Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Binomial Theorem Solutions Exercise 6(b) will help students to clear their doubts quickly.

## Intermediate 2nd Year Maths 2A Binomial Theorem Solutions Exercise 6(b)

I.

Question 1.

Find the set of values of x for which the binomial expansions of the following are valid.

(i) (2 + 3x)^{-2/3}

(ii) (5 + x)^{3/2}

(iii) (7 + 3x)^{-5}

(iv) \(\left(4-\frac{x}{3}\right)^{-1 / 2}\)

Solution:

(i) (2 + 3x)^{-2/3} = \(\left[2\left(1+\frac{3}{2} x\right)\right]^{-2 / 3}\)

Question 2.

Find the

(i) 6th term of \(\left(1+\frac{x}{2}\right)^{-5}\)

Solution:

(ii) 7th term of \(\left(1-\frac{x^2}{3}\right)^{-4}\)

Solution:

(iii) 10th term of (3 – 4x)^{-2/3}

Solution:

(iv) 5th term of \(\left(7+\frac{8 y}{3}\right)^{7 / 4}\)

Solution:

Question 3.

Write down the first 3 terms in the expansion of

(i) (3 + 5x)^{-7/3}

Solution:

(ii) (1 + 4x)^{-4}

Solution:

(iii) (8 – 5x)^{2/3}

Solution:

(iv) (2 – 7x)^{-3/4}

Solution:

Question 4.

Find the general term (r + 1)^{th} term in the expansion of

(i) (4 + 5x)^{-3/2}

Solution:

(ii) \(\left(1-\frac{5 x}{3}\right)^{-3}\)

Solution:

(iii) \(\left(1+\frac{4 x}{5}\right)^{5 / 2}\)

Solution:

(iv) \(\left(3-\frac{5 x}{4}\right)^{-1 / 2}\)

Solution:

II.

Question 1.

Find the coefficient of x^{10} in the expansion of \(\frac{1+2 x}{(1-2 x)^2}\)

Solution:

Question 2.

Find the coefficient of x^{4} in the expansion of (1 – 4x)^{-3/5}

Solution:

Question 3.

(i) Find the coefficient of x^{5} in \(\frac{(1-3 x)^2}{(3-x)^{3 / 2}}\)

Solution:

(ii) Find the coefficient of x^{8} in \(\frac{(1+x)^2}{\left(1-\frac{2}{3} x\right)^3}\)

Solution:

(iii) Find the coefficient of x^{7} in \(\frac{(2+3 x)^3}{(1-3 x)^4}\)

Solution:

Question 4.

Find the coefficient of x^{3} in the expansion of \(\frac{\left(1+3 x^2\right)^{3 / 2}}{(3+4 x)^{1 / 3}}\)

Solution:

III.

Question 1.

Find the sum of the infinite series

(i) \(1+\frac{1}{3}+\frac{1.3}{3.6}+\frac{1.3 .5}{3.6 .9}+\ldots\)

Solution:

(ii) \(1-\frac{4}{5}+\frac{4.7}{5.10}-\frac{4.7 .10}{5.10 .15}+\ldots \ldots\)

Solution:

(iii) \(\frac{3}{4}+\frac{3.5}{4.8}+\frac{3.5 .7}{4.8 .12}+\ldots\)

Solution:

(iv) \(\frac{3}{4.8}-\frac{3.5}{4.8 .12}+\frac{3.5 .7}{4.8 .12 .16}-\ldots \ldots\)

Solution:

Question 2.

If t = \(\frac{4}{5}+\frac{4.6}{5.10}+\frac{4.6 .8}{5.10 .15}+\ldots \ldots \ldots \infty\), then prove that 9t = 16.

Solution:

Question 3.

If x = \(\frac{1.3}{3.6}+\frac{1.3 .5}{3.6 .9}+\frac{1.3 .5 .7}{3.6 .9 .12}+\ldots \ldots\) then prove that 9x^{2} + 24x = 11.

Solution:

⇒ 3x + 4 = 3√3

Squaring on both sides

(3x + 4)^{2} = (3√3)^{2}

⇒ 9x^{2} + 24x + 16 = 27

⇒ 9x^{2} + 24x = 11

Question 4.

If x = \(\frac{5}{(2 !) \cdot 3}+\frac{5 \cdot 7}{(3 !) \cdot 3^2}+\frac{5 \cdot 7 \cdot 9}{(4 !) \cdot 3^3}+\ldots\) then find the value of x^{2} + 4x.

Solution:

Question 5.

Find the sum of the infinite series \(\frac{7}{5}\left(1+\frac{1}{10^2}+\frac{1.3}{1.2} \cdot \frac{1}{10^4}+\frac{1.3 .5}{1.2 .3} \cdot \frac{1}{10^6}+\ldots .\right)\)

Solution:

Question 6.

Show that \(1+\frac{x}{2}+\frac{x(x-1)}{2.4}+\frac{x(x-1)(x-2)}{2.4 .6}+\ldots .\) = \(1+\frac{x}{3}+\frac{x(x+1)}{3.6}+\frac{x(x+1)(x+2)}{3.6 .9}+\ldots\)

Solution: