Question 1: Find the $11^{th}$ term from the beginning and the $11^{th}$ term from the end in the expansion of  $\Big( 2x-$ $\frac{1}{x^2}$ $\Big)^{25}$.

Given expression: $\Big( 2x-$ $\frac{1}{x^2}$ $\Big)^{25}$

Therefore there are $26$ terms in the expansion.

$11^{th}$ term from the end would be $(26-11+1)$ i.e. $16^{th}$ term from the beginning for expression $\Big( 2x-$ $\frac{1}{x^2}$ $\Big)^{25}$

We know that for $(x+a)^n : T_{r+1} = ^{n} C_{r} (x)^{n-r} a^{r}$

Therefore $T_{16} = T_{15+1} = ^{25} C_{15} (2x)^{25-15} \Big(-$ $\frac{1}{x^2}$ $\Big)^{15}$

$= ^{25} C_{15} (2x)^{10} \Big(-$ $\frac{1}{x^{30}}$ $\Big)$

$= - ^{25} C_{15} \Big($ $\frac{2^{10}}{x^{20}}$ $\Big)$

Also $T_{11} = T_{10+1} = ^{25} C_{10} (2x)^{25-10} \Big(-$ $\frac{1}{x^2}$ $\Big)^{10}$

$= ^{25} C_{10} (2x)^{15} \Big($ $\frac{1}{x^{20}}$ $\Big)$

$= ^{25} C_{10} \Big($ $\frac{2^{15}}{x^{5}}$ $\Big)$

$\\$

Question 2: Find the $7^{th}$ term in the expansion  $\Big( 3x^2-\frac{1}{x^3} \Big)^{10}$.

Given expression: $\Big( 3x^2- \frac{1}{x^3} \Big)^{10}$

We know that for $(x+a)^n : T_{r+1} = ^{n} C_{r} (x)^{n-r} a^{r}$

$T_{7} = T_{6+1} = ^{10} C_{6} (3x^2)^{10-6} \Big(-$ $\frac{1}{x^3}$ $\Big)^{6}$

$= ^{10} C_{6} \Big($ $\frac{3^4 x^8}{x^{18}}$ $\Big)$

$= ^{10} C_{6} \Big($ $\frac{3^4 }{x^{10}}$ $\Big)$

$=$ $\frac{10 \cdot 9 \cdot 8 \cdot 7 }{4 \cdot 3 \cdot 2 \cdot 1 }$ $\times \Big($ $\frac{3^4}{x^{10}}$ $\Big)$

$=$ $\frac{17010}{x^{10}}$

$\\$

Question 3: Find the $5^{th}$ terms from the end in the expansion of $\Big( 3x-$ $\frac{1}{x^2}$ $\Big)^{10}$.

Given expression: $\Big( 3x-$ $\frac{1}{x^2}$ $\Big)^{10}$

Therefore there are $11$ terms in the expansion.

$5^{th}$ term from the end would be $(11-5+1)$ i.e. $7^{th}$ term from the beginning for expression $\Big( 3x-$ $\frac{1}{x^2}$ $\Big)^{10}$

$T_7 = T_{6+1} = ^{10} C_6 (3x)^{10-6} \Big($ $\frac{-1}{x^2}$ $\Big)^6$

$=$ $\frac{10 \cdot 9 \cdot 8 \cdot 7 }{4 \cdot 3 \cdot 2 \cdot 1 }$ $3^4 \cdot x^4 \cdot$ $\frac{1}{x^{12}}$

$=$ $\frac{17010}{x^8}$

$\\$

Question 4: Find the $8^{th}$ term in the expansion of $( x^{\frac{3}{2}} y^{\frac{1}{2}} - x^{\frac{1}{2}} y^{\frac{3}{2}} )^{10}$.

Given expression: $( x^{\frac{3}{2}} y^{\frac{1}{2}} - x^{\frac{1}{2}} y^{\frac{3}{2}} )^{10}$.

$T_8 = T_{7+1} = ^{10} C_7$ $( x^{\frac{3}{2}} y^{\frac{1}{2}})^{10-7} ( - x^{\frac{1}{2}} y^{\frac{3}{2}} )^7$

$= - \Big($ $\frac{10 \cdot 9 \cdot 8 }{3 \cdot 2 \cdot 1}$ $\Big)$ $x^{\frac{9}{2}} \cdot y^{\frac{3}{2}} \cdot x^{\frac{7}{2}} \cdot y^{\frac{21}{2}}$

$= - 120 x^8y^{12}$

$\\$

Question 5: Find the $7^{th}$ term in the expansion of  $\Big($ $\frac{4x}{5}$ $+$ $\frac{5}{2x}$ $\Big)^8$.

Given expression: $\Big($ $\frac{4x}{5}$ $+$ $\frac{5}{2x}$ $\Big)^8$

$T_7 = T_{6+1} = ^8 C_6 \Big($ $\frac{4x}{5}$ $\Big)^{8-6} \Big($ $\frac{5}{2x}$ $\Big)^{6}$

$=$ $\frac{8 \cdot 7}{2 \cdot 1}$ $\times$ $\frac{4^2 x^2}{5^2}$ $\times$ $\frac{5^6}{2^6 x^6}$

$=$ $\frac{4375}{x^4}$

$\\$

Question 6: Find the $4^{th}$ term in the beginning  and the $4^{th}$ term from the end in the expansion $\Big( x +$ $\frac{2}{x}$ $\Big)^9$.

Given expression: $\Big( x +$ $\frac{2}{x}$ $\Big)^9$

Therefore there are $10$ terms in the expansion.

$4^{th}$ term from the end would be $(10-4+1)$ i.e. $7^{th}$ term from the beginning for expression $\Big( x +$ $\frac{2}{x}$ $\Big)^9$

$T_7 = T_{6+1} =^9 C_6 ( x)^{9-6} \Big($ $\frac{2}{x}$ $\Big)^6$

$=$ $\frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1}$ $\cdot x^3\cdot$ $\frac{2^6}{x^6}$

$=$ $\frac{5376}{x^3}$

$T_4 = T_{3+1} =^9 C_3 ( x)^{9-3} \Big($ $\frac{2}{x}$ $\Big)^3$

$=$ $\frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1}$ $\cdot x^6\cdot$ $\frac{2^3}{x^3}$

$= 672x^3$

$\\$

Question 7: Find the $4^{th}$ term from the end in the expansion of $\Big($ $\frac{4x}{5}$ $-$ $\frac{5}{2x}$ $\Big)^9$.

Given expression: $\Big($ $\frac{4x}{5}$ $-$ $\frac{5}{2x}$ $\Big)^9$

Therefore there are $10$ terms in the expansion.

