Practice the AP 10th Class Maths Bits with Answers Chapter 5 Quadratic Equations on a regular basis so that you can attempt exams with utmost confidence.

## AP State Syllabus 10th Class Maths Bits 5th Lesson Quadratic Equations with Answers

Question 1.

If sin α, cos α are roots of the equation ax2 + bx + c = 0, then ………………

A) a^{2} + b^{2} – 2ac = 0

B) (a + c)^{2} = b^{2} + c^{2}

C) a^{2} – b^{2} + 2ac = 0

D) (a + c)^{2} = b^{2} – c^{2}

Answer:

D) (a + c)^{2} = b^{2} – c^{2}

Question 2.

If px^{2} + qx + r = 0 has equal roots , then r = ……………….

A) \(\frac{\mathrm{q}^{2}}{2 \mathrm{p}}\)

B) \(\frac{\mathrm{q}}{2 \mathrm{p}}\)

C) \(\frac{-q^{2}}{4 p}\)

D) \(\frac{q^{2}}{4 p}\)

Answer:

D) \(\frac{q^{2}}{4 p}\)

Question 3.

If the equations x^{2} + bx + c = 0, x^{2} + cx + b = 0 [b ≠ c] have a common root, then …………………

A) b + c = 0

B) b + c = 1

C) b + c + 1 = 0

D) None of these

Answer:

C) b + c + 1 = 0

Question 4.

The roots of x – \(\frac{3}{x}\) = 2 are

A) 3, 2

B) 3,-2

C) 3, 1

D) 3, – 1

Answer:

D) 3, – 1

Question 5.

The roots of the quadratic equation 2x^{2} -2√2x + 1 = 0 are ……………….

A) √2, \(\frac{1}{\sqrt{2}}\)

B) \(\frac{1}{2}, \frac{1}{2}\)

C) \(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\)

D) √2, √2

Answer:

C) \(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\)

Question 6.

If α, β are the roots of x^{2} + 2x + 5 = 0 then \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\) =

A) \(\frac{6}{5}\)

B) \(\frac{4}{5}\)

C) \(-\frac{6}{5}\)

D) \(-\frac{4}{5}\)

Answer:

C) \(-\frac{6}{5}\)

Question 7.

If the roots of a(b – c)x^{2} + b(c – a) x + c(a – b) = 0 are equal, then a, b, c are in

A) AP

B) GP

C) HP

D) None

Answer:

C) HP

Question 8.

The roots of are \(\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}\)

A) -1, 2

B) 1, 2

C) 1, -2

D) -1, -2

Answer:

B) 1, 2

Question 9.

If A is the solution set of x^{2} – 5x + 6 = 0 and B is the solution set of x – \(\sqrt{3 x-6}=\) = 2, then A ∩ B =

A) Φ

B) A

C) B

D) {2}

Answer:

D) {2}

Question 10.

If the sum of the squares of the roots of x^{2} + px – 3 = 0 is 10, then p =

A) ± 2

B) ± 3

C) ± 5

D) ± 6

Answer:

A) ± 2

Question 11.

If one root of x^{2} – 8x + 13 = 0 is 4 + √3,then the other root is …………….

A) 2 + √3

B) 2 – √3

C) -4 + √3

D) 4 – √3

Answer:

D) 4 – √3

Question 12.

If α and β are the roots of a quadratic a P equation x^{2} – px + q = 0, then \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\) =

A) \(\frac{p^{2}-2 q}{q}\)

B) \(\frac{p^{2}+2 q}{q}\)

C) \(\frac{p^{2}-q}{q}\)

D) \(\frac{p^{2}+q}{q}\)

Answer:

D) \(\frac{p^{2}+q}{q}\)

Question 13.

If x = \(\sqrt{3+\sqrt{3+\sqrt{3+\ldots \ldots \infty} \infty}}\) then

A) x^{2} – x + 3 = 0

B) x^{2} + x + 3 = 0

C) x^{2} – x – 3 = 0

D) x^{2} + x – 3 = 0

Answer:

C) x^{2}– x – 3 = 0

Question 14.

If the difference of two numbers is 5 and their product is 84, then the numbers

A) 14, 6

B) 12, 7

C) 21, 4

D) 14, 9

Answer:

B) 12, 7

Question 15.

\(\sqrt{6+\sqrt{6+\sqrt{6+\ldots \ldots \infty} \infty}}\) =

A) 1

B) 2

C) 3

D) 4

Answer:

C) 3

Question 16.

If x + \(\frac{1}{x}\) = 2, then x^{2} + \(\frac{1}{x^{2}}\) =

A) 0

B) 2

C) 4

D) 8

Answer:

B) 2

Question 17.

The discriminant of the quadratic equation ax^{2} + bx + c = 0is

A) b^{2} – 4ac

B) a^{2} – 4bc

C) c^{2} – 4ab

D) None

Answer:

A) b^{2} – 4ac

Question 18.

A ball is thrown vertically upward from the ground. The distance s in t seconds is given by s = 4t^{2} + t – 3. After how many seconds does the ball come to rest?

A) 2s

B) 1s

C) \(\frac{3}{4}\)s

D) \(\frac{4}{3}\)s

Answer:

A) 2s

Question 19.

The product of the roots of √2x^{2} – 10x + 5√2 = 0 is

A) 5√2

B) 5

C) 2√5

D) 2

Answer:

A) 5√2

Question 20.

The discriminant of \(\sqrt{x^{2}-5 x-1}\) = 2 is

A) 21

B) 23

C) 27

D) 45

Answer:

C) 27

Question 21.

The equation x^{2} + x + 1 = 0 has

A) real equal roots

B) no real roots

C) real and unequal roots

D) All of the above a

Answer:

B) no real roots

Question 22.

If \(\frac{\mathbf{x}}{\mathbf{a}-\mathbf{b}}=\frac{\mathbf{a}}{\mathbf{x}-\mathbf{b}}\) then x = …………….

A) (b – a), -a

B) (b – a), \(\frac{1}{a}\)

C) (a – b), b^{2}

D) (b – a), a

Answer:

D) (b – a), a

Question 23.

The discriminant of the equation x^{2} + 4x + 2 = 0 is ……………….

A) 8

B) -8

C) 10

D) 4

Answer:

A) 8

Question 24.

The discriminant of the equation px^{2} + qx + r = 0 is …………………..

A) q^{2} + 4pr

B) q^{2} – 4pr

C) p^{2} – 4qr

D) r^{2} – 4pq

Answer:

B) q^{2} – 4pr

Question 25.

The degree of a quadratic equation is

A) 1

B) 0

C) 2

D) 3

Answer:

C) 2

Question 26.

In a quadratic equation ax^{2} + bx + c = 0, if b^{2} – 4ac > 0, then the roots are ……………….

A) real arid distinct

B) real and equal

C) imaginary

D) none

Answer:

A) real arid distinct

Question 27.

A quadratic equation, whose roots are 2 + √3 and 2 – √3 = ……………….

A) x^{2} – x – 4 = 0

B) x^{2} – 4x + 1 = 0

C) x^{2} + 4x + 3 = 0

D) x^{2} + x – 3 = 0

Answer:

B) x^{2} – 4x + 1 = 0

Question 28.

The adjacent diagram indicates …………………

A) b^{2} – 4ac >0

B) b^{2} – 4ac = 0

C) b^{2} – 4ac < 0

D) None of the given

Answer:

A) b^{2} – 4ac >0

Question 29.

If \(\sqrt{x}+\frac{58}{\sqrt{x}}\) = 31, then x =

A) 529

B) 933

C) 729

D) 841

Answer:

D) 841

Question 30.

Which one of the following figures shows the quadratic equation

ax^{2} + bx + c = 0 (a ≠ 0) having distinct roots ?

Answer:

Question 31.

If 1 is a common root of ax^{2} + ax + 2 = 0 and x^{2} + x + b = 0, then a. b = ………………

A) 2

B) -2

C) 3

D) -3

Answer:

A) 2

Question 32.

Find the zeros of the quadratic polynomial x^{2} – 5x – 14.

A) (+7,-2)

B) (-7,-2)

C) (-7, +2)

D) (7, 2)

Answer:

A) (+7,-2)

Question 33.

If one root of the quadratic equation x^{2} – 4x + 1 = 0 is 2 + √3, then the other

root is ……………………

A) 7 + √3

B) 7 – 4√3

C) 1 + √3

D) 2 – √3

Answer:

D) 2 – √3

Question 34.

If α, β are the roots of x^{2} – 10x + 9 = 0, then | α – β|=

A) 9

B) 8

C) -10

D) 10

Answer:

B) 8

Question 35.

The number of diagonals in a polygon having ‘n’ sides is ………………..

A) \(\frac{n(n+1)}{2}\)

B) \(\frac{n(n-1)}{2}\)

C) \(\frac{n(n-3)}{2}\)

D) \(\frac{n(n+3)}{2}\)

Answer:

C) \(\frac{n(n-3)}{2}\)

Question 36.

If x^{2}– px + q = 0(p,q ∈ Rand p ≠ 0,q ≠ 0) has distinct real roots, then………………

A) p^{2} < 4q

B) p^{2} > 4q

C) p^{2} = 4q

D) p^{2} + 4q = 0

Answer:

B) p^{2} > 4q

Question 37.

If 5 is a root of x^{2} – (K – 1)x + 10 = 0, then the value of K is

A) -8

B) 7

C) 8

D) 12

Answer:

C) 8

Question 38.

Sum of the roots of x^{2} – 16 = 0 is

A) \(\frac{1}{16}\)

B) 1

C) 0

D) 16

Answer:

C) 0

Question 39.

The sum of squares of the roots of x^{2} + 8x + 15 = 0 is …………………

A) 30

B) 34

C) 40

D) 44

Answer:

B) 34

Question 40.

The value of \(\sqrt{2+\sqrt{2+\sqrt{2+\ldots \ldots \ldots}}}\) is……………….

A) 3

B) 4

C) 2

D) 8

Answer:

C) 2

Question 41.

The roots of (b – c) x^{2} + (c – a) x + (a – b) = 0 are equal and kb = a + c, then the value of k is………………..

A) 1

B) 2

C) 3

D) 4

Answer:

B) 2

Question 42.

The roots of x – \(\frac{3}{x}\) = 2 are

A) 1, 3

B) 3, -1

C) 2, 2

D) 1, 2

Answer:

B) 3, -1

Question 43.

If one root of 3x^{2} + 2x + k = 0 is the reciprocal of the other, then the value of

k is

A) 3

B) -3

C) 2

D) 6

Answer:

A) 3

Question 44.

If the roots of 2x^{2} + kx + 3 = 0 are real and equal, then the value of k is

A) ± 2√6

B) ± 6√2

C) ± 4

D) ± 5

Answer:

C) ± 4

Question 45.

