Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b)

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 = \(\left[2\left(1+\frac{3}{2} x\right)\right]^{-2 / 3}\)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q1
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q1.1

Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b)

Question 2.
Find the
(i) 6th term of \(\left(1+\frac{x}{2}\right)^{-5}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q2(i)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q2(i).1

(ii) 7th term of \(\left(1-\frac{x^2}{3}\right)^{-4}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q2(ii)

(iii) 10th term of (3 – 4x)-2/3
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q2(iii)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q2(iii).1

(iv) 5th term of \(\left(7+\frac{8 y}{3}\right)^{7 / 4}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q2(iv)

Question 3.
Write down the first 3 terms in the expansion of
(i) (3 + 5x)-7/3
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q3(i)

(ii) (1 + 4x)-4
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q3(ii)

(iii) (8 – 5x)2/3
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q3(iii)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q3(iii).1

(iv) (2 – 7x)-3/4
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q3(iv)

Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b)

Question 4.
Find the general term (r + 1)th term in the expansion of
(i) (4 + 5x)-3/2
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q4(i)

(ii) \(\left(1-\frac{5 x}{3}\right)^{-3}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q4(ii)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q4(ii).1

(iii) \(\left(1+\frac{4 x}{5}\right)^{5 / 2}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q4(iii)

(iv) \(\left(3-\frac{5 x}{4}\right)^{-1 / 2}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) I Q4(iv)

II.

Question 1.
Find the coefficient of x10 in the expansion of \(\frac{1+2 x}{(1-2 x)^2}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q1

Question 2.
Find the coefficient of x4 in the expansion of (1 – 4x)-3/5
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q2

Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b)

Question 3.
(i) Find the coefficient of x5 in \(\frac{(1-3 x)^2}{(3-x)^{3 / 2}}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q3(i)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q3(i).1

(ii) Find the coefficient of x8 in \(\frac{(1+x)^2}{\left(1-\frac{2}{3} x\right)^3}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q3(ii)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q3(ii).1

(iii) Find the coefficient of x7 in \(\frac{(2+3 x)^3}{(1-3 x)^4}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q3(iii)

Question 4.
Find the coefficient of x3 in the expansion of \(\frac{\left(1+3 x^2\right)^{3 / 2}}{(3+4 x)^{1 / 3}}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q4
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) II Q4.1

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:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q1(i)

(ii) \(1-\frac{4}{5}+\frac{4.7}{5.10}-\frac{4.7 .10}{5.10 .15}+\ldots \ldots\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q1(ii)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q1(ii).1

(iii) \(\frac{3}{4}+\frac{3.5}{4.8}+\frac{3.5 .7}{4.8 .12}+\ldots\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q1(iii)

(iv) \(\frac{3}{4.8}-\frac{3.5}{4.8 .12}+\frac{3.5 .7}{4.8 .12 .16}-\ldots \ldots\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q1(iv)
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q1(iv).1

Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b)

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:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q2
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q2.1

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 9x2 + 24x = 11.
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q3
⇒ 3x + 4 = 3√3
Squaring on both sides
(3x + 4)2 = (3√3)2
⇒ 9x2 + 24x + 16 = 27
⇒ 9x2 + 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 x2 + 4x.
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q4

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:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q5
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q5.1

Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b)

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:
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q6
Inter 2nd Year Maths 2A Binomial Theorem Solutions Ex 6(b) III Q6.1