally surrounded by four Oxygen atoms.
  • Each Oxygen at the vertex of the tetrahedron is shared by two Silicon atoms.
  • In SiO2 Silicon atom under goes sp3 hybridisation.
    AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 19

a) Silica reacts with NaOH and forms Sodium Silicate (Na2SiO3)
SiO2 + 2 NaOH → Na2SiO3 + H2O

b) SiO2 is treated with HF to form SiF4. Which on hydrolysis to form H4SiO4 and H2SiF6.
SiO2 + 4HF → SiF4 + 2H2O
SiF4 + 4H2O → H4SiO4 + 2H2SiF6

Question 3.
Write a note on the allotropy of carbon.
Answer:
The property of an element to exist in two or more physical forms due to difference in the arrangement of atoms is called Allotropy. Allotropes have more or less similar chemical properties but different physical properties.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 20
Diamond, graphite and fullerenes are the crystalline allotropes of carbon.
Structure of Diamond : In diamond, each carbon atom bonded to four carbon atoms situated tetrahydrally around it.
In diamond, each carbon atom is in sp3 hybridisation and is linked to four carbon atoms by single covalent bonds.
C – C bond distance in diamond is 1.54 A°.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 21 bond angle in diamond is 109° 28″.
Uses of Diamond :

  1. Diamonds are used as precious stones for jewellery because of their ability to reflect light.
  2. Diamonds are used for cutting glass and drilling rocks due to their remarkable hardness.

Structure of Graphite : Graphite consists of a series of layers in which hexagonal rings made up of carbon atoms.
In Graphite, each carbon atom undergo sp2 hybridisation and forms three covalent bonds with three other carbon atoms.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 22
The fourth electron present in the pure ‘p orbitaI which is unhybridised. The electron become Free Electron.
The C — C bond length in graphite is 1.42 A°.
AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 23
The distance between the two layers in graphite is 3,4 A°.
These layers are held together by Vander Waal’s forces which are weak.
Graphite is a layer lattice structure.
Uses of Graphite:

  1. Graphite is used as a lubricant.
  2. It is used in the manufacturing of lead pencils.
  3. It is used in the manufacturing of Electrodes and Refractory crucibles.

Fullerenes :

  • Fullerenes are one type of crystalline allotropes of carbon.
  • These are formed by heating graphite in an electric arc in presence of inertgases such as Helium (or) Argon.
  • These have smooth structure without dangling bonds. Hence Fullerenes are the only pure forms of carbon.
  • C60 molecule is called Buck minster fullerene and it’s shave was like a soccer ball.
  • C60 contains twenty 6 – membered rings and twelve 5 – membered rings.
  • In C60 6 – membered rings can combine with 5 (or) 6 – membered rings while 5 – membered rings only combine with 6 – membered rings.
  •  In Fullerens each carbon undergoes sp2 hybridisation.
  • Fullere has aromatic nature due to delocalisation of electrons in unhybrid p-orbitals.
  • The C – C bond length in these compound lies between single and double bond lengths.
  • Spherical fullerene are also named as bucky balls.
  • The ball shaped molecule has 60 vertices.
  • The C – C bond distances are 1.43A° and 1.38A° respectively.
  • Amorphous allotropes of carbon are coal, coke, animal charcoal wood charcoal, lamp black, carbon black, gas carbon, petroleum coke and sugar charcoal.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 4.
Write a note on
a) Silicates
b) Zeolites
c) Fullerenes.
Answer:
a) Silicates : Many building materials are silicates.
Ex : Granites, slates, bricks and cement. Ceramics and glass are also silicates. The Si – O bonds in silicates are very strong. They do not dissolve in any of the common solvents nor do they mix with other substances readily.

The silicates can be divided into six types. They are mentioned here.

  1. Orthosilicate or Nesosilicates : Their general formula may be M211 (SiO4).
    Ex : Willemite Zn2 (SiO4).
  2. Pyrosilicates or sorosilicates or Disilicates : These contain SiO7-6 units. Pyrosilicates are rare.
    Ex : Thorveitite Ln2 (Si2O7).
  3. Chain silicates : They have the units (SiO3)n2n-
    Ex : Spodumene LiAl (SiO3)2.
    Amphiboles are one type chain silicates. Generally double chains are formed in them.
  4. Cyclic silicates : They are silicates having ring structures. They may be formed of general formula (SiO3)n2n-. Rings containing three, four, six and eight tetrahedral units are known. But rings with three and six are the most common.
    Ex : Beryl Be3Al2 (Si6O18)
  5. Sheet silicates : When SiO4 units share three corners the structure formed is an inifinite two dimensional sheet. The empirical formula (Si2O5)n2n-. These compounds appear in layer struc-tures. They can be cleaved.
    Ex : Kaolin Al2 (OH)4 Si2O5.
  6. Frame work silicates or three dimensional silicates : Sharing all the four corners of a SiO4 tetrahedron results in three dimensional lattice of formula SiO2.
    Ex : Quartz, Tridymite; Cristobalite; Feldspar and ultramarine (Na8 [Al6 Si6 O24] S2), zeolites.

b) Zeolites : The three dimensional structure contain no metal ions. If some of the Si+4 in them is replaced by Al+3 and an additional metal ion an infinite three dimensional lattice is formed. Replacements of one or two silicon atoms in [Si2O8]-2n form zeolites. Zeolites act as ion ex-changes and as molecular sieves. The structures of zeolites permit the formation of cavities of different sizes. Water molecules and a variety of other molecules like NH3, CO2 and ethanol can be trapped in these cavities. Then zeolites serve as molecular sieves. They trap Ca+2 ions from hard water and replace them by Na+2 ions.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Uses of Zeolites :

  • Zeolites are used as catalysts in petrochemical industries for cracking of hydrocarbons and in Isomerisation.
  • Zeolite ZSM – 5 is used to convert alcohols directly into gasoline.

c) Fullerenes :

  • Fullerenes are one type of crystalline allotropes of carbon.
  • These are formed by heating graphite in an electric arc in presence of inertgases such as Helium (or) Argon.
  • These have smooth structure without dangling bonds. Hence Fullerenes are the only pure forms of carbon. .
  • C60 molecule is called Buck minster fullerene and it’s shave was like a soccer ball.
  • C60 contains twenty 6 – membered rings and twelve 5 – mem- bered rings.
    AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 12
  • In C60. 6 – membered rings can combine with 5 (or) 6 – membered rings while 5 – membered rings only combine with 6 – membered rings.
  • In Fullerens each carbon undergoes sp2 hybridisation.
  • Fullere has aromatic nature due to delocalisation of electrons in unhybrid p-orbitals.
  • The C – C bond length in these compound lies between single and double bond lengths.
  • Spherical fullerene are also named as bucky balls.
  • The ball shaped molecule has 60 vertices.
  • The C – C bond distances are 1.43A° and 1.38A° respectively.

Solved Problems

Question 1.
Select the member(s) of group 14 that

  1. forms the most acidic dioxide
  2. is commonly found in +2 oxidation state
  3. used as semiconductor.

Solution:

  1. Carbon
  2. lead
  3. Silicon and germanium

Question 2.
[SiF6]2- is known whereas [SiCl6]2- not. Give possible reasons.
Solution:
The main reasons are :

  1. Six large chloride ions cannot be accommodated around Si4+ due to limitation of its size.
  2. Interaction between lone pair of chloride ion and Si4+ is not very strong.

AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14

Question 3.
Diamond is covalent, yet it has high melting point. Why ?
Solution:
Diamond has a three – dimensional network involving strong C – C bonds, which are very difficult to break and in turn has high melting point.

Question 4.
What are Silicones ?
Solution:
Simple Silicones consist of AP Inter 1st Year Chemistry Study Material Chapter 11 The p-Block Elements - Group 14 24 chains in which alkyl or phenyl groups occupy the remaining bonding positions on each silicon. They are hydrophobic (water repellant) in nature.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements – Group 13

Andhra Pradesh BIEAP AP Inter 1st Year Chemistry Study Material 10th Lesson The p-Block Elements – Group 13 Textbook Questions and Answers.

AP Inter 1st Year Chemistry Study Material 10th Lesson The p-Block Elements – Group 13

Very Short Answer Questions

Question 1.
Discuss the pattern of variation in the oxidation states of Boron to Thallium.
Answer:

  • Boron exhibits – 3 oxidation state due to its small size and non-metalic nature.
  • Aluminium exhibits +3 oxidation state.
  • Gallium, Indium and Thallium exhibits both +1 and +3 oxidation states.
  • In Thallium +1 oxidation state is more stable than +3 due to inert pair effect.

Question 2.
How do you explain higher stability of TlCl3 ?
Answer:
[TlCl3 is not stable because Tl doesnot exist in Tl3 stable state.] [TlCl is stable because of inertpair effect Tl+1 is stable].

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 3.
Why does BF3 behave as a Lewis acid ?
Answer:
BF3 is a electron deficient molecule. It has the tendency to accept an electron pair. Electron pair acceptors are Lewis acids. Hence BF3 behave as a Lewis acid.