The quadratic equation px^{2} + qx + r = 0 has imaginary roots, if ………………

A) q^{2} > 4 pr

B) q^{2} < 4 pr

C) q^{2} = 4 pr

D) p = q + r

Answer:

B) q^{2} < 4 pr

Question 46.

Which of the following is a quadratic equation ?

A) x^{3} – 6x^{2} + 2x -1 = 0

B) x^{2} + \(\frac{1}{x^{2}}\) = 2

C) x + \(\frac{1}{x}\) = 3

D) (x + 1) (x + 2) (x + 3) = 0

Answer:

C) x + \(\frac{1}{x}\) = 3

Question 47.

The roots of the quadratic equation x^{2} – 5x + 6 = 0 are ………………..

A) -2, 3

B) -2, -3

C) 2, -3

D) 2, 3

Answer:

D) 2, 3

Question 48.

The roots of the quadratic equation x^{2} – 2x + 1 = 0 are

A) -1, -1

B) 1, -1

C) 1, 1

D) 2, 2

Answer:

C) 1, 1

Question 49.

The roots of the quadratic equation 3(x + 3)^{2} = 48 are

A) 1, 7

B) 1, -7

C) -1, 7

D) -1, -7

Answer:

B) 1, -7

Question 50.

The roots of x^{2} – 2x – (r^{2} – 1) = 0 are

A) 1 – r, r + 1

B) 1, r

C) 1 – r, r

D) 1 – r, -r – 1

Answer:

A) 1 – r, r + 1

Question 51.

If 3 is a solution of 3x^{2} + (k -1) x + 9 = 0, then k =

A) 13

B) 11

C) -11

D) -13

Answer:

C) -11

Question 52.

The roots of the quadratic equation \(\frac{x^{2}-8}{x^{2}+20}=\frac{1}{2}\) are ………………….

A) ± 2

B) ± 4

C) ± 3

D) ± 6

Answer:

D) ± 6

Question 53.

If x^{2} + 2kx + 4 = 0 has a root x = 2, then ‘ the value of k is

A) 2

B) -2

C) -1

D) -4

Answer:

B) -2

Question 54.

The roots of the quadratic equation (3x – 2) (2x + 1) = 0 are

A) \(\frac{2}{3}, \frac{-1}{2}\)

B) \(\frac{-2}{3}, \frac{-1}{2}\)

C) \(\frac{-2}{3}, \frac{1}{2}\)

D) \(\frac{2}{3}, \frac{1}{2}\)

Answer:

A) \(\frac{2}{3}, \frac{-1}{2}\)

Question 55.

The roots of the quadratic equation (√3x – √2) (√3x – √2) = 0 are

A) \(\frac{\sqrt{2}}{\sqrt{3}}, \frac{\sqrt{2}}{\sqrt{3}}\)

B) \(\frac{-\sqrt{2}}{\sqrt{3}}, \frac{-\sqrt{2}}{\sqrt{3}}\)

C) \(\frac{\sqrt{2}}{\sqrt{3}}, \frac{-\sqrt{2}}{\sqrt{3}}\)

D) \(\frac{-\sqrt{2}}{\sqrt{3}}, \frac{\sqrt{2}}{\sqrt{3}}\)

Answer:

A) \(\frac{\sqrt{2}}{\sqrt{3}}, \frac{\sqrt{2}}{\sqrt{3}}\)

Question 56.

If \(\frac {1}{2}\) is a root of the equation x2 + kx – \(\frac {5}{4}\) = 0 then the value of k is

A) \(\frac {1}{2}\)

B) \(\frac {1}{4}\)

C) 2

D) -2

Answer:

C) 2

Question 57.

The roots of the quadratic equation \(\frac{x}{a}=\frac{a}{x}\) are

A) a, – a

B) -a^{2}, a^{2}

C) -a, -a

D) a, a

Answer:

A) a, – a

Question 58.

The roots of the quadratic equation \(\frac{x^{2}}{a^{2}-b^{2}}=\frac{a+b}{a-b}\) are

A) ± a

B) a, b

C) ± (a + b)

D) ± b

Answer:

C) ± (a + b)

Question 59.

The roots of the quadratic equation \(\frac{11}{3+x}\) = 4(3 – x)are

A) ± 2

B) ± \(\frac{5}{2}\)

C)± \(\frac{1}{5}\)

D) ± 5

Answer:

A) ± 2

Question 60.

The roots of the quadratic equation \(\frac{9}{x^{2}-27}=\frac{25}{x^{2}-11}\) are

A) ± 6

C) ± 3

B) ± 4

D) ± 5

Answer:

A) ± 6

Question 61.

The roots of the quadratic equation x^{2} + 7x = 0 are

A) 0, 7

B) 7, -7

C) -7, -7

D) 0, -7

Answer:

D) 0, -7

Question 62.

If (x + 4) (x – 4) = 9, then the values of x are

A) ± \(\frac {1}{2}\)

B) ± 5

C) \(\frac{1}{5}, \frac{1}{5}\)

D) 5, 5

Answer:

B) ± 5

Question 63.

The roots of the quadratic equation (2x + b + a) (2x + b – a) = 0 are

A) \(\frac{a-b}{2}, \frac{a-b}{2}\)

B) \(\frac{a+b}{2}, \frac{a+b}{2}\)

C) \(-\frac{a+b}{2}, \frac{a-b}{2}\)

D) \(\frac{a+b}{2}, \frac{a-b}{2}\)

Answer:

C) \(-\frac{a+b}{2}, \frac{a-b}{2}\)

Question 64.

The roots of the equation \(\sqrt{2 x^{2}+9}\) = 9 are

A) x = 0

B) x = ± 6

C) x = 6

D) x = -6

Answer:

B) x = ± 6

Question 65.

The value of k for which 3 is a root of the equation kx^{2} – 7x + 3 = 0 is

A) -3

B) 3

C) 2

D) -2

Answer:

C) 2

Question 66.

The two roots of a quadratic equation are 2 and – 1. The equation is

A) x^{2} + x + 2 = 0

B) x^{2} – 2x + 2 = 0

C) x^{2} + 2x – 2 = 0

D) x^{2} – x – 2 = 0

Answer:

D) x^{2} – x – 2 = 0

Question 67.

Which constant must be added and subtracted to solve the quadratic equation 9x^{2} + 3x – 8 = 0 by the method of completing the square ?

A) \(\frac {1}{36}\)

B) \(\frac {9}{64}\)

C) \(\frac {1}{3}\)

D) \(\frac {1}{4}\)

Answer:

D) \(\frac {1}{4}\)

Question 68.

If the sum of the roots of the quadratic equation 3x^{2} + (2k + 1) x – (k + 5) = 0 is equal to the product of the roots, then the value of k is

A) 3

B) 5

C) 4

D) 2

Answer:

C) 4

Question 69.

The quadratic equation whose one root is 3 + √5 is

A) x^{2} – 6x – 4 = 0

B)x^{2} + 6x + 4 = 0

C) x^{2} – 6x + 4 = 0

D) x^{2} + 6x + 5 = 0

Answer:

C) x^{2} – 6x + 4 = 0

Question 70.

If the two roots of the two quadratic equations x^{2} + mx + 1 = 0 and ax^{2} + bx + a = 0 are common, then the value of m is

A) \(\frac{a}{b}\)

B) b

C) \(\frac{b}{a}\)

D) ab

Answer:

C) \(\frac{b}{a}\)

Question 71.

The roots of the quadratic equation x^{2} + 8x + 7 = 0 are

A) 1, 7

B) -1, -7

C) -1, 7

D) 1, -7

Answer:

B) -1, -7

Question 72.

The value of k for which roots of the quadratic equation kx^{2} + 2x + 3 = 0 are equal is …………………..

A) \(\frac{1}{3}\)

B) -3

C) \(\frac{-1}{3}\)

D) 3

Answer:

A) \(\frac{1}{3}\)

Question 73.

If the roots of the equation 12x^{2} + mx + 5 = 0 are real and equal then m is equal to

A) 4\(\sqrt{15}\)

B) 2\(\sqrt{15}\)

C) 8\(\sqrt{15}\)

D) 10√5

Answer:

A) 4\(\sqrt{15}\)

Question 74.

If the roots of the quadratic equation x^{2} + px + 16 = 0 are equal then the value of pis

A) ± 7

B) ± 6

C) ± 8

D) ± 9

Answer:

C) ± 8

Question 75.

Which of the following equations has the equal roots ?

A) 6x^{2} – x – 2 = 0

B) x^{2} – 8x + 16 = 0

C) 10x – \(\frac{1}{x}\) = 3

D) x^{2} + 6x + 5 = 0

Answer:

B) x^{2} – 8x + 16 = 0

Question 76.

Discriminant of the quadratic equation 2x^{2} + x – 8 = 0 is

A) 65

B) -15

C) -127

D) 15

Answer:

A) 65

Question 77.

The discriminant of x^{2} + 5x + 5 = 0 is

A) \(\frac{-5}{2}\)

B) \(\frac{5}{2}\)

C) – 5

D) 5

Answer:

D) 5

Question 78.

The value (s) of k such that the equation kx^{2} + 6x + k = 0 has equal roots is ………………..

A) ± 3

B) ± 6

C)-6 and 3

D) 9

Answer:

A) ± 3

Question 79.

If the discriminant of the equation 6x^{2} – bx + 2 = 0 is 1, then the value of ‘b’ is

A) ± 7

B)± √7

C) -7

D) 7

Answer:

A) ± 7

Question 80.

The discriminant of the quadratic equation 7√3x^{2} + 10x – √3 = 0 is :

A) \(\frac{-10}{7 \sqrt{3}}\)

B) 142

C) 184

D) 26

Answer:

C) 184

Question 81.

Discriminant of the quadratic equation 2x^{2} + x – 8 = 0 is

A) -65

B) 65

C) -127

D) -15

Answer:

B) 65

Question 82.

The roots of (x^{2} – 3x + 2) (x) (x – 4) = 0 are

A) 0 and 4

B) 0, 1, 2 and 4

C) 4

D) 1, 2 and 4

Answer:

B) 0, 1, 2 and 4

Question 83.

The difference of the roots of x^{2} – 7x – 9 = 0 is

A) \(\sqrt{85}\)

B) 7

C) -9

D) 8

Answer:

A) \(\sqrt{85}\)

Question 84.

The equation \(\sqrt{x+10}-\frac{6}{\sqrt{x+10}}\) = 5 has

A) an extraneous root between -5 and -1

B) two extraneous roots

C) an extraneous root between -10 and -6

D) a real root between 20 and 25

Answer:

C) an extraneous root between -10 and -6

Question 85.

If log_{10}(x^{2} – 3x + 6) = 1, then the value of x is

A) 4 or -1

B) 4 only

C) 10 or 2

D) 4 or -2

Answer:

A) 4 or -1

Question 86.