$4^{th}$ term from the end would be $(10-4+1)$ i.e. $7^{th}$ term from the beginning for expression $\Big($ $\frac{4x}{5}$ $-$ $\frac{5}{2x}$ $\Big)^9$

$T_7 = T_{6+1} = ^9 C_6 \Big($ $\frac{4x}{5}$ $\Big)^{9-6} \Big($ $\frac{-5}{2x}$ $\Big)^{6}$

$=$ $\frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1}$ $\cdot$ $\frac{4^3x^3}{5^3}$ $\cdot$ $\frac{5^6}{2^6x^6}$

$=$ $\frac{10500}{x^3}$

$\\$

Question 8: Find the $7^{th}$ term from the end in the expansion of $\Big( 2x^2 -$ $\frac{3}{2x}$ $\Big)^8$.

Given expression: $\Big( 2x^2 -$ $\frac{3}{2x}$ $\Big)^8$

Therefore there are $10$ terms in the expansion.

$7^{th}$ term from the end would be $(9-7+1)$ i.e. $3^{rd}$ term from the beginning for expression $\Big( 2x^2 -$ $\frac{3}{2x}$ $\Big)^8$

$T_3 = T_{2+1} = ^8 C_2 (2x^2)^{8-2} \Big($ $\frac{-3}{2x}$ $\Big)^2$

$=$ $\frac{8 \cdot 7 }{2 \cdot 1}$ $\cdot 2^6 x^{12} \cdot \Big($ $\frac{3^2}{2^2x^2}$ $\Big)$

$= 4032 x^{10}$

$\\$

Question 9: Find the coefficient of:

i) $x^{10}$ in the expansion of $\Big( 2x^2 -$ $\frac{1}{x}$ $\Big)^{20}$

ii) $x^{7}$ in the expansion of $\Big( x -$ $\frac{1}{x^2}$ $\Big)^{40}$

iii) $x^{-15}$ in the expansion of $\Big( 3x^2 -$ $\frac{a}{3x^3}$ $\Big)^{10}$

iv) $x^{9}$ in the expansion of $\Big( x^2 -$ $\frac{1}{3x}$ $\Big)^{9}$

v) $x^{m}$ in the expansion of $\Big( x +$ $\frac{1}{x}$ $\Big)^{n}$

vi) $x$ in the expansion of $( 1- 2x^3 + 3x^5) \Big( 1+$ $\frac{1}{x}$ $\Big)^{8}$

vii) $a^5b^7$ in the expansion of $(a-2b)^{12}$

viii) $x$ in the expansion of $( 1 - 3x + 7x^2)( 1 - x)^{16}$

i) The given expression: $\Big( 2x^2 -$ $\frac{1}{x}$ $\Big)^{20}$.

Let $(r+1)^{th}$ term has $x^{10}$

$T_{r+1} = ^{20} C_{r} \cdot (2x^2)^{20-r} \cdot \Big($ $\frac{-1}{x}$ $\Big)^r$

$= ^{20} C_{r} \cdot 2^{20-r} \cdot x^{40-2r} \cdot \Big($ $\frac{-1^r}{x^r}$ $\Big)$

$= (-1)^r \cdot {^{20} C_{r}} \cdot 2^{20-r} \cdot x^{40-3r}$

Therefore $40-3r= 10 \Rightarrow r = 10$

Therefore coefficient of $x^{10} = (-1)^{10} \cdot {^{20} C_{10}} \cdot 2^{20-10} = ^{20} C_{10} \cdot 2^{10}$

$\\$

ii) The given expression: $\Big( x -$ $\frac{1}{x^2}$ $\Big)^{40}$.

Let $(r+1)^{th}$ term has $x^{7}$

$T_{r+1} = ^{40} C_{r} \cdot (x)^{40-r} \cdot \Big($ $\frac{-1}{x^2}$ $\Big)^r$

$= (-1)^r {^{40} C_{r}} \cdot \Big($ $\frac{x^{40-r}}{x^{2r}}$ $\Big)$

$= (-1)^r \cdot {^{40} C_{r}} \cdot x^{40-3r}$

Therefore $40-3r= 7 \Rightarrow r = 11$

Therefore coefficient of $x^{7} = (-1)^{11} \cdot {^{40} C_{11}} = - ^{40} C_{11}$

$\\$

iii) The given expression: $\Big( 3x^2 -$ $\frac{a}{3x^3}$ $\Big)^{10}$

Let $(r+1)^{th}$ term has $x^{-15}$

$T_{r+1} = ^{10} C_{r} \cdot (3x^2)^{10-r} \cdot \Big($ $\frac{-a}{3x^3}$ $\Big)^r$

$= (-1)^r {^{10} C_{r}} \cdot 3^{10-r} x^{20-2r} \Big($ $\frac{a^r}{3^r x^{3r}}$ $\Big)$

$= (-1)^r \cdot {^{10} C_{r}} 3^{10-2r} a^r \cdot x^{20-5r}$

Therefore $20-5r= -15 \Rightarrow r = 7$

Therefore coefficient of $x^{7} = (-1)^{7} \cdot {^{10} C_{7}} 3^{10-14} a^7 = - ^{10} C_{7}$ $\frac{a^7}{3^4}$ $=$ $\frac{-40}{9}$ $a^7$

$\\$

iv) The given expression: $\Big( x^2 -$ $\frac{1}{3x}$ $\Big)^{9}$.

Let $(r+1)^{th}$ term has $x^{9}$

$T_{r+1} = ^{9} C_{r} \cdot (x^2)^{9-r} \cdot \Big($ $\frac{-1}{3x}$ $\Big)^r$

$= (-1)^r \cdot {^{9} C_{r}} \cdot x^{18-2r} \Big($ $\frac{1}{3^r x^{r}}$ $\Big)$

$= (-1)^r \cdot {^{9} C_{r}} \cdot$ $\frac{1}{3^r}$ $\cdot x^{18-3r}$

Therefore $18-3r= 9 \Rightarrow r = 3$

Therefore coefficient of $x^{9} = (-1)^{3} \cdot {^{9} C_{3}} \cdot$ $\frac{1}{3^3}$ $= (-1) \cdot$ $\frac{9 \cdot 8 \cdot 7 }{3 \cdot 2 \cdot 1 }$ $\cdot$ $\frac{1}{3^3}$ $=$ $\frac{-28}{9}$

$\\$

v) The given expression: $\Big( x +$ $\frac{1}{x}$ $\Big)^{n}$.