Question 4.
Is boric acid a protic acid ? Explain.
Answer:
Boric acid is a weak mono basic acid. In Boric acid plannar BO3 units are joined by hydrogen bonds. It has layer structure (polymeric). Hence it is not a protic acid. It does not give up a proton.

Question 5.
What happens when boric acid is heated ?
Answer:
Boric acid when heated above 370 K forms meta boric acid. This on heating forms Boric oxide.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 1

Question 6.
Describe the shapes of BF3 and BH4. Assign the hybridization of boron in these species.
Answer:

  • Shape of BF3 molecule is Trigonal planar
    Hybridisation of ‘B’ in BF3 is sp2
  • Shape of BH4 molecule is Tetrahedral
    Hybridisation of ‘B’ in BH4 is sp3

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 7.
Explain why atomic radius of Ga is less than that of ‘Al’.
Answer:
In Gallium penultimate shell contains 10-d electrons. Due to this 10-d electrons shielding effect becomes poor on outer most electrons. So nuclear charge increased in Gallium. Hence atomic radius of Ga is lessthan that of ‘Al’.

Question 8.
Explain inert pair effect.
Answer:
The reluctance of ‘ns’ pair of electrons to take part in bond formation is called inert pair effect.
(or)
The occurrence of oxidation states two unit lessthan the group oxidation states is called inert pair effect.
Eg : In Group – 13 Tl exhibits +1 oxidation state instead of +3 oxidation state due to inert pair effect.

Question 9.
Write balanced equations for
a) BF3 + LiH →
b) B2H6 + H2O →
c) NaH + B2H6
d) H3BO3 AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 2
e) B2H6 + NH3 AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 2
Answer:
a) 2BF3 + 6LiH → B2H6 + 6 LiF
b) B2H6 + 6H2O → 2B(OH)3 + 6H2
c) B2H6 + 2NaH → 2NaBH4
d) H3BO3 AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 2 HBO2 AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 2 B2O3
e) B2H6 + 6NH3 → 3[BH2(NH3)2]+ (BH4) AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 3 2B3N3H6 + 12H2

Question 10.
Why is boric acid polymeric ?
Answer:
Boric acid has layer like lattice. In this structure planar BO3 units are joined by hydrogen bonds and forms a polymeric structure.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 11.
What is the hybridization of B in diborane and borazine ?
Answer:

  • In diborane ‘B’ hybridisation is sp3
  • In Borazine ‘B’ hybridisation is sp2

Question 12.
Write the electronic configuration of group – 13 elements.
Answer:
General outer electronic configuration of Group – 13 elements is ns2np1

  • B – 1s22s22p1
  • Al – [Ne] 3s23p1
  • Ga – [Ar] 3d104s24p1
  • In – [Kr] 4d105s25p1
  • Tl – [Xe] 5d10 6s26p1

Question 13.
Give the formula of borazine. What is its common name ?
Answer:

  • The formula of borazine is B3N3H6.
  • It’s common name is “In organic benzene” because it is – iso structural with benzene.

Question 14.
Give the formulae of
a) Borax
b) Colemanite.
Answer:
a) Formula of Borax is Na2B4O7. 10H2O
b) Formula of Colemanite is Ca2B6O11.5H2O

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 15.
Give two uses of aluminium.
Answer:
Uses of Aluminium :

  • Aluminium is used in packing.
  • Aluminium is used in utensil making.
  • Aluminium alloys are used in shaping of pipes, tubes, wires etc.
  • Aluminium alloys are used in making air craft bodies.

Question 16.
What happens when
a) LiAlH4 and BCl3 mixture in dry ether is warmed and
b) Borax is heated with H2SO4 ?
Answer:
a) When LiAlH4 and BCl3 mixture is warmed in dry ether diborane is formed.
4BF3 + 3 LiAlH4 → 2B2H6 + 3 LiF + 3 AlF3
b) Borax is heated with H2S04 then boric acid is formed
Na2B4O7 + H2SO4 + 5H2O → Na2SO4 + 4H3BO3

Question 17.
Sketch the structure of Orthoboric acid.
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 4

Question 18.
Write the structure of AlCl3 as a climer.
Answer:
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 5

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 19.
Metal borides (having 10B) are used as protective shield – Why ?
Answer:
Boron – 10 (10B) has the capacity to absorb neutrons. Hence metal borides (having 10B) are used as protective shields in nuclear industry.

Short Answer Questions

Question 1.
Write reactions to justify amphoteric nature of aluminium.
Answer:

  • Amphoteric nature means having acidic as well as basic nature.
  • Aluminium reacts with both mineral acids as well as aqueous alkalis.

a) Reaction with mineral acid :
‘Al’ reacts with dil.HCl and liberates hydrogen gas.
2Al + 6HCl → 2AlCl3 + 3H2

b) Reaction with aqueous alkali :
‘Al’ reacts with aqueous alkali (NaOH) and liberates hydrogen gas.
2Al + 2NaOH + 6H2O → Na2[Al(OH)4] + 3H2

Question 2.
What are electron deficient compounds ? Is BCl3 an electron deficient species ? Explain.
Answer:
These are the compounds in which the available no.of valence electrons is lessthan the number required for normal covalent bond formation (or) for writting the Lewis structure of the molecule.

  • These compounds are electron pair acceptors and acts as Lewis acids.
  • BCl3 is an electron deficient compound.
  • In BCl3 ‘B’ contains only six electrons instead of eight electrons.
  • BCl3 has the tendency to accept an electron pair and acts as Lewis acid.
    Eg : Formation of BCl3 . NH3 :-
    BCl3 accepts an electron pair from NH3 and forms the compound BCl3.NH3 (Tetrahedral)
    AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 6

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 3.
Suggest reasons why the B – F bond lengths in BF3 (130 pm) and BF4 (143 pm) differ.
Answer:
About BF3 :

  • In BF3 the central atom ‘B’ contains three bond pairs in the valency shell.
  • ‘B’ under goes sp2 hybridisation.
  • Shape of the molecule is trigonal planar.

About BF4 :

  • In BF4 the central atom ‘B’ contains four bond pairs in the valency shell.
  • ‘B’ under goes sp3 hybridisation.
  • Shape of the molecule is tetrahedral.
  • The above reasons suggent that the difference in bond lengths of BF3 (130 pm) and BF4 (143 pm).

Question 4.
B – Cl bond has a bond moment. Explain why BCl3 molecule has zero dipolemoment.
Answer:

  • B – Cl bond is a polar bond so it has bond moment.
  • BCl3 molecule is non-polar because of its symmetrical structure. (Trigonal planar structure)
  • Symmetrical molecules has zero dipole moment.
    ∴ Dipole moment of BCl3 (μ) = 0

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 5.
Explain the structure of boric acid.
Answer:

  • Boric acid has a layer lattice.
  • In this layer lattice planar BO3 units are joined by hydrogen bonds.
  • The structure of Boric acid is polymeric as shown in following figure.
    AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 7
  • In the above structure dotted lines represents the hydrogen bonds

Question 6.
What happens when
a) Borax is heated strongly
b) Boric acid is added to water
c) Aluminium is heated with dilute NaOH
d) BF3 is treated with ammonia
e) Hydrated alumina is treated with aq.NaOH solution.
Answer:
a) Borax on heating first loses water molecules and forms sodium tetraborate. This on further heating forms a mixture of sodium metaborate and boric an hydride. This mixture is solidifies into glass like substance.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 8

b) Boric acid is added to water, boric acid accepts a hydroxyl ion from water.
B(OH)3 + 2H2O → [B(OH)4] + H3O+

c) Aluminium is heated with dilute NaOH, sodium metaluminate is formed with the liberation of hydrogen gas.
2Al + 2NaOH → 2NaAlO2 + H2

d) BF3 is treated with NH3 an addition compound BF3. NH3 is formed. BF3 accepts an electron pair from NH3 and forms a dative bond.
BF3 + AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 9 → [BF3 ← NH3] → [BF3.NH3]

e) Hydrated Alumina is treated with aq.NaOH to form sodium metaluminate.
Al2O3.2H2O + 2NaOH(aq) → 2NaAlO(aq) + 3H2O

Question 7.
Give reasons
a) Conc.HNO3 can be transported in aluminium container.
b) A mixture of dil. NaOH and aluminium pieces is used to open drain.
c) Aluminium alloys are used to make aircraft body.
d) Aluminium utensils should not be kept in water overnight.
e) Aluminium wire is used to make transmission cables.
Answer:
a) Cone. HNO3 can be transported in Aluminium containers because Al is passive towards Conc.HNO3 due to the formation of thin layer of Al2O3 on the surface.

b) A mixture of dil.NaOH and aluminium pieces is used to open drain because it acts as cleaning agent.
2Al + 2NaOH → 2NaAlO2 + H2

c) Aluminium alloys are used to make air craft body because it is a light metal, soft, malleable, ductile and tenacious. It shows resistance to atmosphere corrosion.

d) Aluminium utensils should not be kept in water overnight because Aluminium reacts with water and liberates hydrogen gas and heat. It makes colour dissolving and sometimes Aluminium compounds are toxic in nature.

e) Aluminium wire used to make transmission cables because of it’s good conductivity (electrical) and resistance to atmospheric corrosion.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 8.
Explain why the electronegativity of Ga, In and Tl will not vary very much.
Answer:

  1. Ga, In and Tl have the electro negativity values 1.6, 1.7 and 1.8 respectively.
  2. In Ga, In and Tl the d-electrons (d10) in penultimate shell do not shield the outer most electrons from nuclear attraction effectively.
  3. The reason for the above fact is.the shielding effect of various electrons in the orbitals follows the order
    s > p > d > f.
  4. Hence the outer electrons are held more firmly by the nucleus. Because of this, atoms with d- electrons in the penultimate shell (d10) are smaller in size. Ga, In and Tl has same number of penultimate shell electrons.
  5. So, Ga, In and Tl will not vary very much in their electronegativities.

Question 9.
Explain Borax bead test with a suitable example. [T.S. Mar. 16] [Mar. 13]
Answer:
Borax bend test: This test is useful in the identification of basic radicals in qualitative analysis. On heating borax swells into a white, opaque mass of anhydrous sodium tetra borate. When it is fused, borax glass is obtained. Borax glass is sodium meta borate and B203. The boric anhydride, B203, combined with metal oxides to form metal metaborates as coloured beads. The reactions are as follows :
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 10

Question 10.
Explain the structure of diborane. [A.P. Mar. 16] [A.P. & T.S. Mar. 15]
Answer:
Diborane is an electron deficient compound. It has ’12’ valency electrons for bonding purpose instead of ’14’ electrons.

In diborane each boron atom undergoes sp3 hybridization out of the four hybrid orbitals one is vacant.

Each boron forms two, σ – bonds (2 centred – 2 electron bonds) bonds with two hydrogen atoms by overlapping with their ‘1s’ orbital.

The remaining hybrid orbitals of boran used for the formation of B-H-B bridge bonds.

In the formation of B-H-B bridge, half filled sp3 hybrid ofbital of one boron atom and vacant sp3 hybrid orbital of second boron atom overlap with 1s orbital of H-atom.

These three centred two electron bonds are also called as banana bonds. These bonds are present above and below the plane of BH2 units.

Diborane contains two coplanar BH2 groups. The four hydrogen atoms are called terminal hydrogen atoms and the remaining two hydrogens are called bridge hydrogen atoms.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 11
Bonding in diborane, Each B atom uses sp3 hybrids for bonding. Out of the four sp3 hybrids on each B atom, one is without an electron shown in broken lines. The terminal B-H bonds are normal 2-centre-2- electron bonds but the two bridge bonds are 3-centre-2- electron bonds. The 3-centre-2-electron bridge bonds are also referred to as banana bonds.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 11.
Explain the reactions of aluminium with acids.
Answer:
Reactions of ‘Al’ with adds :
i) DiS. (or) cone. HCl dissolves Al and gives H2.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 12

ii) a) Dil.H2SO4 liberates H2.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 13
b) Cone. H2SO4 dissolves the metal ‘Al’ and gives SO2.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 14

iii) a) Very dil.HNO3 is reduced to NH4 NO3 by Al.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 14
b) Cone. HNO3 makes ‘Al’ passive due to the formation of a thin film of oxide layer on the metal surface.

Question 12.
Write a short note on the anamalous behaviour of boron in group – 13.
Answer:

  • Among Group – 13 elements ‘B’ is only the non metal.
  • ‘B’ forms only covalents compounds.
  • ‘B’ doesnot displaces hydrogen from acids.
  • ‘B’ shows diagonal relationship with ‘Si’.
  • ‘B’ forms acidic oxide where as other elements of group forms amphoteric oxides and basic oxides.
  • ‘B’ has only two electrons in the penultimate shell.
  • ‘B’ has covalency ‘4’ where as other elements has covalency of maximum ‘6’.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 13.
Aluminium reacts with dil.HNO3 but not with conc.HNO3 – explain.
Answer:

  • Dilute HNO3 reacts with Aluminium slowly and forms aluminium nitrate and ammonium nitrate.
    8Al + 30 HNO3 → 8 Al(NO3)3 + 3NH4NO3 + 9H2O
  • Aluminium doesnot react with cone. HNO3.

Reasons :

  • Aluminium is passive towards cone. HNO3 due to the formation of thin film of Al2O3 layer on the surface.
  • Because of this passivity between Al and conc.HNO3, conc.HNO3 is transported in Aluminium containers.

Question 14.
Give two methods of preparation of diborane.
Answer:

  1. In industries diborane is prepared by the reaction between boron tri fluoride and lithium hydride.
    AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 16
  2. Boron trichloride and hydrogen mixture subjected to silent electric discharge at low pressure to from diborane.
    2BCl3 + 6H2 → B2H6 + 6HCl
  3. Boron trichloride undergo reduction with LiAlH4 to form diborane.
    AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 17

Question 15.
How does diborane react with
a) H2O
b) CO
c) N(CH3)3 ?
Answer:
a) Diborane reacts with water to form boric acid and hydrogen.
B2H6 + 6H2O → 2H3BO3 + 6H2

b) Diborane reacts with CO at 100° C and 20 atm. pressure to form borane carbonyl.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 18
c) Diborane reacts with N(CH3)3 and form a adduct.
B2H6 + 2N(CH3)3 → 2BH3.N(CH3)3 (adduct)
Reactions b, c are cleavage reactions.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 16.
Al2O3 is amphoteric – explain with suitable reactions.
Answer:

  • Amphoteric oxides are the oxides which possess both acidic as well as basic nature.
    Al2O3 possess both acidic as well as basic behaviour.
  • Al2O3 react with both acids as well as bases to produce salts and water.
    Supporting reactions for amphoteric nature of Al2O3
    a) With acids:
    AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 19
    b) With bases:
    AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 20

Question 17.
Na2B4O7 + Cone. H2SO4 → A AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 21 > B (Green edged flame) Identify A and B
Hint: A = H3BO3 B = (C2H5)3 BO3.
Answer:
Na2B4O7 + Cone. H2SO4 + 5H2O → 4H3BO3 A) AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 21 B) 4(C2H5)3BO3

  • ‘A’ is H3BO3
  • ‘B’ is (C2H5)3 BO3.

Long Answer Questions

Question 1.
How are borax and boric acid prepared ? Explain the action of heat on them.
Answer:
Preparation of Borax:
Boric acid on heating first forms tetraboric acid. This on reaction with sodium hydroxide to form borax.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 22
Preparation of Boric acid : Borax is treated with conc.H2SO4 boric acid is formed.
Na2B4O7 + H2SO4 + 5H2O → 4H3BO3 + Na2SO4.

Heating of Borax :
Borax on heating first loses the water molecules and forms sodium tetraborate. This on further heating forms a mixture of sodium meta borate and boric anhydride.
Na2B4O7.10H2O AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 2 Na2B4O7 imgg 2 2 NaBO2 + B2O3.

Heating of Boric acid :
Boric acid on heating forms boric anhydride. The reaction depends on the temperature used.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 23

Question 2.
How is diborane prepared ? Explain its structure.
Answer:
In industries diborane is prepared by the reaction between boran tri fluoride and lithium hydride.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 24
Boron trichioride and hydrogen mixture subjected to silent electric discharge at low pressure to from diborane.
2BCl3 + 6H2 → B2H6 + 6HCl
Boron trichloride undergo reduction with LiAlH4 to form diborane.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 25
Diborane is an electron deficient compound. It has ’12’ valency electrons for bonding purpose instead of ’14’ electrons.

In diborane each boron atom undergoes sp3 hybridization out of the four hybrid orbitals one is vacant.

Each boron forms two, σ – bonds (2 centred – 2 electron bonds) bonds with two hydrogen atoms by overlapping with their ‘1s’ orbital.

The remaining hybrid orbitals of boran used for the formation of B-H-B bridge bonds.

In the formation of B-H-B bridge, half filled sp3 hybrid ofbital of one boron atom and vacant sp3 hybrid orbital of second boron atom overlap with 1s orbital of H-atom.

These three centred two electron bonds are also called as banana bonds. These bonds are present above and below the plane of BH2 units.