If y = x + \(\) then x^{4} + x^{3} – 4x^{2} + x + 1 = 0 becomes

A) x^{2}(y^{2} + y – 6) = 0

B) x^{2}(y^{2} + y – 3) = 0

C) x^{2}(y^{2} + y – 4) = 0

D) x^{2}(y^{2} + y – 2) = 0

Answer:

A) x^{2}(y^{2} + y – 6) = 0

Question 87.

The roots of the equations \(2 \sqrt{x}+2 x^{\frac{1}{2}}\) = 5 can be found by solving

A) 4x^{2} – 25x -4 = 0

B) 4x^{2} – 17x + 4 = 0

C) 4x^{2} – 25x + 4 = 0

D) 4x^{2} – 17x -4 = 0

Answer:

B) 4x^{2} – 17x + 4 = 0

Question 88.

The values of ’k’ for which the equation 2x^{2} – kx + x + 8 = 0 will have real and

equal roots are …………………

A) 9 and -7

B)-9 and 7

C) only 9

D) only -7

Answer:

A) 9 and -7

Question 89.

Two numbers whose sum is 6 and the absolute value of whose difference is 8 are the roots of the equation

A) x^{2} – 6x + 8 = 0

B) x^{2} – 6x – 7 = 0

C) x^{2} – 6x + 7 = 0

D) x^{2} + 6x – 8 = 0

Answer:

B) x^{2} – 6x – 7 = 0

Question 90.

If a and b are the roots of x^{2} – px + q = 0, then a^{2} + b^{2} =…………………

A) p^{2} + 2q

B) p^{2} – q^{2}

C) p^{2} – 2q

D) p^{2} + q^{2}

Answer:

C) p^{2} – 2q

Question 91.

The solution of \(\sqrt{5 x-1}+\sqrt{x-1}\) = 2 is

A) x = 2

B) x = 2, x = 1

C) x = 1

D) x = 1, x = 4

Answer:

C) x = 1

Question 92.

The’ roots of \(\frac{15}{x^{2}-4}-\frac{2}{x-2}\) = = 1 are………………

A) -3 and 5

B) 3 only

C) -5 and 3

D) -5 only

Answer:

C) -5 and 3

Question 93.

In the equation 2x^{2} – hx + 2k = 0, the sum of the roots is 4 and the product of the roots is -3. Then h and k have the values ………………… respectively.

A) 8 and -3

B) 8 and 6

C) 4 and -3

D) -3 and 8

Answer:

A) 8 and -3

Question 94.

If \(\sqrt{x-1}-\sqrt{x+1}\) + 1 = 0, then 4x equal to

A) 0

B) 1 \(\frac{1}{4}\)

C) 5

D) 4\(\sqrt{-1}\)

Answer:

C) 5

Question 95.

The number of roots satisfying the equation \(\sqrt{5-x}=x \sqrt{5-x}\) – x is/are

A) 2

B) 3

C) 1

D) unlimited

Answer:

B) 3

Question 96.

If α and β are the roots of the quadratic equation x^{2} + px + 12 = 0 with the condition α – β = 1, then the value of ‘p’ is

A) 7 or -7

B) 1

C) -7

D) 7

Answer:

A) 7 or -7

Question 97.

The coefficient of x in the quadratic equation x^{2} + px + q = 0 was taken as 17 in the place of 13 and its roots were found to be-2 and-15. The roots of the original equation are

A) 3 or -10

B) -3 or 10

C) -3 or -10

D) 3 or 10

Answer:

C) -3 or -10

Question 98.

The roots of \(\sqrt{2 x-3}+\sqrt{3 x-5}-\sqrt{5 x-6}\) = 0 are

A) 2 or \(\frac{7}{6}\)

B) 2 only

C) \(\frac{7}{6}\) only

D) 2 and \(\frac{7}{6}\)

Answer:

A) 2 or \(\frac{7}{6}\)

Question 99.

If one of roots of x^{2} + ax + 4 = 0 is twice the other root, then the value of ‘a’ is

A) 8√2

B) √2

C) -3√2

D) -2√2

Answer:

C) -3√2

Question 100.

If the roots of the equation

(a – b)x^{2} + (b – c) x + (c – a) = 0 are equal, then

A) 2b = a + c

B) 2c = a + b

C) 2a = b + c

D) 2a = b – c

Answer:

C) 2a = b + c

Question 101.

If a = b = c, then the roots of the equa-tion (x – a) (x – b) + (x – b)(x – c) +

(x – c) (x – a) = 0 are

A) imaginary

B) real and equal

C) real and unequal

D) real

Answer:

B) real and equal

Question 102.

If the sum of the roots of the quadratic equation 3x^{2} + (2k + 1) x – (k + 5) = 0 is equal to the product of the roots, then the value of ‘k’ is

A) 0

B) 1

C) -4

D) 4

Answer:

D) 4

Question 103.

For a quadratic equation, 3 + √5 is one root. The other root must be

A) 3 + √5

B) 3 – √5

C) 3 + 2√5

D) 3 – 2√5

Answer:

B) 3 – √5

Question 104.

If one root of the equation ax^{2} + bx + c = 0 is three times the other, then …………………

A) b^{2} = 16ac

B) 3b^{2} = 16ac

C) 2b^{2} = 9ac

D) b^{2} = ac

Answer:

B) 3b^{2} = 16ac

Question 105.

