KC Sinha Mathematics Solution Class 11 Chapter 16 Binomial Theorem Exercise 16.2

 Exercise 16.2

Page no 16.40

Type 2

Question 1

Find the 7 th term in the expansion of $\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{9}$.

Question 2

Find the 9 th term in the expansion of $\left(\frac{x}{a}-\frac{3 a}{x^{2}}\right)^{12}$.

Question 3

Find the 5 th term of (i) $\left(\frac{a}{3}-3 b\right)^{7}$

(ii) $\left(2 x^{2}-\frac{1}{3 x^{3}}\right)^{10}$

Question 4

Find $a$ if the 17 th and 18 th terms of the expansion $(2+a)^{50}$ are equal.

Question 5

Find the $r$ th term from end in $(x+a)^{n}$.

Question 6

Find the 4th term from the end in $\left(\frac{x^{3}}{2}-\frac{2}{x^{2}}\right)^{9}$.

Question 7

Find $n$, if the ratio of the fifth term from the beginning to the fifth term end in the expansion $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n}$ is $\sqrt{6}: 1$.

Question 8

Write the general term in the following expansions :

(i) $\left(1-x^{2}\right)^{12}$

(ii) $\left(x-\frac{3}{x^{2}}\right)^{10}$

(iii) $\left(x^{2}-\frac{1}{x}\right)^{12}, x \neq 0$

Type 3

Question 9

Find the middle term in the expansion of :

(i) $\left(\frac{2 x}{3}-\frac{3 y}{2}\right)^{20}$

(ii) $\left(\frac{2 x}{3}-\frac{3}{2 x}\right)^{6}$

(iii) $\left(\frac{x}{y}-\frac{y}{x}\right)^{7}$

(iv) $(1+x)^{2 n}$

(v) $\left(1-2 x+x^{2}\right)^{n}$

(vi) $\left(3-\frac{x^{3}}{6}\right)^{7}$

Question 10

Find the general and middle term in the expansion of $\left(\frac{x}{y}+\frac{y}{x}\right)^{2 n+1}, n$ being a positive integer and show that there is no term free from $\frac{x}{y}$.

Page no 16.41

Question 11

(i) Show that the middle term in the expansion of $(1+x)^{2 n}$ is $\frac{1.3 .5 \ldots(2 n-1)}{n !} \cdot 2^{n} x^{n}$

(ii) Show that the middle term in the expansion of $\left(x-\frac{1}{x}\right)^{2 n}$ is $\frac{1.3 .5 \ldots(2 n-1)}{n !}(-2)^{n}$

Type 4

Question 12

Find the coefficient of $x$ in the expansion of $\left(2 x-\frac{3}{x}\right)^{9}$.

Question 13

Find the coefficient of $x^{7}$ in the expansion of $\left(3 x^{2}+\frac{1}{5 x}\right)^{11}$.

Question 14

Find the coefficient of $x^{9}$ in the expansion of $\left(2 x^{2}-\frac{1}{x}\right)^{20}$.

Question 15

Find the coefficient of $x^{24}$ in the expansion of $\left(x^{2}-\frac{3 a}{x}\right)^{15}$.

Question 16

Find the coefficient of $x^{9}$ in the expansion of $\left(x^{2}-\frac{1}{3 x}\right)^{9}$.

Question 17

Find the coefficient of $x^{-7}$ in the expansion of $\left(2 x-\frac{1}{3 x^{2}}\right)^{11}$.

Question 18

Find the coefficient of $x^{5}$ in the expansion of :

(i) $(x+3)^{8}$

(ii) $(x+3)^{9}$

Question 19

Find the coefficient of $a^{5} b^{7}$ in $(a-2 b)^{12}$.