Let $(r+1)^{th}$ term has $x^{m}$

$T_{r+1} = ^{n} C_{r} \cdot (x)^{n-r} \cdot \Big($ $\frac{1}{x}$ $\Big)^r$

$= {^{n} C_{r}} \cdot x^{n-2r}$

Therefore $n-2r= m \Rightarrow r =$ $\frac{n-m}{2}$

Therefore coefficient of $x^{m} = ^{n} C_{\frac{n-m}{2}} =$ $\frac{n!}{(\frac{n-m}{2})! (\frac{n+m}{2})!}$

$\\$

vi) Given expression: $( 1- 2x^3 + 3x^5) \Big( 1+$ $\frac{1}{x}$ $\Big)^{8}$

$= ( 1- 2x^3 + 3x^5) \Big[ ^8 C_0 \Big( \frac{1}{x} \Big)^0 + ^8 C_1 \Big( \frac{1}{x} \Big)^1 +^8 C_2 \Big( \frac{1}{x} \Big)^2 +^8 C_3 \Big( \frac{1}{x} \Big)^3 +^8 C_4 \Big( \frac{1}{x} \Big)^4 \\ \\ { \hspace{5.0cm}+^8 C_5 \Big( \frac{1}{x} \Big)^5 +^8 C_6 \Big( \frac{1}{x} \Big)^6 +^8 C_7 \Big( \frac{1}{x} \Big)^7 + ^8 C_8 \Big( \frac{1}{x} \Big)^8} \Big]$

$= ( 1- 2x^3 + 3x^5) \Big[ ^8 C_0 + ^8 C_1 \Big( \frac{1}{x} \Big) +^8 C_2 \Big( \frac{1}{x^2} \Big) +^8 C_3 \Big( \frac{1}{x^3} \Big) +^8 C_4 \Big( \frac{1}{x^4} \Big) \\ \\ { \hspace{5.0cm}+^8 C_5 \Big( \frac{1}{x^5} \Big) +^8 C_6 \Big( \frac{1}{x^6} \Big) +^8 C_7 \Big( \frac{1}{x^7} \Big) + ^8 C_8 \Big( \frac{1}{x^8} \Big)} \Big]$

Therefore terms with $x = -2x^3 \cdot ^8 C_2 \cdot$ $\frac{1}{x^2}$ $+ 3x^5 \cdot ^8 C_4 \cdot$ $\frac{1}{x^4}$

$= -2 \cdot ^8 C_2 \cdot x + 3 \cdot ^8 C_4 \cdot x$

$= \Big( -2 \cdot$ $\frac{8 \cdot 7}{2 \cdot 1}$ $+ 3 \cdot$ $\frac{8 \cdot7 \cdot 6 \cdot 5}{4 \cdot 3 \cdot 2 \cdot 1}$ $\Big) x$

$= (-56+210) x$

$= 154x$

Hence the coefficient of the term with $x = 154$

$\\$

vii) Given expression: $(a-2b)^{12}$

$= ^{12} C_0 (a)^{12} (-2b)^0 + ^{12} C_1 (a)^{11} (-2b)^1 + ^{12} C_2 (a)^{10} (-2b)^2 \\ \\ { \hspace{3.0cm} + ^{12} C_3 (a)^{9} (-2b)^3 + ^{12} C_4 (a)^{8} (-2b)^4 + ^{12} C_5 (a)^{7} (-2b)^5} \\ \\ {\hspace{3.0cm} + ^{12} C_6 (a)^{6} (-2b)^6 + ^{12} C_7 (a)^{5} (-2b)^7 + \ldots }$

Therefore coefficient of $a^5b^7 = ^{12} C_7 (-2)^7 = -$ $\frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }$ $\cdot 2^7 = - 101376$

$\\$

viii) Given expression: $( 1 - 3x + 7x^2)( 1 - x)^{16}$

$= ( 1 - 3x + 7x^2) [ ^{16} C_0 (-x)^0 + ^{16} C_1 (-x)^1 + \ldots ]$

$= ( 1 - 3x + 7x^2) [ 1 + 16(-x) + \ldots ]$

Therefore coefficient of $x = -3 + ( -16) = - 19$

$\\$

Question 10: Which term in the expansion of $\Big\{ \Big( \frac{x}{\sqrt{y}} \Big)^{\frac{1}{3}} + \Big( \frac{y}{x^{\frac{1}{3}}} \Big)^{\frac{1}{2}} \Big\}^{21}$ contains $x$ and $y$ to one and the same power?

Given expression: $\Big\{ \Big( \frac{x}{\sqrt{y}} \Big)^{\frac{1}{3}} + \Big( \frac{y}{x^{\frac{1}{3}}} \Big)^{\frac{1}{2}} \Big\}^{21}$

$T_{r+1} = ^{21} C_r \cdot$ $\Big[ \Big( \frac{x}{\sqrt{y}} \Big)^{\frac{1}{3}} \Big]^{21-r} \cdot \Big[ \Big( \frac{y}{x^{\frac{1}{3}}} \Big)^{\frac{1}{2}} \Big]^{r}$

$= ^{21} C_r \cdot$ $\Big[ \frac{x^{\frac{21-r}{3}}}{y^{\frac{21-r}{6}}} \Big] \cdot \Big[ \frac{y^{\frac{r}{2}}}{x^{\frac{r}{6}}} \Big]$

$= ^{21} C_r \cdot$ $\frac{ x^{\frac{21-r}{3} - \frac{r}{6}}}{y^{\frac{21-r}{6} - \frac{r}{2}}}$

$= ^{21} C_r \cdot$ $\frac{x^{\frac{42-3r}{6}}}{y^{\frac{21-4r}{6}}}$

$= ^{21} C_r \cdot$ $x^{7 - \frac{r}{2}} \cdot y^{ \frac{2}{3} r - \frac{7}{2}}$

Therefore $7 -$ $\frac{r}{2} = \frac{2}{3} r - \frac{7}{2}$

$\frac{21}{2} = \frac{7r}{6}$  $\Rightarrow r = 9$

Therefore the required term $= 10^{th}$ term.

$\\$

Question 11: Does the expansion of $\Big( 2x^2 -$ $\frac{1}{x}$ $\Big)^{20}$ contain any term involving $x^{9}$.

General expression: $\Big( 2x^2 -$ $\frac{1}{x}$ $\Big)^{20}$

Suppose $x^9$ occurs in the given expression at the $( r+1)^{th}$ term.

$T_{r+1}= ^{20} C_r \cdot (2x^2)^{20-r} \cdot \Big($ $\frac{-1}{x}$ $\Big)^r$

$= (-1)^r \cdot {^{20} C_r} \cdot 2^{20-r} \cdot x^{40-2r-r}$

Fr the term to contain $x^9$,

$40-3r = 9 \Rightarrow r =$ $\frac{31}{3}$

Since $r$ can only be an integer, this is not possible. Hence there is no term in the expansion of $\Big( 2x^2 -$ $\frac{1}{x}$ $\Big)^{20}$ that contains $x^9$.