The root of the quadratic equation 3x^{2} – kx + 14 = 0 are in the ratio 7 : 6 then k = ………………….

A) 12

B) -3

C) 1

D) 13

Answer:

D) 13

Question 106.

If α, β are the roots of the equation x^{2} – px + q = 0, then the quadratic equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\boldsymbol{\beta}}{\alpha}\) is ……………………

A) qx^{2} + (2q – p^{2}) x + q = 0

B) qx^{2} – px + 1 = 0

C) qx^{2} + (p^{2} – 2q) x + q = 0

D) qx^{2} – px – 1 = 0

Answer:

A) qx^{2} + (2q – p^{2}) x + q = 0

Question 107.

Find the value of k, if the expression x^{2} + kx + 1 is factorizable into two linear factors ………………

A) either k ≥ 2 or k ≤ -2

B) k ≥ 2

C) k ≤ -2

D) neither k ≥ 2 nor k ≤ – 2

Answer:

A) either k ≥ 2 or k ≤ -2

Question 108.

The product of two consecutive odd numbers is 143. The numbers are

A) 13 and 15

B) 11 and 13

C) -11 and 13

D) -13 and 15

Answer:

B) 11 and 13

Question 109.

The perimeter of a rectangular room is 34m and the length of a diagonal is 13m. The dimensions of the room are …………………..

A) 17m and 7m

B) 12m and 5m

C) 9m and 2m

D) 15m and 2m

Answer:

B) 12m and 5m

Question 110.

If x^{2} + 4ax + 3 = 0 and 2x^{2} + 3ax – 9 = 0 have a common root, then the value of ‘a’is ……………………

A) 1

B) ± 3

C) ± 1

D) -3

Answer:

C) ± 1

Question 111.

The product of two consecutive natural numbers is 600. The numbers are …………………….

A) 35, 36

B) 16, 25

C) 26, 25

D) 24, 25

Answer:

D) 24, 25

Question 112.

The roots of the quadratic equation x^{2} – 5x + 6 = 0 are

A) 3, -2

B) -3, 2

C) 2, 3

D) -2, -3

Answer:

C) 2, 3

Question 113.

If the roots of a quadratic equation are equal, then

A) b^{2} = 4ac

B) b^{2} – 4ac < 0

C) b^{2} – 4ac > 0

D) b^{2} – 4ac = 0

Answer:

A) b^{2} = 4ac

Question 114.

The roots of the quadratic equation abx^{2} – (a + b) x + 1 are

A) a + b, a – b

B) \(\frac{1}{a}, \frac{1}{b}\)

C) a, b

D) \(\frac{1}{a+b}, \frac{1}{a-b}\)

Answer:

B) \(\frac{1}{a}, \frac{1}{b}\)

Question 115.

If α, β be the roots of the quadratic equation x^{2} – 2x + 1 = 0, then the quadratic equation whose roots are α + β and αβ is ………………

A) x^{2} – 2x – 1 = 0

B) x^{2} + 2x + 1 = 0

C) x^{2} + 2x -1 = 0

D) x^{2} – 2x + 1 = 0

Answer:

D) x^{2} – 2x + 1 = 0

Question 116.

The roots of the equation x + \(\) = 5\(\) are

A) 5, -5

B) 5, \(\frac{1}{5}\)

C) -5, -5

D) 5, 5

Answer:

B) 5, \(\frac{1}{5}\)

Question 117.

Which of the following is not a quadratic equation ?

A) x (2x + 3) = x^{2} + 1

B) (x – 2)^{2} + 1 = 2x – 3

C) x(x + 1) + 8 = (x + 2) (x – 2)

D) (x + 2)^{3} = x^{3} – 4

Answer:

C) x(x + 1) + 8 = (x + 2) (x – 2)

Question 118.

Is – √5x^{2} + 6x + 3 = 0 a quadratic equation ?

A) Yes

B) No

C) can’t say

D) It is a linear equation

Answer:

A) Yes

Question 119.

Which of the following equation has \(\frac {1}{5}\) as a root ?

A) 2x^{2} – 7x + 6 = 0

B) 35x^{2} – 2x – 1 = 0

C) 35x^{2} + 12x + 1 = 0

D) 10x^{2} – 3x – 1 = 0

Answer:

B) 35x^{2} – 2x – 1 = 0

Question 120.

If x = 1 is a common root of the equations ax^{2} + ax + 3 = 0 and x^{2} + x + b = 0, then the value of ab is :

A) 3

B) 6

C) 3.5

D) -3

Answer:

A) 3

Question 121.

Which of the following equations has the sum of roots as 3 ?

A) 2x^{2} – 3x + 6 = 0

B) 3x^{2} – 3x + 3 = 0

C) √2 x^{2}–\(\frac{3}{\sqrt{2}}\)x + = 0

D) -x^{2} + 3x – 3 = 0

Answer:

D) -x^{2} + 3x – 3 = 0

Question 122.

Which constant should be added and substracted to solve the quadratic equation 4x^{2} – √3x – 5 = 0 by the method of completing the square?

A) \(\frac{3}{16}\)

B) \(\frac{\sqrt{3}}{4}\)

C) \(\frac{9}{16}\)

D) \(\frac{3}{4}\)

Answer:

A) \(\frac{3}{16}\)

Question 123.

Which of the following equations has the product of its roots as 4 ?