Question 20

Find the coefficient of $x^{6} y^{3}$ in (i) $(x+y)^{9}$

(ii) $(x+2 y)^{9}$

Question 21

Find a positive value of $m$ for which the coefficient of $x^{2}$ in the expansion of $(1+x)^{m}$ is 6 

Question 22

Prove that the coefficient of $x^{n}$ in $(1+x)^{2 n}$ is twice the coefficient of $x^{n}$ in $(1+x)^{2 n-1}$

Question 23

Find the coefficient of $x^{7}$ in the expansion of $\left(a x^{2}+\frac{1}{b x}\right)^{11}$ and the coefficient of $x^{-7}$ in the expansion of $\left(a x-\frac{1}{b x^{2}}\right)^{11}$. Also find the relation between $a$ and $b$ so that these coefficients are equal.

Question 24

If the coefficients of $x, x^{2}$ and $x^{3}$ in the binomial expansion $(1+x)^{2 n}$ are in arithmetic progression, then prove that $2 n^{2}-9 n+7=0$.

Question 25

Find the term independent of $x$ in the following Binomial expansion $(x \neq 0)$ :

(i) $\left(x+\frac{1}{x}\right)^{2 n}$

(ii) $\left(x-\frac{1}{x}\right)^{14}$

(iii) $\left(2 x^{2}+\frac{1}{x}\right)^{13}$

Page no 16.42

(iv) $\left(x^{2}+\frac{1}{x}\right)^{12}$

(v) $\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^{2}}\right)^{10}$

(vi) $\left(2 r^{2}-\frac{1}{x}\right)^{12}$

(vii) $\left(2 x^{2}-\frac{3}{x^{3}}\right)^{25}$

(viii) $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{6}$

(ix) $\left(x^{3}-\frac{3}{x^{2}}\right)^{15}$

$(x)\left(x^{2}-\frac{3}{x^{3}}\right)^{10}$

(xi) $\left(\frac{1}{2} x^{\frac{1}{3}}+x^{-\frac{1}{3}}\right)^{8}$

(xii) $\left(x-\frac{1}{x}\right)^{12}$

(xiii) $\left(\sqrt[3]{x}+\frac{1}{2, \sqrt[3]{x}}\right)^{18}$

Question 26

(i) If there is a term independent of $x$ in $\left(x+\frac{1}{x^{2}}\right)^{n}$, show that it is equal to

$\frac{n !}{\left(\frac{n}{3}\right) !\left(\frac{2 n}{3}\right) !}$

(ii) If $x p$ occurs in the expansion of $\left(x^{2}+\frac{1}{x}\right)^{2 n}$, show that its coefficient is

$\frac{(2 n) !}{\left(\frac{4 n-p}{3}\right) !\left(\frac{2 n+p}{3}\right) !}$

Question 27

Find the coefficient of $a^{4}$ in the expansion of the product $(1+2 a)^{4}(2-a)^{5}$.

Type 5

Question 28

If the $r$ th term in the expansion of $(1+x)^{20}$ has its coefficient equal to that of the $(r+4)$ th term, find $r$.

Question 29

(i) If the coefficient of $(2 r+4)$ th term and $(r-2)$ th term in the expansion of $(1+x)^{18}$ are equal, find $r$.

(ii) If the coefficients of $(r-5)$ th and $(2 r-1)$ th term in the expansion of $(1+x)^{34}$ are equal, find $r$.

Question 30

If the coefficient of $(2 r+5) t h$  term and $(r-6)$ th term in the expansion of $(1+x)^{39}$ are equal find ${ }^{r} C_{12}=$

Question 31

Given positive integers $r>1, n>2, n$ being even and the coefficient of (3r)th term and $(r+2)$ th term in the expansion of $(1+x)^{2 n}$ are equal; find $r$.

Question 32

If the coefficient of $(p+1)$ th term in the expansion of $(1+x)^{2 n}$ be equal to that of the $(p+3)$ th term, show that $p=n-1$.