$\\$

Question 12: Show that the expansion of  $\Big( x^2 +$ $\frac{1}{x}$ $\Big)^{12}$ does not contain any term involving $x^{-1}$.

Given expression: $\Big( x^2 +$ $\frac{1}{x}$ $\Big)^{12}$

Suppose $x^{-1}$ occurs in the given expression at the $( r+1)^{th}$ term.

$T_{r+1}= ^{12} C_r \cdot (x^2)^{12-r} \cdot \Big($ $\frac{1}{x}$ $\Big)^r$

$= ^{12} C_r \cdot x^{24-2r-r}$

For the term to contain $x^{-1}$,

$24-3r = -1 \Rightarrow r =$ $\frac{25}{3}$

Since $r$ can only be an integer, this is not possible. Hence there is no term in the expansion of $\Big( x^2 +$ $\frac{1}{x}$ $\Big)^{12}$.

$\\$

Question 13: Find the middle term in the expansion of:

i) $\Big($ $\frac{2x}{3}$ $-$ $\frac{3}{2x}$ $\Big)^{20}$     ii)  $\Big($ $\frac{a}{x}$ $+ bx \Big)^{12}$     iii)  $\Big( x^2 -$ $\frac{2}{x}$ $\Big)^{10}$     iv) $\Big($ $\frac{x}{a}$ $-$ $\frac{a}{x}$ $\Big)^{10}$

i) $\Big($ $\frac{2x}{3}$ $-$ $\frac{3}{2x}$ $\Big)^{20}$

Here $n = 20$ is even. Therefore $\Big($ $\frac{20}{2}$ $+1\Big)$ i.e. $11^{th}$ term is the middle term.

$T_{11} = T_{10+1} = ^{20} C_{10} \cdot \Big($ $\frac{2}{3}$ $x \Big)^{20-10} \cdot \Big($ $\frac{-3}{2x}$ $\Big)^{10}$

$= ^{20} C_{10} \cdot \Big($ $\frac{2}{3}$ $\Big)^{10} x^{10} \cdot \Big($ $\frac{3}{2}$ $\Big)^{10}$ $\frac{1}{x^{10}}$

$= ^{20} C_{10}$

ii)  $\Big($ $\frac{a}{x}$ $+ bx \Big)^{12}$

Here $n = 12$ is even. Therefore $\Big($ $\frac{12}{2}$ $+1\Big)$ i.e. $7^{th}$ term is the middle term.

$T_{7} = T_{6+1} = ^{12} C_{6} \cdot \Big($ $\frac{a}{x}$ $\Big)^{12-6} \cdot (bx)^6$

$= ^{12} C_{6} a^6b^6$

$= 924 a^6b^6$

iii)  $\Big( x^2 -$ $\frac{2}{x}$ $\Big)^{10}$

Here $n = 10$ is even. Therefore $\Big($ $\frac{10}{2}$ $+1\Big)$ i.e. $6^{th}$ term is the middle term.

$T_{6} = T_{5+1} = ^{10} C_{5} \cdot (x^2)^{10-5} \cdot \Big($ $\frac{-2}{x}$ $\Big)^{5}$

$= - ^{10} C_{5} \cdot x^{10} \cdot \Big($ $\frac{2^5}{x^5}$ $\Big)$

$= -$ $\frac{ 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ $\times 2^5 \times x^5$

$= -8064 x^5$

iv) $\Big($ $\frac{x}{a}$ $-$ $\frac{a}{x}$ $\Big)^{10}$

Here $n = 10$ is even. Therefore $\Big($ $\frac{10}{2}$ $+1\Big)$ i.e. $6^{th}$ term is the middle term.

$T_{6} = T_{5+1} = ^{10} C_{5} \cdot \Big($ $\frac{x}{a}$ $\Big)^{10-5} \cdot \Big($ $\frac{-a}{x}$ $\Big)^{5}$

$= - ^{10} C_{5} \cdot \Big($ $\frac{x^5}{a^5}$ $\Big) \cdot \Big($ $\frac{a^5}{x^5}$ $\Big)$

$= - ^{10} C_{5} = -252$

$\\$

Question 14: Find the middle term in the expansion of:

i) $\Big( 3x -$ $\frac{x^3}{6}$ $\Big)^{9}$     ii)  $\Big( 2x^2-$ $\frac{1}{x}$ $\Big)^{7}$     iii)  $\Big( 3x -$ $\frac{2}{x^2}$ $\Big)^{15}$     iv) $\Big( x^4 -$ $\frac{1}{x^3}$ $\Big)^{11}$

i) Given expression: $\Big( 3x -$ $\frac{x^3}{6}$ $\Big)^{9}$

Here $n = 9$ is odd. Therefore $\Big($ $\frac{9+1}{2}$ $\Big)$ and $\Big($ $\frac{9+1}{2}$ $+ 1 \Big)$ i.e. $5^{th}$ and $6^{th}$ terms are the middle term of the given expression.

$T_{5} = T_{4+1} = ^{9} C_{4} \cdot (3x)^{9-4} \cdot \Big($ $\frac{-x^3}{6}$ $\Big)^4$

$= ^{9} C_{4} \cdot 3^5 \cdot x^5 \cdot$ $\frac{x^{12}}{6^4}$

$=$ $\frac{9 \cdot 8 \cdot 7 \cdot 6}{ 4 \cdot 3 \cdot 2 \cdot 1}$ $\times$ $\frac{3^5}{6^4}$ $\times x^{17}$

$=$ $\frac{189}{8}$ $x^{17}$

$T_{6} = T_{5+1} = ^{9} C_{5} \cdot (3x)^{9-5} \cdot \Big($ $\frac{-x^3}{6}$ $\Big)^5$

$= - ^{9} C_{5} \cdot 3^4 \cdot x^4 \cdot$ $\frac{x^{15}}{6^5}$

$= -$ $\frac{9 \cdot 8 \cdot 7 \cdot 6}{ 4 \cdot 3 \cdot 2 \cdot 1}$ $\times$ $\frac{3^4}{6^5}$ $\times x^{19}$

$= -$ $\frac{21}{16}$ $x^{19}$

ii)  Given expression: $\Big( 2x^2-$ $\frac{1}{x}$ $\Big)^{7}$

Here $n = 7$ is odd. Therefore $\Big($ $\frac{7+1}{2}$ $\Big)$ and $\Big($ $\frac{7+1}{2}$ $+ 1 \Big)$ i.e. $4^{th}$ and $5^{th}$ terms are the middle term of the given expression.