Diborane contains two coplanar BH2 groups. The four hydrogen atoms are called terminal hydrogen atoms and the remaining two hydrogens are called bridge hydrogen atoms.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 11
Bonding in diborane, Each B atom uses sp3 hybrids for bonding. Out of the four sp3 hybrids on each B atom, one is without an electron shown in broken lines. The terminal B-H bonds are normal 2-centre-2- electron bonds but the two bridge bonds are 3-centre-2- electron bonds. The 3-centre-2-electron bridge bonds are also referred to as banana bonds.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 3.
Write any two methods of preparation of diborane. How does it react with
a) Carbon monoxide and
b) Ammonia ?
Answer:
Preparation of diborane:
In industries diborane is prepared by the reaction between boroh tri fluoride and lithium hydride.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 24
Boron trichloride and hydrogen mixture subjected to silent electric discharge at low pressure to from diborane.
2BCl3 + 6H2 → B2H6 + 6HCl
Boron trichloride undergo reduction with LiAlH4 to form diborane.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 25

a) Reaction with carbon monoxide :
Diborane reacts with CO at 100° C and 20 atm. pressure to form borane carbonyl.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 26

b) Reaction with ammonia
Diborane reacts with ammonia at 120° C first forms B2H6.2NH3 (or) [BH2(NH3)2]+[BH4] and on further heating forms borazole (or) borazine. Which is also called as “In organic benzene”. It has iso structural with benzene. Flence it is named as “Inorganic Benzene”.
AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13 27

Solved Problems

Question 1.
Standard electrode potential values, EΘ for Al3+ / Al is – 1.66 V and that of Tl3+ / Tl is + 1.26 V. Predict about the formation of M3+ ion in solution and compare the electropositive character of the two metals.
Solution:
Standard electrode potential values for two half cell reactions suggest that aluminium has high tendency to make Al3+ (aq) ions, whereas Tl3+ is not only unstable in solution but is a powerful oxidising agent also. Thus Tl+ is more stable in solution than Tl3+. Aluminium being able to form +3 ions easily, is more electropositive than thallium.

Question 2.
White fumes appear around the bottle of anhydrous aluminium chloride. Give reason.
Solution:
Anhydrous aluminium chloride is partially hydrolysed with atmospheric moisture to liberate HCl gas. Moist HCl appears white in colour.

AP Inter 1st Year Chemistry Study Material Chapter 10 The p-Block Elements - Group 13

Question 3.
Boron is unable to form [latex]\mathrm{BF}_6^{3-}[/latex] ion. Explain.
Solution:
Due to non – availability of d orbitals, boron is unable to expand its octet. Therefore, the maximum covalence of boron cannot exceed 4.

Question 4.
Why is boric acid considered as a weak acid ?
Solution:
Because it is not able to release H+ ions on its own. It receives OH+ ions from water molecule to complete its octet and in turn releases H+ ions.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Andhra Pradesh BIEAP AP Inter 1st Year Chemistry Study Material 9th Lesson The s-Block Elements Textbook Questions and Answers.

AP Inter 1st Year Chemistry Study Material 9th Lesson The s-Block Elements

Very Short Answer Questions

Question 1.
Give reasons for the diagonal relationship observed in the periodic table.
Answer:

  • Diagonal relationship is due to similar sizes of atoms (or) ions
  • Diagonal relationship is due to similar electro negativities of the respective elements. Diagonally similar elements possess same polarising power.
  • Polarizing Power = [latex]\frac{\text { ionic charge }}{\text { (ionic radius) }^2}[/latex]

Question 2.
Write completly the electronic configurations of K and Rb.
Answer:
The electronic configuration of ‘K’ is 1s2 2s2 2p6 3s2 3p6 4s2
The electronic configuration of ‘Rb’ is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s1.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 3.
Lithium salts are mostly hydrated. Why ?
Answer:
Hydration enthalpy of Li+ ion is very high. It has very high degree of hydration. So ‘Li’ salts are mostly hydrated.
Eg : LiCl . 2H2O.

Question 4.
Which of the alkali metals shows abnormal density ? What is the order of the varia-tion of density among the IA group elements.
Answer:

  • ‘K’ has abnormal density among alkalimetals due to high inter atomic distances in crystal lattice.
  • The order of the variation of density among the IA group elements as follows.
    Li < Na > K < Rb < Cs.

Question 5.
Lithium reacts with water less vigorously than sodium. Give your reasons.
Answer:
Lithium reacts with water less vigorously than sodium.

Reasons :

  • Lithium has small size.
  • Lithium has very high hydration energy.

Question 6.
Lithium Iodide is the most covalent among the alkali metal halides. Give the reasons.
Answer:
Lithium iodide is the most covalent among the alkalimetal halides.

Reasons :

  • The polarising capability of lithium ion is high.
  • Li+ ion has very small size.
  • Li+ ion has high tendency to distort electron cloud around the iodide ion.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 7.
In what respects lithium hydrogen carbonate differs from other alkali metal hydrogen carbonates.
Answer:
Lithium hydrogen carbonate cannot exist in solid form but remaining alkali metal hydrogen carbonates exist as solids.

Question 8.
Write the complete electronic configurations of any two alkaline earth metals.
Answer:

  1. The electronic configuration of ‘Mg’ is 1s2 2s2 2p6 3s2
  2. The electronic configuration of ‘Ca’ is 1s2 2s2 2p6 3s2 3p6 4s2.

Question 9.
Tell about the variation of m.pts., and b.pts among the alkaline earth metals.
Answer:

  1. The m.pts and b.pts of alkaline earth metals are higher than corresponding alkali metals due to smaller sizes.
  2. Due to low I.P. values the variation of m.pts and b.pts among alkaline earth metals is not sys-tematic.

Question 10.
What are the characterstic colours imparted by the HA elements ?
Answer:
Elements – Imparted colours towards flame
Calcium – Brick red
Strontium – Crimson red
Barium – Apple green
Beryllium – No colour
Magnesium – No colour

Question 11.
What happens when magnesium metal is burnt in air ?
Answer:
Magnesium metal burns with dazzling brilliance in air to give MgO and Mg3N2.
2 Mg + O2 → 2 MgO
3 Mg + N2 → Mg3N2.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 12.
Lithium carbonate is not so stable to heat as the other alkali metal carbonates. Explain.
Answer:
Lithium carbonate is not so stable to heat because Lithium has very small size and it polarises the large CO3-2 ion which leads to the formation of more stable Li2O and CO2.
As the electro positive character increases down the group, the stability of carbonats increase.

Question 13.
Write a balanced equation for the formation of ammoniated IIA metal ions from the metals in liquid ammonia.
Answer:
Alkaline earth metals dissolve in liquid ammonia to give deep blue black solutions forming ammoniated ions.
M + (x + y) NH3 → [M(NH3)x]2+ + 2 [e(NH3)y]
From the above solutions, ammoniates [M(NH3)x]2+ can be recovered.

Question 14.
The fluorides of alkaline earth metals are relatively less soluble than their respective chlorides in water. Why ?
Answer:
Because of their high lattice energies fluorides of alkaline earth metals are relatively less soluble than their respective chlorides in water.

Question 15.
What happens when hydrated Mg (NO3)2 is heated ? Give the balanced equation.
Answer:
When hydrated Mg(NO3)2 is heated, it first loses the six water molecules and on further heating
decomposes to give the oxide.
2 Mg (NO3)2 → 2 MgO + 4NO2 + O2.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 16.
Why does the solubility of alkaline earth metal hydroxides in water increases down the group ?
Answer:
Among alkaline earth metal hydroxides, the anion being common the cationic radius will influence the lattice enthalpy. Since lattice enthalpy decreases much more than the hydration enthalpy with increasing ionic size, the solubility increases as we go down the group.

Question 17.
Why does the solubility of alkaline earth metal Carbonates and sulphates in water decrease down the group ?
Answer:
The size of anions being much larger compared to cations, the lattice enthalpy will remain almost constant within a particular group. Since the hydration enthalpies decrease down the group, solubility will decrease as found for alkaline earth metal carbonates and sulphates.

Question 18.
Write the average composition of Portland cement.
Answer:
Composition of port land cement is
Cao – 50 – 60%
Sio2 – 20 – 25%
Al2O3 – 5 – 10%
Mgo – 2 – 3%
Fe2O3 – 1 – 2% and
SO2 – 1 – 2%

Question 19.
Why is gypsum added to cement ?
Answer:
Gypsum is added to cement to slow down the process of setting of the cement and to get sufficiently hardened cement.

Question 20.
Why are alkali metals not found in the free state in nature ? [Mar. 13]
Answer:
Alkali metals are not found in the free state in nature because they readily lose their valency electron to form M+ ion (a nonvalent ion).

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 21.
Potassium carbonate cannot be prepared by Solvay process. Why ?
Answer:
Potassium carbonate cannot be prepared by solvay process because potassium bi carbonate is more soluble and to be precipitated by the addition of ammonium bi carbonate to a saturated solution of potassium chloride.

Question 22.
Describe the important uses of caustic soda.
Answer:
Uses:

  • It is used in petrol refining
  • It is used in the purification of bauxite.
  • It is used in manufacturing of soap, paper.
  • It is used in manufacturing of antificial silk.
  • It is used in manufacturing of so many chemically.
  • It is used in textile industries for mercerising cotton fabrics.
  • It is used in preparation of pure fats and oils.
  • It is used in as laboratory reagent.

Question 23.
Describe the important uses of sodium carbonate.
Answer:
Uses:

  • Na2CO3 is used in the manufacturing of glass.
  • Na2CO3 is used in the manufactuing of borax, caustic soda.
  • Na2CO3 is used in paper, paints and textile industries.
  • Na2CO3 is used in softening of water.
  • Na2CO3 is used in laundries.
  • Na2CO3 is used an important laboratory reagent both in qualitative and quantitative analysis.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 24.
Describe the important uses of quick line.
Answer:
Uses:

  • Quick lime is used in the purification of sugar.
  • Quick lime is used in the manufacture of dyestuffs.
  • Quick lime is used in the manufacture of Na2CO3 from NaOH.
  • It is an important material for manufacturing of cement and it is the cheapest form of alkali.