A) x^{2} + 4x – 4 = 0

B) x^{2} + 4x + 4 = 0

C) x^{2} + 4x – 24 = 0

D) -x^{2} + 4x + 4 = 0

Answer:

B) x^{2} + 4x + 4 = 0

Question 124.

ax^{2} + bx + c = 0, a > 0,b = 0, c > 0 has;

A) Two equal roots

B) One real root

C) Two distinct real roots

D) No real roots

Answer:

D) No real roots

Question 125.

If the equation x^{2} – 4x + a = 0 has no real roots, then

A) a ≤ 4

B) a < 2

C) a < 4 D) a > 4

Answer:

D) a > 4

Question 126.

If the discriminant of the quadratic equation ax^{2} + bx + c = 0, then the roots of the equation

A) are irrational and equal

B) are rational and equal

C) do not exist and real

D) are real and equal

Answer:

D) are real and equal

Question 127.

The least value of ax^{2} + bx + c (a > 0) is

A) \(\frac{-b}{2 a}\)

B) \(\frac{4 a^{2}-b^{2}}{4 a}\)

C) \(\frac{4 a^{2}+b^{2}}{4 a}\)

D) none of the above

Answer:

B) \(\frac{4 a^{2}-b^{2}}{4 a}\)

Question 128.

The equation x + \(\sqrt{x-2}\) = 4 has

A) two real roots

B) two imaginary roots

C) one real root

D) one real and one imaginary root

Answer:

C) one real root

Question 129.

If one root of the equation a(b – c) x^{2} + b(c – a) x + c(a – b) = 0 is 1, then the other root is

A) \(\frac{a(b-c)}{c(a-b)}\)

B) \(\frac{a(b-c)}{b(c-a)}\)

C) \(\frac{c(a-b)}{a(b-c)}\)

D) \(\frac{b(c-a)}{a(b-c)}\)

Answer:

C) \(\frac{c(a-b)}{a(b-c)}\)

Question 130.

The roots of the equation x^{2} + 2√3 x + 3 = 0 are

A) rational and equal

B) real and equal

C) rational and unequal

D) imaginary

Answer:

B) real and equal

Question 131.

The roots of the equation ax^{2} + bx + c = 0 will be reciprocal if

A) a = bc

B) b = c

C) a = b

D) c = a

Answer:

D) c = a

Question 132.

The discriminant of ax^{2} – 2√2 x + c = 0 with a, c and real constants is zero. The roots must be

A) rational and equal

B) real and equal

C) imaginary

D) equal and integral

Answer:

B) real and equal

Question 133.

The equation \(\sqrt{x+4}-\sqrt{x-3}\) + 1 = 0 has

A) one real root

B) two imaginary roots

C) no root

D) one real and one imaginary root

Answer:

C) no root

Question 134.

The sum of the reciprocals of the roots of the equation x^{2} + px + q = 0 is

A) \(\frac{q}{p}\)

B) \(\frac{-\mathrm{p}}{\mathrm{q}}\)

C) \(\frac{-\mathrm{q}}{\mathrm{p}}\)

D) \(\frac{\mathrm{p}}{\mathrm{q}}\)

Answer:

D) \(\frac{\mathrm{p}}{\mathrm{q}}\)

Question 135.

The equation \(x-\frac{7}{x-3}=3-\frac{7}{x-3}\) has

A) no roots

B) infinitely many roots

C) one integral root

D) two real roots

Answer:

A) no roots

Question 136.

If \(\frac{b}{x-a}=\frac{x+a}{b}\) then the value of x in terms of a and b is

A) \(\sqrt{a^{2}+b^{2}}\)

B) \(\pm \sqrt{a^{2}+b^{2}}\)

C) \(-\sqrt{a^{2}+b^{2}}\)

D) \(\pm \sqrt{a^{2}-b^{2}}\)

Answer:

B) \(\pm \sqrt{a^{2}+b^{2}}\)

Question 137.

The standard form of a quadratic equation is

A) ax + b = 0, a ≠ 0

B) ax^{2} + bx + c = 0, a ≠ 0

C) ax^{3} + bx^{2} + cx + d = 0, a ≠ 0

D) ax^{4} + bx^{3} + cx^{2} + dx + e = 0, a ≠ 0.

Answer:

B) ax^{2} + bx + c = 0, a ≠ 0

Question 138.

Which of the following is a quadratic equation?

A) (x – 2) (x + 1) = (x – 1) (x + 3)

B) x^{4} – 1 = 0

C) (x + 1)^{2} = 2(x – 3)

D) x^{2} + 3x + 1 = (x – 2)^{2}

Answer:

C) (x + 1)^{2} = 2(x – 3)

Question 139.

A cottage industry produces a certain number of toys in a day. The cost of production of each toy was found to be 55 rupees. On a particular day, the total cost of production was ? 750. Represent this situation in the form of a quadratic equation.

A) x (55 – x) = 750

B) 55x = 750

C) x(x – 55) = 750

D) x (x + 55) = 750

Answer:

A) x (55 – x) = 750

Question 140.

The sum of the squares of two consecutive natural numbers is 25. Represent this situation in the form of a quadratic equation.

A) x^{2} + (x + 1)^{2} + 25 = 0

B) x^{2} – (x + 1)^{2} = 25

C) x^{2} + (x + 1)^{2} = 25

D) (x + 1)^{2} – x^{2} = 25

Answer:

C) x^{2} + (x + 1)^{2} = 25

Question 141.

If one root of the two quadratic equations x^{2} + ax + b = 0 and x^{2} + bx + a = 0 is common, then

A) a + b = -1

B) ab = -1

C) ab = 1

D) a + b = 1

Answer:

A) a + b = -1

Question 142.