$T_{4} = T_{3+1} = ^{7} C_{3} \cdot (2x^2)^{7-3} \cdot \Big($ $\frac{-1}{x}$ $\Big)^3$

$= - ^{7} C_{3} \cdot 2^4 \cdot x^8 \cdot$ $\frac{1}{x^3}$

$= -$ $\frac{7 \cdot 6 \cdot 5}{ 3 \cdot 2 \cdot 1}$ $\times 2^4 \times x^5$

$= - 560x^5$

$T_{5} = T_{4+1} = ^{7} C_{4} \cdot (2x^2)^{7-4} \cdot \Big($ $\frac{-1}{x}$ $\Big)^4$

$= ^{7} C_{4} \cdot 2^3 \cdot x^6 \cdot$ $\frac{1}{x^4}$

$=$ $\frac{7 \cdot 6 \cdot 5}{ 3 \cdot 2 \cdot 1}$ $\times 2^3 \times x^2$

$= 280x^2$

iii)  Given expression: $\Big( 3x -$ $\frac{2}{x^2}$ $\Big)^{15}$

Here $n = 15$ is odd. Therefore $\Big($ $\frac{15+1}{2}$ $\Big)$ and $\Big($ $\frac{15+1}{2}$ $+ 1 \Big)$ i.e. $8^{th}$ and $9^{th}$ terms are the middle term of the given expression.

$T_{8} = T_{7+1} = ^{15} C_{7} \cdot (3x)^{15-7} \cdot \Big($ $\frac{-2}{x^2}$ $\Big)^7$

$= - ^{15} C_{7} \cdot 3^8 \cdot x^8 \cdot$ $\frac{2^7}{x^{14}}$

$= -$ $\frac{15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 }{ 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ $\times 3^8 \times 2^7 \times$ $\frac{1}{x^6}$

$= -$ $\frac{6435 \times 3^8 \times 2^7}{x^6}$

$T_{9} = T_{8+1} = ^{15} C_{8} \cdot (3x)^{15-8} \cdot \Big($ $\frac{-2}{x^2}$ $\Big)^8$

$= ^{15} C_{8} \cdot 3^7 \cdot x^7 \cdot$ $\frac{2^8}{x^{16}}$

$=$ $\frac{15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 }{ 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ $\times 3^7 \times 2^8 \times$ $\frac{1}{x^9}$

$=$ $\frac{6435 \times 3^7 \times 2^8}{x^9}$

iv) Given expression: $\Big( x^4 -$ $\frac{1}{x^3}$ $\Big)^{11}$

Here $n = 11$ is odd. Therefore $\Big($ $\frac{11+1}{2}$ $\Big)$ and $\Big($ $\frac{11+1}{2}$ $+ 1 \Big)$ i.e. $6^{th}$ and $7^{th}$ terms are the middle term of the given expression.

$T_{6} = T_{5+1} = ^{11} C_{5} \cdot (x^4)^{11-5} \cdot \Big($ $\frac{-1}{x^3}$ $\Big)^5$

$= - ^{11} C_{5} \cdot x^{24} \cdot$ $\frac{1}{x^{15}}$

$= -$ $\frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{ 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ $\times x^9$

$= - 462x^9$

$T_{7} = T_{6+1} = ^{11} C_{6} \cdot (x^4)^{11-6} \cdot \Big($ $\frac{-1}{x^3}$ $\Big)^6$

$= ^{11} C_{6} \cdot x^{20} \cdot$ $\frac{1}{x^{18}}$

$=$ $\frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{ 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ $x^2$

$= 462x^2$

$\\$

Question 15: Find the middle term in the expansion of:

i) $\Big( x -$ $\frac{1}{x}$ $\Big)^{10}$     ii)  $(1-2x+x^2)^n$     iii)  $(1+3x+3x^2+x^3)^{2n}$

iv) $\Big( 2x -$ $\frac{x^2}{4}$ $\Big)^{9}$     v) $\Big( x -$ $\frac{1}{x}$ $\Big)^{2n+1}$      vi) $\Big($ $\frac{x}{3}$ $+9y \Big)^{10}$

vii) $\Big( 3 -$ $\frac{x^3}{6}$ $\Big)^{7}$      viii) $\Big( 2ax -$ $\frac{b}{x^2}$ $\Big)^{12}$      ix) $\Big($ $\frac{p}{x}$ $+$ $\frac{x}{p}$ $\Big)^{9}$     x) $\Big($ $\frac{x}{a}$ $-$ $\frac{a}{x}$ $\Big)^{10}$

i) Given expression: $\Big( x -$ $\frac{1}{x}$ $\Big)^{10}$

Here $n = 10$ is even. Therefore $\Big($ $\frac{10}{2}$ $+1\Big)$ i.e. $6^{th}$ term is the middle term.

$T_6 = T_{5+1} = ^{10} C_5 \cdot (x)^{10-5} \cdot \Big($ $\frac{-1}{x}$ $\Big)^5$

$= - ^{10} C_5 \cdot$ $\frac{x^5}{x^5}$

$= -$ $\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 }{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

$= - 252$

ii) Given expression: $(1-2x+x^2)^n = (1-x)^{2n}$

Here $2n$ is even. Therefore $\Big($ $\frac{2n}{2}$ $+1\Big)$ i.e. $(n+1)^{th}$ term is the middle term.

$T_{n+1} = ^{2n} C_n \cdot (1)^{2n-n} \cdot ( -x )^n$

$= (-1)^n \cdot {^{2n} C_n} \cdot x^n$

$= (-1)^n$ $\frac{(2n)!}{(n!)^2}$ $x^n$

iii) Given expression:   $(1+3x+3x^2+x^3)^{2n} = {(1+x)^3}^{2n} = (1+x)^{6n}$

Here $6n$ is even. Therefore $\Big($ $\frac{6n}{2}$ $+1\Big)$ i.e. $(3n+1)^{th}$ term is the middle term.

$T_{3n+1} = ^{6n} C_{3n} \cdot (1)^{6n-3n} \cdot (x)^{3n}$

$= ^{6n} C_{3n} \cdot (x)^{3n}$

$=$ $\frac{(6n)!}{[(3n)!]^2}$ $(x)^{3n}$

iv) Given expression: $\Big( 2x -$ $\frac{x^2}{4}$ $\Big)^{9}$

Here $n = 9$ is odd. Therefore $\Big($ $\frac{9+1}{2}$ $\Big)$ and $\Big($ $\frac{9+1}{2}$ $+ 1 \Big)$ i.e. $5^{th}$ and $6^{th}$ terms are the middle term of the given expression.