Question 25.
Draw the structures of

  1. BeCl2 (vapour) and
  2. BeCl2 (Solid).

Answer:

  1. In vapour phase BeCl2 forms a bridged dimer which disociates into monomer at high temperatures (around 1200 k)
    Cl – Be – Cl
  2. In solid state BeCl2 has a chain structure.
    AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 1

Question 26.
Describe the importance of Plaster of Paris.
Answer:

  1. Plaster of paris has an important property of setting with water.
  2. It forms a hard solid in 5 to 15 min. When it is mixed with suitable quantity of water.
  3. It is majorly used in building industry and as well as plasters.
  4. It is used in the bone fractures (or) sprain conditions.
  5. It is used in dentistry.
  6. It is used in manufacturing status and busts.

Question 27.
Which of the alkaline earth metal carbonates is thermally the most stable ? Why ?
Answer:
Among Alkaline earth metal carbonates BaCO3 is thermally most stable.
Reason :
As the cationic size increases thermal stability also increases. So BaCO3 is most stable thermally.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 28.
Write balanced equations for the reactions between

  1. Na2O2 and water
  2. K2O and water.

Answer:

  1. Na2O2 + 2H2O → 2 NaOH + H2O2
  2. K2O + H2O → 2 KOH.

Short Answer Questions

Question 1.
Alkali metals and their salts impart characteristic colours to an oxidizing flame. Explain the resonizing flame. Explain the reason.
Answer:
Alkali metals and their salts impart characterstic colours to an oxidizing flame.

Reasons :
The heat from the flame excites the outer most orbital electron to a higher energy level. When the excited electron emitts the radiation and comes back to the ground state. This falls in the visible region.

Question 2.
What makes caesium and potassium useful as electrodes in photoelectric cells ?
Answer:

  • Alkali metals can be detected by the respective flame tests and can be determined by the flame photo metry (or) atomic absorption spectroscopy.
  • Alkali metals when irradiated with light, the light energy absorbed may be sufficient to make an atom lose electron.
  • This makes caesium and potassium useful as electrodes in photo electric cells.

Question 3.
Write a short note on the reactivity of alkali metals towards air.
Answer:

  • The alkali metals forms their oxides in presence of dry air and tarnished.
  • These oxides reacts with moisture to form hydroxides.
  • They burn vigorously in oxygen and forms oxides.
    • Lithium forms Lithium monoxide.
    • Sodium forms monoxide with limited supply of oxygen and peroxide with excess of oxygen.
    • Other metals of this group forms super oxides. The super oxide ion (O2) is stable only in presence of large cations.

Reactions :
4Li + O2 → 2 Li2O
4Na + O2 (Limited) → 2 Na2O2
2Na + O2 (Excess) → Na2O2
K + O2 (Excess) → KO2
Lithium shows a different character. It reacts directly with nitrogen of air and forms Li3N (Lithium nitride).

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 4.
Give any two uses for each of the following metals.

  1. Lithium
  2. Sodium.

Answer:

  1. Uses of Lithium : –
  • ‘Li’ metal is used to make alloys.
    Eg: 1) Lithium with lead forms an alloy which is used for making white metal bearings for motor engines.
    2) Lithium with aluminium forms alloys which are used to make air craft parts.
  • ‘Li’ metal is used in thermo nuclear reactions.
  • ‘Li’ metal is used to make electro chemical cells.

2) Uses of sodium : –

  • Sodium metal is used to make Na and Pb alloy needed to make TEL. this TEL (tetra ethyl lead) is used as antiknock additives to petrol.
  • Liquid ‘Na’ metal is used as a coolant in fast breeder nuclear reactors.
  • ‘Na’ metal is used in the manufacturing of rabber.

Question 5.
Give an account of properties of washing soda.
Answer:
Properties of washing soda : –

  • Na2CO3 is a white crystalline solid.
  • Na2CO3 exists as a decahydrate Na2CO3. 10H2O which is called washing soda.
  • Na2CO3 is readily soluble in water.
  • Na2CO3 (deca hydrate) when heated it loses the water molecules and forms monohydrate. This monohydrate on heating above 373 K it forms anhydrous form which is called soda ash, a white powder.
    Reactions : –
    AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 2
  • Na2CO3 (aq) solution is alkaline (basic) in nature Because it under goes anionic hydrolysis. (PH > 7).
    CO3-2 + H2O → HCO3 + OH

Question 6.
Mention some uses of sodium carbonate.
Answer:
Uses :

  • Na2CO3 is used in the manufacturing of glass.
  • Na2CO3 is used in the manufacturing of borax, caustic soda.
  • Na2CO3 is used in paper, paints and textile industries. .
  • Na2CO3 is used in softening of water.
  • Na2CO3 is used in laundries.
  • Na2CO3 is used an important laboratory reagent both in qualitative and quantitative analysis.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 7.
How do you obtain pure sodium chloride from a crude sample ?
Answer:

  1. Crude sodium chloride is obtained by the crystallisation of brine solution.
  2. Crude sodium chloride contains sodium sulphate, calcium sulphate, calcium chloride and magnesium chloride.
  3. CaCl2 and MgCl2 are the impurities in the Crude NaCl because these absorb moisture easily from the atmosphere.
  4. Pure sodium chloride’s obtained by dissolving Crude NaCl in minimum amount of water and filtered to remove insoluble impurities.
  5. This solution is saturated with HCl gas. Then crystals of pure NaCl are separated out.
  6. Ca and Mg chloride are more soluble in solution than NaCl and these remained in the solution.

Question 8.
What do you know about Castner – Kellner process ? Write the principle involved in it.
Answer:

  • Castner – Kellner process is a commercial method used for the preparation of sodium hydroxide.
  • In this process sodium hydroxide is prepared by the electrolysis of sodium chloride in Castner – Kellner cell.
  • Brine solution is electrolysed using a mercury cathode and a carbon anode.
  • Sodium metal is formed at cathode and it combine with mercury to form sodium amalgam. Chlorine gas is evolved at anode.
  • The amalgam is treated with water to form, sodium hydroxide.
    Cell Reactions :
    2NaCl → 2Na+ + 2Cl
    2Na+ + 2e AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 3 2Na – amalgam
    2Cl → Cl2 + 2e
    2Na – amalgam + 2H2O → 2NaOH + 2Hg + H2
  • This process is also called as mercury cathode process.

Question 9.
Write a few applications of caustic soda.
Answer:
Uses :

  • It is used in petrol refining
  • It is used in the purification of bauxite.
  • It is used in manufacturing of soap, paper.
  • It is used in manufacturing of antificial silk.
  • It is used in manufacturing of so many chemically.
  • It is used in textile industries for mercerising cotton fabrics.
  • It is used in preparation of pure fats and oils.
  • It is used in as laboratory reagent.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 10.
Give an account of the biological importance of Na+ and K+ ions.
Answer:

  • Na+ ions participate in the transmission of nerve signals.
  • Na+ ions regulates the flow of water accross cell membranes.
  • Na+ ions responsible for transport of sugars and amino acids into cells.
  • K+ ions are useful in activating enzymes.
  • K+ ions participate in the oxidation of glucose to produce ATP.
  • K+ along with Na+ responsible for the transmission of nerve signals.

Question 11.
Mention the important uses of Mg metal.
Answer:

  1. Magnesium forms so many useful alloys with Al, Zn, Mn and Sn.
  2. Mg – Al alloys are useful in air – craft construction.
  3. Mg powder and ribbon is used in flash powders bulbs.
  4. Mg is used in incendiary bombs and signals.
  5. Milk of magnesice is used as antacid in medicine.
  6. MgCO3 is the main ingradient in tooth pastes.

Question 12.
Show that Be(OH)2 is amphoteric in nature.
Answer:

  1. Be(OH)2 is amphoteric in nature. This can be evidented by the following reactions.
  2. Be(OH)2 reacts with both acids and alkalis.
    Be(OH)2 + 2OH [Be(OH)4]2- (Beryllation)
    Be(OH)2 + 2HCl + 2H2O → [Be(OH)4]Cl2
  3. Hence Be(OH)2 is amphoteric in nature.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 13.
Write a note on anomalous behaviour of beryllium.
Answer:
Anomalous characters of Be:
As was already discussed in the earlier sections, the first element shows some differences from the properties of the other elements in the group. Be differs from the other alkaline earth metals because of its small size and high electronegativity. Be differs from the other elements in the following aspects.

  1. Be compounds are predominantly covalent due to its high polarizing power and its salts are readily Hydrolyzed.
  2. Be is not easily affected by dry air and does not decompose water at ordinary temperature.
  3. Be is an amphoteric metal. It dissolves in alkali solutions forming beryllates.
  4. Be SO4 is soluble in water whereas the sulphates of Ca, Sr and Ba are not soluble.
  5. Be and its salts do not respond to flame test while Ca, Sr and Ba give characteristic flame colours.
  6. Be forms many complexes while the heavier elements do not show a great tendency to form complexes.
  7. Be has a maximum covalency of 4 while other can have a maximum covalency of 6.