Question 33

Find the two consecutive coefficients in the expansion of  $(3 x-2)^{75}$ Whose values are equal

Question 34

Show that the coefficient of  $(r+1)$ th Term in the expansion of $(1+x)^{n+1}$ is equal to the sum of the coefficients of the r th and $(r+1) t h$ in the expansion of $(1+x)^{n}$

Page no 16.43

Type 6

Question 35

Find the greatest term in the expansion of

(i) $(2+3 x)^{10}$, when $x=\frac{3}{5}$

(ii) $(4-3 x)^{7}$, when $x=\frac{2}{3}$

(iii) $(a+x)^{13}$, when $a=5, x=2$.

Question 36

Find the limits between which $x$ must lie in order that the greatest term in the expansion of $(1+x)^{30}$ may have the greatest coefficient.

Type 7

Question 37

If $n$ is a positive integer, show that

(i) $4^{n}-3 n-1$ is divisible by 9

(ii) $2^{5 n}-31 n-1$ is divisible by 961

(iii) $3^{2 n+2}-8 n-9$ is divisible by 64

(iv) $2^{5 n+6}-3 \ln -32$ is divisible by 961 if $n>1$

(v) $3^{2 n}-1+24 n-32 n^{2}$ is divisible by 512 if $n>2$.

Type 8

Question 38

If three consecutive coefficients in the expansion of $(1+x)^{n}$ be 56,70 and 56 , find $n$ and the position of the coefficients.

Question 39

If three successive coefficients in the expansion of $(1+x)^{n}$ be 220,495 and 972 , find $n$.

Question 40

(i) If $3 \mathrm{rd}, 4$ th, 5 th terms in the expansion of $(a+x)^{n}$ be 84,280 and 560 , find $x, a$ and $n$.

(ii) Find $a, b$ and $n$ in the expansion of $(a+b)^{n}$ if the first three terms of the expansion are $729,7290,30375$ respectively.

Question 41

If the 6 th, 7 th, 8 th terms in the expansion of $(x+y)^{n}$ be 112,7 and $\frac{1}{4}$, find $x, y$ and $n$.

Question 42

The coefficient of the $(r-1)$ th $r$ th and $(r+1)$ th terms in the expansion of $(x+1)^{n}$ are in the ratio $1: 3: 5$. Find both $n$ and $r$.

Question 43

If in any binomial expansion $a, b, c$ and $d$ be the 6 th, 7 th, 8 th and 9 th terms respectively, prove that $\frac{b^{2}-a c}{c^{2}-b d}=\frac{4 a}{3 c}$.

[HOTS]

Question 44

If the four consecutive coefficients in any binomial expansion be $a, b, c, d$, then prove that

(i) $\frac{a+b}{a}, \frac{b+c}{b}, \frac{c+d}{c}$ are in H.P.

(ii) $(b c+a d)(b-c)=2\left(a c^{2}-b^{2} d\right)$.

Question 45

If $a, b, c$ and $d$ are the coefficients of $2 \mathrm{nd}, 3 \mathrm{rd}, 4$ th and 5 th terms respectively in the binomial expansion of $(1+x)^{n}$, then prove that

$\frac{a}{a+b}+\frac{c}{c+d}=\frac{2 b}{b+c}$

Page no 16.44

Question 46

The coefficients of 5 th, 6 th and 7 th terms in the expansion of $(1+x)^{n}$ are in A.P. Find the value of $n$.

Question 47

If the coefficients of second, third and fourth terms in the expansion of $(1+x)^{2}$ are in A.P. show that $2 n^{2}-9 n+7=0$.

Question 48

If the coefficients of $r t h,(r+1)$ th and $(r+2)$ th terms in the expansion of $(1+x)^{n}$ are in A.P. show that $n^{2}-n(4 r+1)+4 r^{2}-2=0$

Question 49

The coefficients of three consecutive terms in the expansion of $(1+x)^{n}$ are in the ratio $182: 84: 30$, prove that $n=18$

Question 50

The sum of the coefficients of the first three terms of the expansion $\left(x-\frac{3}{x^{2}}\right)^{m}$, $x \neq 0, m$ being a natural number, is 559 . Find the term of the expansion containing $x^{3}$.