$T_5 = T_{4+1} = ^{9} C_4 \cdot (2x)^{9-4} \cdot \Big($ $\frac{-x^2}{4}$ $\Big)^4$

$= ^{9} C_4 \cdot 2^5 x^5 \cdot$ $\frac{x^8}{4^4}$

$=$ $\frac{9 \cdot 8\cdot 7\cdot 6}{4\cdot 3\cdot 2\cdot 1}$ $\times$ $\frac{2^5}{4^4}$ $x^{13}$

$=$ $\frac{63}{4}$ $x^{13}$

$T_6 = T_{5+1} = ^{9} C_5 \cdot (2x)^{9-5} \cdot \Big($ $\frac{-x^2}{4}$ $\Big)^5$

$= - ^{9} C_5 \cdot 2^4 x^4 \cdot$ $\frac{x^{10}}{4^5}$

$= -$ $\frac{9\cdot 8\cdot 7\cdot 6}{4\cdot 3\cdot 2 \cdot 1}$ $\times$ $\frac{2^4}{4^5}$ $x^{14}$

$= -$ $\frac{63}{32}$ $x^{14}$

v) Given expression: $\Big( x -$ $\frac{1}{x}$ $\Big)^{2n+1}$

Here $n = (2n+1)$ is odd. Therefore $\Big($ $\frac{2n+1+1}{2}$ $\Big)$ and $\Big($ $\frac{2n+1+1}{2}$ $+ 1 \Big)$ i.e. $(n+1)^{th}$ and $(n+2)^{th}$ terms are the middle term of the given expression.

$T_{n+1} = ^{2n+1} C_n \cdot (x)^{2n+1-n} \cdot \Big($ $\frac{-1}{x}$ $\Big)^n$

$= {^{2n+1} C_n} \cdot x^{n+1} \cdot$ $\frac{(-1)^n}{x^n}$

$= (-1)^n \cdot {^{2n+1} C_n} x$

$T_{n+2} = T_{n+1+1} = ^{2n+1} C_{n+1} \cdot (x)^{2n+1-(n+1)} \cdot \Big($ $\frac{-1}{x}$ $\Big)^{n+1}$

$= (-1)^{n+1} \cdot {^{2n+1} C_n} \cdot x^{n} \cdot$ $\frac{1}{x^{n+1}}$

$= (-1)^{n+1} \cdot {^{2n+1} C_n} \cdot$ $\frac{1}{x}$

vi) Given expression: $\Big( \frac{x}{3}+9y \Big)^{10}$

Here $n = 10$ is even. Therefore $\Big($ $\frac{10}{2}$ $+1\Big)$ i.e. $6^{th}$ term is the middle term.

$T_6 = T_{5+1} = ^{10} C_5 \cdot \Big($ $\frac{x}{3}$ $\Big)^{10-5} \cdot (9y)^5$

$= ^{10} C_5 \cdot$ $\frac{x^5}{3^5}$ $\cdot 9^5 \cdot y^5$

$= -$ $\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 }{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot \frac{x^5}{3^5}$ $\cdot 9^5 \cdot y^5$

$= 61236 x^5y^5$

vii) Given expression: $\Big( 3 -$ $\frac{x^3}{6}$ $\Big)^{7}$

Here $n = 7$ is odd. Therefore $\Big($ $\frac{7+1}{2}$ $\Big)$ and $\Big($ $\frac{7+1}{2}$ $+ 1 \Big)$ i.e. $4^{th}$ and $5^{th}$ terms are the middle term of the given expression.

$T_4 = T_{3+1} = ^{7} C_7 \cdot (3)^{7-3} \cdot \Big($ $\frac{-x^3}{6}$ $\Big)^3$

$= (-1) {^{7} C_3} \cdot 3^4 \cdot$ $\frac{x^9}{6^3}$

$= -$ $\frac{105}{8}$ $x^{9}$

$T_5 = T_{4+1} = ^{7} C_4 \cdot (3)^{7-4} \cdot \Big($ $\frac{-x^3}{6}$ $\Big)^4$

$= {^{7} C_4} \cdot 3^3 \cdot$ $\frac{x^{12}}{6^4}$

$=$ $\frac{35}{48}$ $x^{12}$

viii) Given expression: $\Big( 2ax -$ $\frac{b}{x^2}$ $\Big)^{12}$

Here $n = 12$ is even. Therefore $\Big($ $\frac{12}{2}$ $+1\Big)$ i.e. $7^{th}$ term is the middle term.

$T_7 = T_{6+1} = ^{12} C_6 \cdot (2ax)^{12-6} \cdot \Big($ $\frac{-b}{x^2}$ $\Big)^6$

$= ^{12} C_6 \cdot 2^6 a^6 x^6 \cdot$ $\frac{b^6}{x^{12}}$

$= 59136 \Big($ $\frac{a^6 b^6 }{x^6}$ $\Big)$

ix) Given expression: $\Big($ $\frac{p}{x}$ $+$ $\frac{x}{p}$ $\Big)^{9}$

Here $n = 9$ is odd. Therefore $\Big($ $\frac{9+1}{2}$ $\Big)$ and $\Big($ $\frac{9+1}{2}$ $+ 1 \Big)$ i.e. $5^{th}$ and $6^{th}$ terms are the middle term of the given expression.

$T_5 = T_{4+1} = ^{9} C_4 \cdot \Big($ $\frac{p}{x}$ $\Big)^{9-4} \cdot \Big($ $\frac{x}{p}$ $\Big)^4$

$= {^{9} C_4} \cdot \Big($ $\frac{p}{x}$ $\Big)^5 \cdot \Big($ $\frac{x}{p}$ $\Big)^4$

$= {^{9} C_4} \Big($ $\frac{p}{x}$ $\Big)$

$= 126 \Big($ $\frac{p}{x}$ $\Big)$

$T_6 = T_{5+1} = ^{9} C_5 \cdot \Big($ $\frac{p}{x}$ $\Big)^{9-5} \cdot \Big($ $\frac{x}{p}$ $\Big)^5$

$= {^{9} C_5} \cdot \Big($ $\frac{p}{x}$ $\Big)^4 \cdot \Big($ $\frac{x}{p}$ $\Big)^5$

$= {^{9} C_4} \cdot \Big($ $\frac{x}{p}$ $\Big)$

$= 126 \Big($ $\frac{x}{p}$ $\Big)$

x) Given expression: $\Big($ $\frac{x}{a}$ $-$ $\frac{a}{x}$ $\Big)^{10}$

Here $n = 10$ is even. Therefore $\Big($ $\frac{10}{2}$ $+1\Big)$ i.e. $6^{th}$ term is the middle term.