Question 14.
Be shows diagonal relationship with Al. Discuss.
Answer:

  1. ‘Be’ shows diagonal relation ship with ‘Al’.
  2. The ionic radius of Be+2 is nearly same as that of Al+3 so ‘Be’ resembles ‘Al’ in several ways.
  3. Al, Be both not readily reacts with acids. This is due to the presence of an oxide film on the surface of metal.
  4. Al(OH)3, Be(OH)2 both similarly dissolves in excess of alkali and forms Beryllate ion [Be(OH)3]2- and Aluminate [Al(OH)4] ions respectively.
  5. The chlorides of Be, Al have bridged chloride structures in vapour phase.
  6. Both the chlorides of Be, Al used as strong Lewis acids.
  7. Both the chlorides of Be, Al used in Friedal craft catalysts.
  8. Be, Al ions have strong tendency to form complexes.

Question 15.
What is Plaster of Paris ? Write a short note on it. [T.S. Mar. 16]
Answer:
Plaster of paris is the hemi hydrate of CaSO4 with formula CaSO4. [latex]\frac{1}{2}[/latex]H2O.
Preparation: –
Plaster of paris is obtained by heating gypsum at 393 K.
AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 4

  1. If temperature is used greater than 393 K then an hydrous CaSO4 is formed which is called ‘dead burnt plaster’.
  2. Plaster of paris has an important property of setting with water.
  3. It forms a hard solid in 5 to 15 min. when it is mixed with suitable quantity of water.
  4. It is majorly used in building industry and as well as plasters.
  5. It is used in the bone fractures (or) sprain conditions.
  6. It is used in dentistry.
  7. It is used in manufacturing status and busts.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 16.
In what ways lithium shows similarities to magnesium in its chemical behaviour ?
Answer:
Diagonal relationship of Li : In the periodic table an element of a group in the 2nd period shows similar properties with the second element of the next group in the third period. This relation is known as diagonal relationship. For examples. Lithium and Magnesium show diagonal relationship. The elements show that diagonal relationship have similar polarizing powers, electronegativities, nature of the compounds. The diagonal similarity may be due to the effects of size and charge. For example, charge per unit area.
Lithium shows similarity to Magnesium in the following respects.
a) Lithium is slow to react with water. Magnesium decomposes water only in the hot condition.
2Li + 2H2O → + H2;
Mg + 2H2O → Mg(OH)2 + H2

b) Lithium combines directly with N2 forming nitride.
6Li + N2 → 2Li3N

c) Both Lithium and Magnesium give only monoxides Li2O, MgO.

d) Lithium chloride is deliquescent like MgCl2, LiCl undergoes hydrolysis to a smaller extent in hot water in a similar way to MgCl2.

e) Due to their covalent nature, the halides Lithium and Magnesium are soluble in organic solvents.

f) Both Li+ and Mg+2 are highly hydrated.

g) The Carbonates, Phosphates and Fluorides of both Li and Mg are sparingly soluble in water.

h) Lithium alkyls (Li+ R) are chemically similar to Grignard reagents in organic synthesis.

Question 17.
When an alkali metal dissolves in liquid ammonia the solution can acquire different colours. Explain the reasons for this type of colour change.
Answer:

  • The alkali metals dissolve in liquid NH3 and gives deep blue solutions. These are conducting in nature.
  • The blue colour of the solution is due to the ammoniated electrons which absorbs energy in the visible region of light and thus imparts blue colour to the solution.
  • These solutions are paramagnetic and on standing liberate hydrogen resulting in the formation of amide.
    M + (x + y) NH3 → [M(NH3)x]+ + [e(NH3)y]
    M(am)+ + e + NH3 → MNH2(am) + 1/2 H2(g)
  • In concentrated solution the blue colour Changes to bronze.colour on warming and becomes did magnetic.

Question 18.
What happens when

  1. Sodium metal is dropped in water ?
  2. Sodium metal is heated in a free supply of air ?
  3. Sodium peroxide dissolves in water ?

Answer:

  1. Sodium metal when dropped in water it reacts with water vigourously and liberates H2 gas.
    2Na + 2H2O → 2NaOH + H2
  2. Sodium metal is heated in free supply of air to form sodium peroxide.
    2Na + O2 → Na2O2 (sodium peroxide)
  3. Sodium peroxide dissolves in water and forms NaOH and hydrogen peroxide.
    Na2O2 + 2H2O → 2NaOH + H2O2

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 19.
States as to why
i) An aqueous solution of Na2CO3 is alkaline ;
ii) Alkali metals are prepared by the electrolysis of their fused chlorides ?
Answer:
i) An aqueous solution of Na2CO3 is alkaline. This is due to anionic (CO3-2) hydrolysis.
CO3-2 + H2O → HCO3-2 + OH
∴ PH > 7. So the solution is alkaline in nature.

ii) Chemically alkali metals are highly reactive and they are placed at top in the electro chemical series.
∴ Common methods of extraction of the metals are not applicable for the alkali metals. So electrolytic reduction of their used chlorides is the possible method for extracting alkali metals.
Eg: ‘Na’ metal obtained from fused ‘NaCl’.

Question 20.
How would you explain the following observations ?

  1. BeO is almost insoluble but BeSO4 is soluble in water ?
  2. BaO is soluble but BaSO4 is insoluble in water ?

Answer:

  1. BeO has amphoteric nature and the solubility in water is low because of its covalent nature.
    BeSO4 is soluble in water. This is due to greater hydration energy of Be+2 ion.
  2. BaO is soluble in water because of its ionic nature.
    BaSO4 is insoluble in water because of low hydration energy of Ba+2 ion.

Long Answer Questions

Question 1.
Justify the inclusion of alkali metals in the same group of the periodic table with reference to the following :
i) Electronic configuration
ii) Reducing nature
iii) Oxides and hydroxides.
Answer:
i) Electronic configuration : All the alkali metals have one valence electron, ns1.
AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 5
ii) Reducing nature :

  • Alkali metals are strong reducing agents.
  • ‘Li’ is most powerful reducing agent and ‘Na’ is poor reducing agent. .
    The standard electrode potential (E°) is the measure of reducing power.
  • ‘Li’ has highest hydration enthalpy. It has high negative S.E.P. (E°) hence it has high reducing power.

iii) Oxides and hydroxides : –

  • The alkali metals forms their oxides in presence of dry air and tarnished.
  • These oxides reacts with , .noisture to form hydroxides.
  • They burn vigorously in oxygen and forms oxides.
    1. Lithium forms lithium monoxide.
    2. Sodium forms monoxide with limited supply of oxygen and peroxide with’ excess of oxygen
    3. Other metals of this group forms super oxides. The super oxide ion (O2) is stable only in presence of large cations.

Reactions:
4Li + O2 → 2 Li2O
4Na + O2 (Limited) → 2 Na2O
2 Na + O2 (Excess) → Na2O2
K + O2 (Excess) → KO2
Lithium shows a different character. It reacts directly with nitrogen of air and forms Li3N (Lithium nitride).

  • Alkali metal oxides easily hydrolysed by water to form the hydroxides.
    Monoxide : M2O + H2O → 2MOH
    Peroxide : M2O2 + 2H2O → 2MOH + H2O2
    Superoxide : 2MO2 + 2H2O → 2MOH + H2O2 + O2
  • Oxides, peroxides are colourless whereas superoxides are coloured because of their para mag-netic property.
  • Hydroxides are white crystalline solids.
  • Hydroxides are strong bases and dissolved freely in water and evolve much heat.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 2.
Write an essay on the differences between lithium and other alkali metals.
Answer:
Anomalous properties of Lithium : In the periodic table some representative elements of different series show similarities, known as diagonal relationship. Li, of the alkali metals, show such a similarity with Mg of II group. That means it differs from alkali metals. Some of the important abnormal characters of lithium are given below.
a) Lithium is hard metal while other alkali metals are soft and can be cut with the knife. Its melting point and boiling point are high.

b) Lithium directly unites with N2 while no other alkali metal combines directly.
6Li + N2 → 2Li3N

c) Lithium element forms a carbide on direct combination. Group IA elements do not form directly. But all these elements are known to give carbides.

d) The solubilities of Lithium Hydroxide (LiOH), Lithium Carbonate (Li2CO3), Lithium Phosphate (Li3PO4) and Lithium Fluoride (LiF), are very less compared to the high solubilities of the other alkali metal compounds.

e) Lithium Hydroxide is a weaker alkali than the alkali metal Hydroxides. Basic nature of other alkali metal Hydroxides is more than Li(OH). Because of this Lithium Hydroxide Carbonate, nitrates are thermally unstable.