The condition that both the roots of the two quadratic equations

a_{1}x2 + b_{1}x + c_{1} = 0 and a_{2}x2 + b_{2}x + c_{2} = 0 are common is

A) a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

B) a_{1}^{2} + b_{1}^{2} + c_{1}^{2} = a_{2}^{2} + b_{2}^{2} + c_{2}^{2}

C) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

D) a_{1} + b_{1} + c_{1} = a_{2} + b_{2} + c_{2}

Answer:

C) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

Question 143.

If x = \(\sqrt{1+\sqrt{1+\sqrt{1+\ldots \ldots \ldots . .}}}\) , then

A) 1 < x < 2

B) x = 1

C) 0 < x < 1

D) x is infinite

Answer:

A) 1 < x < 2

Question 144.

If \(\) has roots which are numerically equal but of opposite signs, the value of ‘m’ must be

A) \(\frac{a+b}{a-b}\)

B) 0

C) 1

D) \(\frac{a-b}{a+b}\)

Answer:

D) \(\frac{a-b}{a+b}\)

Question 145.

Two consecutive positive integers differ by

A) 3

B) 4

C) 2

D) 1

Answer:

D) 1

Question 146.

The sum of a number and its reciprocal is \(\frac {5}{2}\) . Represent this, situation in the form of a quadratic equation.

A) x –\(\frac{1}{x}=\frac{5}{2}\)

B) x +\(\frac{1}{x}=\frac{5}{2}\)

C) x +\(\sqrt{x}=\frac{5}{2}\)

D) x^{2}\(\frac{1}{x^{2}}=\frac{5}{2}\)

Answer:

B) x +\(\frac{1}{x}=\frac{5}{2}\)

Question 147.

Two consecutive even integers are

A) x, x + 1

B) x, 2x

C) x, x – 1

D) x, x + 2

Answer:

D) x, x + 2

Question 148.

Two consecutive positive integers are

A) x, x – 2

B) x, x + 1

C) x, x + 2

D) x, 2x

Answer:

B) x, x + 1

Question 149.

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each and the product of the number of marbles they have now is 124. Represent this situation in the form of quadratic equation.

A) (x – 5) (x – 40) = 124

B) (x + 5) (40 + x) + 124 = 0

C) (x – 5) (40 – x) = 124

D) (5 + x) (40 – x) = 124

Answer:

C) (x – 5) (40 – x) = 124

Question 150.

What is the condition that one root of the quadratic equation ax^{2} + bx + c is reciprocal of the other ?

A) a = b

B) b = c

C) a = c

D) a + b + c = 0

Answer:

C) a = c

Question 151.

If the roots of the quadratic equation ax^{2} + bx + c are sinα and cosα, then

A) a^{2} + b^{2} = c^{2}

B) b^{2} – 2ac = a^{2}

C) b^{2} + 2ac = a^{2}

D) a^{2}2 – 2bc = b^{2}

Answer:

B) b^{2} – 2ac = a^{2}

Question 152.

If the difference of the roots of the quadratic equation x^{2} – ax + b is 1, then

A) a^{2} – 4b = -1

B) a^{2} – 4b = 4

C) a^{2} – 4b = 1

D) a^{2} – 4b = 0

Answer:

C) a^{2} – 4b = 1

Question 153.

The value of c for which the equation ax^{2} + bx + c = 0 has equal roots is

A) \(\frac{\mathrm{b}^{2}}{4 \mathrm{a}}\)

B) \(\frac{a^{2}}{b}\)

C) \(\frac{a^{2}}{4 b}\)

D) \(\frac{\mathrm{b}^{2}}{\mathrm{a}}\)

Answer:

A) \(\frac{\mathrm{b}^{2}}{4 \mathrm{a}}\)

Question 154.

The roots of the equation x^{2} – 2x = 0 can be obtained graphically by finding the abscissa of the points of intersection of each of the following pairs of equations except ………………….

A) y = x,y = x – 2

B) y = x^{2} – 2x + 1, y = 1

C) y = x^{2},y = 2x

D) y = x^{2} – 2x, y = 0

Answer:

A) y = x,y = x – 2

Question 155.

If in applying the quadratic f ormula to a quadratic equation f(x) = ax^{2} + bx + c = 0 it happens that c = \(\). Then the graph of y = f(x) will certainly …………….

A) have a minimum

B) be a tangent to the Y – axis

C) have a maximum

D) be a tangent to X-axis

Answer:

D) be a tangent to X-axis

Question 156.

In the equation \(\) the roots are equal when

A) m = 0

B) m = \(\frac {1}{2}\)

C) m = –\(\frac {1}{2}\)

D) m = 1

Answer:

C) m = –\(\frac {1}{2}\)

Question 157.

If b^{2} – 4ac > 0 the graph may be …………………

Answer:

Question 158.

If b^{2} = 4ac then the graph may be

Answer:

Question 159.

The graph of a linear polynomial may be

Answer:

Question 160.

Identify parabola among the following graphs.

Answer:

Question 161.

If a quadratic equation have no real roots then the graph may be ………………

Answer:

Question 162.

If the graph of ax^{2} + bx + c = 0, a ≠ 0 never intersects X axis then the number of real zeroes are ………………..

A) 2

B) 0

C) 3

D) 4

Answer:

B) 0

Question 163.

From the graph given below the nature of roots are ………………….

A) Real and equal

B) Real and not equal

C) Not real

D) 10, 11

Answer:

C) Not real