$T_6 = T_{5+1} = ^{10} C_5 \cdot \Big($ $\frac{x}{a}$ $\Big)^{10-5} \cdot \Big($ $\frac{-a}{x}$ $\Big)^5$

$= - {^{10} C_5} \cdot \Big($ $\frac{x}{a}$ $\Big)^5 \cdot \Big($ $\frac{a}{x}$ $\Big)^5$

$= - {^{10} C_5}$

$= -252$

$\\$

Question 16: Find the term independent of $x$ in the expansion of the following expressions:

i) $\Big($ $\frac{3}{2}$ $x^2 -$ $\frac{1}{3x}$ $\Big)^{9}$     ii) $\Big( 2x +$ $\frac{1}{3x^2}$ $\Big)^{9}$     iii) $\Big( 2x^2 -$ $\frac{3}{x^3}$ $\Big)^{25}$     iv) $\Big( 3x -$ $\frac{2}{x^2}$ $\Big)^{15}$

v) $\Big($ $\sqrt{\frac{x}{3}}$ $+$ $\frac{\sqrt{3}}{2x^2}$ $\Big)^{10}$     vi) $\Big( x -$ $\frac{1}{x^2}$ $\Big)^{3n}$     vii) $\Big($ $\frac{1}{2}$ $x^{\frac{1}{3}} + x^{-\frac{1}{5}} \Big)^{15}$

viii) $(1 + x + 2x^3) \Big($ $\frac{3}{2}$ $x^2 -$ $\frac{1}{3x}$ $\Big)^{9}$     ix) $\Big( \sqrt[3]{x} +$ $\frac{1}{2}$ $\sqrt[3]{x} \Big)^{18} , x > 2$     x) $\Big($ $\frac{3}{2}$ $x^2 -$ $\frac{1}{3x}$ $\Big)^{6}$

i) Given expression: $\Big($ $\frac{3}{2}$ $x^2 -$ $\frac{1}{3x}$ $\Big)^{9}$

Let $T_{r+1}$ term be independent of $x$

Therefore $T_{r+1} = {^9C_{r}} \cdot \Big($ $\frac{3}{2}$ $x^2 \Big)^{9-r} \cdot \Big($ $\frac{-1}{3x}$ $\Big)^{r}$

$= (-1)^r \cdot {^9C_{r}} \cdot \Big($ $\frac{3}{2}$ $\Big)^{9-r} \cdot \Big($ $\frac{x^{18-2r}}{3^r x^r}$ $\Big)$

If $T_{r+1}$ is to be independent of $x$, then

$18-2r-r=0 \Rightarrow r = 6$ Therefore the $7^{th}$ term.

Therefore the required term $T_{6+1} = (-1)^6 \cdot {^9C_{6}} \cdot \Big($ $\frac{3}{2}$ $\Big)^{9-6} \cdot$ $\frac{1}{3^6}$ $= {^9C_{6}} \cdot \Big($ $\frac{3^3}{2^3 3^6}$ $\Big) =$ $\frac{7}{18}$

$\\$

ii) Given expression: $\Big( 2x +$ $\frac{1}{3x^2}$ $\Big)^{9}$

Let $T_{r+1}$ term be independent of $x$

Therefore $T_{r+1} = {^9C_{r}} \cdot (2x)^{9-r} \cdot \Big($ $\frac{1}{3x^2}$ $\Big)^{r}$

$= {^9C_{r}} \cdot 2^{9-r} \cdot \Big($ $\frac{x^{9-r}}{3^r x^{2r}}$ $\Big)$

If $T_{r+1}$ is to be independent of $x$, then

$9-r-2r=0 \Rightarrow r = 3$ Therefore the $4^{th}$ term.

Therefore the required term $T_{3+1} = ^9 C_3 \cdot \Big($ $\frac{2^6}{3^3}$ $\Big) =$ $\frac{64}{27}$ $\cdot ^9 C_3 =$ $\frac{1792}{9}$

$\\$

iii) Given expression: $\Big( 2x^2 -$ $\frac{3}{x^3}$ $\Big)^{25}$

Let $T_{r+1}$ term be independent of $x$

Therefore $T_{r+1} = {^{25}C_{r}} \cdot (2x^2)^{25-r} \cdot \Big($ $\frac{-3}{x^3}$ $\Big)^{r}$

$= (-3)^r \cdot {^{25}C_{r}} \cdot 2^{25-r} \cdot \Big($ $\frac{x^{50-2r}}{x^{3r}}$ $\Big)$

If $T_{r+1}$ is to be independent of $x$, then

$50-2r-3r=0 \Rightarrow r = 10$ Therefore the ${11}^{th}$ term.

Therefore the required term $T_{10+1} =(-3)^{10} \cdot ^{25}C_{10} \cdot 2^{25-10} = 3^{10} \cdot 2^{15} \cdot ^{25}C_{10}$

$\\$

iv) Given expression: $\Big( 3x -$ $\frac{2}{x^2}$ $\Big)^{15}$

Let $T_{r+1}$ term be independent of $x$

Therefore $T_{r+1} = {^{15}C_{r}} \cdot (3x)^{15-r} \cdot \Big($ $\frac{-2}{x^2}$ $\Big)^{r}$

$= (-1)^r \cdot {^{15}C_{r}} \cdot 3^{15-r} \cdot \Big($ $\frac{x^{15-r}2^r}{x^{2r}}$ $\Big)$

If $T_{r+1}$ is to be independent of $x$, then

$15-r-2r=0 \Rightarrow r = 5$ Therefore the ${6}^{th}$ term.

Therefore the required term $T_{5+1} = ((-1)^5 \cdot {^{15}C_{5}} \cdot 3^{15-5} \cdot 2^5 = - ^{15} C_5 \cdot 3^{10} \cdot 2^5 = - 3003 \cdot 3^{10} \cdot 2^5$

$\\$

v) Given expression: $\Big($ $\sqrt{\frac{x}{3}}$ $+$ $\frac{\sqrt{3}}{2x^2}$ $\Big)^{10}$

Let $T_{r+1}$ term be independent of $x$

Therefore $T_{r+1}= ^{10} C_{r} \cdot$ $( \sqrt{\frac{x}{3} })^{10-r} \cdot ( \frac{ \sqrt{3} }{2x^2} )^r$

$= {^{10}C_{r}} \cdot \Big($ $\frac{x^{\frac{10-r}{2} }}{3^{\frac{10-r}{2}} } \cdot \frac{ 3^{ \frac{r}{2} } }{2^r x^{2r}}$ $\Big)$

If $T_{r+1}$ is to be independent of $x$, then

$\frac{10-r}{2}$ $-2r =0 \Rightarrow 10-r-4r = 0 \Rightarrow r = 2$ Therefore the ${3}^{rd}$ term.