Question 3.
Discuss the preparation and properties of sodium carbonate.
Answer:
Preparation :

  • Sodium carbonate is prepared by solvay process.
  • In this process sodium chloride reacts with ammonium bicarbonate and gets precipitated the low soluble sodium bicarbonate.
  • The lather is prepared by passing CO2 into a concentrated solution of NaCl saturated with ammonia. Here (NH4)2CO3 followed by NH4 HCO3 are formed.
    Chemical equations involved :
    2NH3 + H2O + CO2 → (NH4)2 CO3
    (NH4)2 CO3 + H2O + CO2 → 2NH4HCO3
    NH4HCO3 + NaCl → NH4Cl + NaHCO3
  • The separated NaHC03 Crystals heated to get Na2CO3.
    2NaHCO3 → Na2CO3 + H2O + CO2
  • In this process ammonia is regenerated by Ca(OH)2.
    2NH4Cl + Ca(OH)2 → 2NH3 + CaCl2 + H2O

Properties of washing soda :

  • Na2CO3 is a white crystalline solid.
  • Na2CO3 exists as a decahydrate Na2CO3. 10H2O which is called washing soda.
  • Na2CO3 is readily soluble in water.
  • Na2CO3 (decahydrate) when heated it loses the water molecules and forms monohydrate. This monohydrate on heating above 373 K it forms anhydrous form which is called soda ash, a white powder.
    Reactions : –
    AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 2
  • Na2CO3 (aq) solution is alkaline (basic) in nature Because it under goes anionic hydrolysis. (PH > 7).
    CO3-2 + H2O → HCO3 + OH

Uses :

  • Na2CO3 is used in the manufactuing of glass.
  • Na2CO3 is used in the manufactuing of borax, caustic soda.
  • Na2CO3 is used in paper, paints and textile industries.
  • Na2CO3 is used in softening of water.
  • Na2CO3 is used in laundries.
  • Na2CO3 is used an important laboratory reagent both in qualitative and quantitative analysis.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 4.
Discuss the similarities between alkaline earth metals and gradation in the following aspects.
i) Electronic configuration
ii) Hydration enthalpies
iii) Nature of oxides and hydroxides.
Answer:
i) Electronic configuration :
The general electronic configuration of alkaline earth metals represented by [noble gas] ns2.
AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 6

ii) Hydration Enthalpies :

  • The hydration enthalpies of alkaline earth metal ions decrease with increase in ionic size down the group.
    Be+2 > Mg+2 > Ca+2 > Sr+2 > Ba+2
  • Hydration enthalpies of these elements ions are larger than those of alkali metal ions.
    Eg : MgCl2 MgCl2 . 6H2O
    CaCl2 CaCl2 . 6H2O

iii) Nature of oxides and hydroxides : –

  • Alkaline earth metals forms oxides of type MO.
  • These are formed by burning in oxygen.
  • BeO is amphoteric and covalent in nature whereas other oxides are ionic and basic in nature. Other oxides i.e., except BeO forms hydroxides with water.
    Eg : MgO + H2O → Mg(OH)2
  • The solubility, thermal stability and the basic character of these hydroxides increase with increase of atomic no. from Mg(OH)2 to Ba(OH)2. H
  • These hydroxides are less basic, less stable than alkali metal hydroxide.
  • Be(OH)2 is amphoteric in nature. This can be evidented by the following reactions.
  • Be(OH)2 reacts with both acids and alkalis.
    Be(OH)2 + 2OH [Be(OH)4]2- (Beryllation)
    Be(OH)2 + 2HCl + 2H2O → [Be(OH)4]Cl2
  • Hence Be(OH)2 is amphoteric in nature.

Question 5.
Discuss on;
i) Carbonates
ii) Sulphates and
iii) Nitrates of alkaline earth metals.
Answer:
i) Carbonates :

  • Alkaline earth metals forms MCO3 type carbonates.
  • These Carbonates are insoluble in water.
  • The solubility of these carbonates in water decreases as the atomic no.of the element increases.
  •  The thermal stability increases with increasing cationic size.
  • These carbonates decompose on heating to give CO2
    CaCO3 AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 7 Cao + CO2
  • BeCO3 is unstable and kept only in at atmosphere of CO2.

ii) Sulphates :

  • Alkaline earth metals forms MSO4 type sulphates.
  • These are white solids and are stable to heat.
  • BeSO4 and MgSO4 are readily soluble in water due to high hydration enthalies of Be2+, Mg2+.
  • The solubility decrease from CaSO4 to BaSO4.

iii) Nitrates :

  • Alkaline earth metals forms M(NO3)2 type Nitrates.
  • These are formed by the reaction of carbonates in dil.HNO3.
  • Mg(N03)2 crystallises with six water molecules and Ba(NO3)2 is an hydrous.
  • All of these decompose on heating to give the respective oxides.
    2M(NO3)2 → 2MO + 4NO2 + O2
    M = Be, Mg, Ca, Sr, Ba.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 6.
What are the common physical and chemical features of alkali metals ?
Answer:
Physical features :

  • Alkali metals are silvery white, soft and light metals.
  • These elements have low density which increases down the group from Li to Cs. (one exception density of K < density of Na).
  • The m.pts, b.pts of alkali metals are low.
  • Alkali metals and their salts impart characterstic colours to an oxidizing flame.

Reasons :
The heat from the flame excities the outer most orbital electron to a higher energy level. When the excited electron emitts the radiation and comes back to the ground state. This falls in the visible region.
AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 8

  • Alkali metals can be detected by the respective flame tests and can be determined by the flame photo metry (or) atomic absorption spectroscopy.
  • Alkali metals when irradiated with light, the light energy absorbed may be sufficient to make an atom lose electron. .
  • This makes calsium and potassium useful as electrodes in photo electric cells.

Chemical features : –
i) Reactivity towards Air : —

  • The alkali metals forms their oxides in presence of dry air and tarnished.
  • These oxides reacts with moisture to form hydroxides.
  • They burn vigorously in oxygen and forms oxides.
    1. Lithium forms Lithium monoxide.
    2. Sodium forms monoxide with limited supply of oxygen and peroxide with excess of oxygen.
    3. Other metals of this group forms super oxides. The super oxide ion (O2) is stable only in presence of large cations.
  • Reasons :
    4Li + O2 → 2 Li2O
    4Na + O2 (Limited) → 2 Na2O
    2Na + O2 (Excess) → Na2O2
    K + O2 (Excess) → KO2
  • Lithium shows a different character. It reacts directly with nitrogen of air and forms Li3N (Lithium nitride).

ii) Reactivity with H2: Alkali metals react with H2 directly at 300 – 600° C and form hydrides. The reaction can be written as follows :
AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 9
Where M = Li, Na, K, Rb or Cs. These hydrides are ionic in nature. Their ionic nature increases with the metalic nature of alkali metals.

iii) Reactivity with halogens : All the alkali’ metals react with halogens to give the binary compounds. The chemical reactivity in the alkali metals increases with increase in atomic number.
2M + X2 → 2MX (where M is any alkali metal)
All the metal halides are ionic compounds.

iv) Reactivity with water : The alkali metals decompose water vigorously and liberate hydrogen gas! The chemical reactivity of these metals increases as the atomic number increases. The metal hydroxides are formed.
2M + 2H2O → 2MOH + H2
Where M = any one of the alkali metals.

ii) Reducing nature :

  • Alkali metals are strong reducing agents.
  • ‘Li1 is most powerful reducing agent and ‘Na’ is poor reducing agent.
  • The standard electrode potential (E°) is the measure of reducing power.
  • ‘Li’ has highest hydration enthalpy. It has high negative S.E.P. (E°) hence it has high reducing power.
  • The alkali metals dissolves in liquid NH3 and gives deep blue solutions. These are conducting in nature.
  • The blue colour of the solution is due to the ammoniated electrons which absorbs energy in the visible region of light and thus imparts blue colour to the solution.
  • These solutions are paramagnetic and on standing Liberate hydrogen resulting in the formation of amide.
    M + (x + y) NH3 → [M(NH3)x]+ + [e(NH3)y]
    M(am)+ + e + NH3 → MNH2(am) + 1/2 H2(g)
  • In concentrated solution the blue colour Changes to bronze.colour on warming and becomes did magnetic.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 7.
Discuss the general characterstics and gradation in properties of alkaline earth metals.
Answer:
The general characterstics and gradation in properties of alkaline earth metals follows.
i) Oxides and hydroxides :

  • Alkaline earth metals forms oxides of type Mo.
  • These are formed by burning in oxygen.
  • BeO is amphoteric and covalent in nature where as other oxides are ionic and basic in nature.
  • Other oxides i.e., except Beo forms hydroxides with water.
    Eg : MgO + H2O → Mg(OH)2
  • The solubility, thermal stability and the basic character of these hydroxides increase with increase of atomic no. from Mg(OH)2 to Ba(OH)2.
  • These hydroxides are less basic, less stable than alkali metal hydroxide.
  • Be(OH)2 is amphoteric in nature. This can be evidented by the following reactions.
  • Be(OH)2 reacts with both acids and alkalis.
    Be(OH)2 + 2OH [Be(OH)4]2- (Beryllation)
    Be(OH)2 + 2HCl + 2H2O → [Be(OH)4]Cl2
  • Hence Be(OH)2 is amphoteric in nature.

