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 $x^{3}$.
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