ii) Halides:-

  • These forms MX2 type halides.
  • Except Be – halides, all other halides of these metals are ionic.
  • Be – halides are covalent and soluble in organic solvents.
  • In vapour phase BeCl2 forms a bridged dimer which disociates into monomer at high temperatures (around 1200 k)
    Cl – Be – Cl
  • In solid state BeCl2 has a chain structure.
    AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 10
  • The tendency to form halide hydrates gradually decreases – down the group.
    Eg : MgCl2, 8H2O, CaCl2, 6H2O, BaCl2. 2H2O.
  • Ca, Sr and Ba halides, dehydration can be done by heating.
  • Fluorides are less soluble than the chlorides due to their high lattice energies.

i) Carbonates :

  • Alkaline earth metals forms MCO3 type carbonates.
  • These Carbonates are insoluble in water.
  • The solubility of these carbonates in water decreases as the atomic no.of the element increases.
  • The thermal stability increases with increasing cationic size.
  • These carbonates decompose on heating to give CO2
    CaCO3 AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 7 Cao + CO2
  • BeCO3 is unstable and kept only in at atmosphere of CO2.

ii) Sulphates:

  • Alkaline earth metals forms MSO4 type sulphates.
  • These are white solids and are stable to heat.
  • BeSO4 and MgSO4 are readily soluble in water due to high hydration enthalpies of Be2+, Mg2+.
  • The solubility decrease from CaSO4 to BaSO4.

iii) Nitrates:

  • Alkaline earth metals forms M(NO3)2 type Nitrates.
  • These are formed by the reaction of carbonates in dil.HNO3.
  • Mg(NO3)2 crystallises with six water molecules and Ba(NO3)2 is an hydrous.
  • All of these decompose on heating to give the respective oxides.
    2M(NO3)2 → 2MO + 4NO2 + O2
    M = Be, Mg, Ca, Sr, Ba.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 8.
Discuss the various reactions that occur in the solvay process. [A.P. Mar. 16]
Answer:
Preparation:

  • Sodium carbonate is prepared by solvay process.
  • In this process sodium chloride reacts with ammonium bicarbonate and gets precipitated the low soluble sodium bicarbonate. ‘
  • The lather is prepared by passing CO2 in to a concentrated solution of NaCl saturated with ammonia. Here (NH4)2CO3 followed by NH4 HCO3 are formed.
  • Chemical equations involved:
    2NH3 + H2O + CO2 → (NH4)2 CO3
    (NH4)2 CO3 + H2O + CO2 → 2NH4HCO3
    NH4HCO3 + NaCl → NH4Cl + NaHCO3
  • The separated NaHCO3 Crystals heated to get Na2CO3.
    2NaHCO3 → Na2CO3 + H2O + CO2
  • In this process ammonia is regenerated by Ca(OH)2.
    2NH4Cl + Ca(OH)2 → 2NH3 + CaCl2 + H2O

Question 9.
Starting with sodium chloride how would you proceed to prepare
i) Sodium metal
ii) Sodium hydroxide
iii) Sodium peroxide
iv) sodium carbonate.
Answer:
i) Fused NaCl on electrolysis gives sodium metal.
2NaCl → 2Na+ + 2Cl
2Na+ + 2e → 2Na (Cathode)
2Cl → Cl2 + 2e (anode)

ii)

  • Castner-Kellner process is a commercial method used for the preparation of sodium hydroxide.
  • In this process sodium hydroxide is prepared by the electrolysis of sodium chloride in Castner – Kellner cel..
  • Brine solution is electrolysed using a mercury cathode and a carbon anode.
  • Sodium metal is formed at cathode and it combine with mercury to form sodium amalgam. Chlorine gas is evolved at anode.
  • The amalgam is streated with water to form sodium hydroxide.
    Cell Reactions:
    2NaCl → 2Na+ + 2Cl
    2Na+ + 2e AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 11 2Na – amalgam
    2Cl → Cl2 + 2e
    2Na-amalgam + 2H2O → 2NaOH + 2Hg + H2
  • This process is also called as mercury cathode process.

iii) The obtained Na – metal reacts with excess of oxygen to form sodium peroxide.
2Na + O2 → Na2O2 (Sodium peroxide.)

iv) Preparation:

  1. Sodium carbonate is prepared by solvay process.
  2. In this process sodium chloride reacts with ammonium bicarbonate and gets precipitated the low soluble sodium bicarbonate.
  3. The lather is prepared by passing CO2 into a concentrated solution of NaCl saturated with ammonia. Here (NH4)2CO3 followed by NH4 HCO3 are formed.

Question 10.
What happens when
i) Magnesium is burnt in air ?
ii) Quick lime is heated with silica
iii) Chlorine reacts with slaked lime
iv) calcium nitrate is strongly heated.
Answer:
i) Magnesium burns with dazzling brilliance in air to give MgO and Mg3N2.
2Mg + O2 →2MgO
3Mg + N2 → Mg3N2

ii) Quick lime heated with silica to form calcium silicate
Cao + SiO2 → CaSio3

iii) Slaked lime reacts with chlorine gas to form bleaching powder.
AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 12

iv) Calcium nitrate on strong heating to form respective oxide
2Ca(NO3)2 → 2CaO + 4NO2 + O2.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 11.
Explain the significance of sodium, potassium, magnesium and calcium in biological fluids.
Answer:
Biological importance of Na, K.

  • Na+ ions participate in the transmission of nerve signals.
  • Na+ ions regulates the flow of water accross cell membranes.
  • Na+ ions responsible for transport of sugars and amino acids into cells.
  • K+ ions are useful in activating enzymes.
  • K+ ions participate in the oxidation of glucose to produce ATP.
  • K+ along with Na+ responsible for the transmission of nerve signals.

Biological importance of Mg and Ca :
Role of Mg2+ in biology :

  1. Mg2+ ions are concentrated in animal cells.
  2. Enzymes like “phosphohydrolases1 and ‘Phospho transferases’ contain Mg2+ ions. These enzymes participate in ATP reactions and release energy in the process. Mg2+ forms a complex with ATP.
  3. Mg2+ is a constituent of chlorophyll, the green component of plants.

Role of Ca+2:
About 99% of body calcium is present in bones and teeth. It also plays important roles in neuromuscular function, interneuronal transmission, cell membrance integrity and blood coaqulation.

The calcium concentration in plasma is regulated at about 100 mg/Lit. It is maintained by two hormones, calcitonin and parathyroid hormone. Ca2+ ion are necessary for muscle contraction.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 12.
Write few lines about cement ?
Answer:

  • Cement is an important building material.
  • It is also called portland cement.
  • Cement is obtained by combining a material rich in lime, CaO with other material such as clay which contains Sio2 along with the oxides of Al, Fe and Mg.
  • Composition of port land cement is
    Cao – 50 – 60%
    Sio2 – 20 – 25%
    Al2O3 – 5 – 10%
    Mgo – 2 – 3%
    Fe2O3 – 1 – 2%
    and SO2 – 1 – 2%
  • For a good quality of cement the ratio of SiO2 to Al2O3 is between 2.5 and 4 and the ratio of lime (Co) to the total of the oxides of SiO2, Al2O3 and Fe2O3 is as close as ‘2’.
  • The raw materials used for the manufacture of cement are lime stone and clay.
    Clay + lime AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements 13 cement clinker.
  • This cement clinker mixed with 2 – 3% by wt. of gypsum to form cement.
  • Important ingradients in portland cement are
    Ca2SiO4 – 26%, Ca3SiO5 – 51 % and Ca3Al2O6 – 11 %

Setting of Cement: –

  • Cement mixed with water to give a hard mass i.e setting of cement takes place.
  • This is due to the hydration of molecules of the cement.
  • The purpose of adding gypsum is to slow down the process of setting and to get sufficiently hardness.

Uses:

  • It is used in concrete and rein forced concrete.
  • It is used in plastering.
  • It is used in construction of bridges, dams and buildings.

Solved Problems

Question 1.
What is the oxidation state of K in KO2 ?
Solution:
The superoxide species is represented as O2; since the compound is neutral, the oxidation state of potassium is +1.

Question 2.
The EΘ for Cl2 / Cl is + 1.36, for I2/I is + 0.53, for Ag+/Ag is + 0.79, Na+/Na is – 2.71 and for Li+/Li is – 3.04. Arrange the following ionic species in decreasing order of reducing strength : I, Ag, Cl, Li, Na
Solution:
The order is Li > Na > I > Ag > Cl.

AP Inter 1st Year Chemistry Study Material Chapter 9 The s-Block Elements

Question 3.
Why is KO2 paramagnetic ? [T.S. Mar. 16]
Solution:
The superoxide O2 is paramagnetic because of one unpaired electron in π*2p molecular orbital.

Question 4.
Why does the solubility of alkaline earth metal hydroxides in water increases down the group ?
Solution:
Among alkaline earth metal hydroxides, the anion being common the cationic radius will influence the lattice enthalpy. Since lattice enthalpy decreases much more than the hydration enthalpy with increasing ionic size, the solubilit y increases as we go down the group.

Question 5.
Why does the solubility of alkaline earth metal carbonates and sulphates in water decrease down the group ?
Solution:
The size of anions being much larger compared to cations, the lattice enthalpy will remain almost constant within a particular group. Since the hydration enthalpies decrease down the group, solubility will decrease as found for alkaline earth metal carbonates and sulphates.