Therefore the required term $T_{3} = {^{10}C_{2}} \Big($ $\frac{3^{ \frac{2}{2} }}{2^2 \cdot 3^{\frac{10-2}{2} }}$ $\Big) = {^{10} C_{2}} \Big($ $\frac{3}{4 \cdot 3^4}$ $\Big) =$ $\frac{5}{12}$

$\\$

vi) Given expression: $\Big( x -$ $\frac{1}{x^2}$ $\Big)^{3n}$

Let $T_{r+1}$ term be independent of $x$

$T_{r+1} = ^{3n} C_r \cdot (x)^{3n} \cdot \Big($ $\frac{-1}{x^2}$ $\Big)^r$

$= (-1)^r {^{3n} C_r} \cdot$ $\frac{x^{3n-r}}{x^{2r}}$

If $T_{r+1}$ is to be independent of $x$, then,

$3n-r-2r \Rightarrow r = n$  i.e  $(n+1)^{th}$ term

$T_{n+1} = ( -1)^n \cdot {^{3n} C_n}$

$\\$

vii) $\Big($ $\frac{1}{2}$ $x^{\frac{1}{3}} + x^{-\frac{1}{5}} \Big)^{15}$

Let $T_{r+1}$ term be independent of $x$

$T_{r+1} = ^8 C_r \Big($ $\frac{1}{2}$ $x^{\frac{1}{3}}$ $\Big)^{8-r} \cdot ( x^{\frac{-1}{5}})^r$

$= ^8 C_r \Big($ $\frac{x^{\frac{8-r}{3}}}{2^{8-r}}$ $\Big) \cdot x^{\frac{-r}{5}}$

If $T_{r+1}$ is to be independent of $x$, then,

$\frac{8-r}{3}$ $+$ $\frac{-r}{5}$ $= 0 \Rightarrow 40-5r-3r = 0 \Rightarrow r = 5$ Therefore the ${6}^{th}$ term.

$T_{5+1} = ^8 C_5 \cdot$ $\frac{1}{2^{8-5}}$ $= ^8 C_5 \cdot$ $\frac{1}{8}$ $= 7$

$\\$

viii) Given expression: $(1 + x + 2x^3) \Big($ $\frac{3}{2}$ $x^2 -$ $\frac{1}{3x}$ $\Big)^{9}$

$= (1+x+2x^3) \Big[ ^9 C_0 \Big( \frac{3}{2} x^2 \Big)^{9-0} \Big( \frac{-1}{3x} \Big)^0 + ^9 C_1 \Big( \frac{3}{2} x^2 \Big)^{9-1} \Big( \frac{-1}{3x} \Big)^1 + \ldots + ^9 C_9 \Big( \frac{3}{2}x^2 \Big)^{9-9} \Big( \frac{-1}{3x} \Big)^9\Big]$

Now $T_{r+1} = ^9 C_r \cdot \Big($ $\frac{3}{2}$ $x^2 \Big)^{9-r} \cdot \Big($ $\frac{-1}{3x}$ $\Big)^r$

$= (-1)^r \cdot {^9 C_r} \cdot \Big($ $\frac{3}{2}$ $\Big)^{9-r} \cdot x^{18-2r} \cdot \Big($ $\frac{1}{3^r x^r}$ $\Big)$

$= (-1)^r \cdot {^9 C_r} \cdot \Big($ $\frac{(\frac{3}{2})^{9-r}}{3^r}$ $\Big) \cdot x^{18-2r}$

Let $T_{r+1}$ term be independent of $x$

$18-3r = 0 \Rightarrow r = 6$. Therefore it is the $7^{th}$ term.

Coefficient of the $7^{th}$ term $= (-1)^6 \cdot {^9 C_6} \Big($ $\frac{(\frac{3}{2})^{9-6}}{3^6}$ $\Big) =$ $\frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1}$ $\times$ $\frac{3^3}{3^6 \cdot 2^3}$ $=$ $\frac{7}{18}$

Let $T_{r+1}$ term be independent of $x^{-1}$

$18-3r = -1 \Rightarrow r =$ $\frac{17}{3}$. This is not possible. Hence there is no term with $x^{-1}$ in the expansion.

If $T_{r+1}$ is to be independent of $x^{-3}$, then,

$18-3r = -3 \Rightarrow r = 7$. Therefore it is the $8^{th}$ term.

Coefficient of the $8^{th}$ term

$= (-1)^7 \cdot {^9 C_7} \Big($ $\frac{(\frac{3}{2})^{9-7}}{3^7}$ $\Big) = (-1)$ $\frac{9 \cdot 8 }{ 2 \cdot 1}$ $\times$ $\frac{3^2}{3^7 \cdot 2^2}$ $= -$ $\frac{2}{27}$

Therefore the coefficient of the term independent of $x =$ $\frac{7}{18}$ $-$ $\frac{2}{27}$ $=$ $\frac{17}{54}$

$\\$

ix) Given expression: $\Big( \sqrt[3]{x} +$ $\frac{1}{2}$ $\sqrt[3]{x} \Big)^{18} , x > 2$

Let $T_{r+1}$ term be independent of $x$

$T_{r+1}= ^{18} C_r ( x^{\frac{1}{3}} )^{18-r} \Big($ $\frac{1}{2x^{\frac{1}{3}}}$ $\Big)^r$

$= ^{18} C_r \Big($ $\frac{x^{\frac{18-r}{3}}}{2^r - x^{\frac{r}{3}}}$ $\Big)$

If $T_{r+1}$ is to be independent of $x$, then,

$\frac{18-r}{3}$ $-$ $\frac{r}{3}$ $= 0 \Rightarrow r = 9$ Therefore it is the ${10}^{th}$ term.

$T_{9+1} = ^{18} C_{9} \cdot$ $\frac{1}{2^9}$ $=$ $\frac{ 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10}{9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot \frac{1}{2^9}$ $=$ $\frac{12155}{128}$

$\\$

x) Given expression: $\Big($ $\frac{3}{2}$ $x^2 -$ $\frac{1}{3x}$ $\Big)^{6}$

Let $T_{r+1}$ term be independent of $x$

$T_{r+1} = ^6C_r \Big($ $\frac{3}{2}$ $x^2 \Big)^{6-r} \cdot \Big($ $\frac{-1}{3x}$ $\Big)^r$

$= (-1)^r \cdot {^6C_r} \cdot \Big($ $\frac{3}{2}$ $\Big)^{6-r} \cdot \Big($ $\frac{x^{12-2r}}{3^r x^r}$ $\Big)$

If $T_{r+1}$ is to be independent of $x$, then,

$12 - 2r - r = 0 \Rightarrow r = 4$. Therefore it is the ${5}^{th}$ term.

$T_{4+1} = (-1)^4 \cdot {^6 C_4} \cdot \Big($ $\frac{3}{2}$ $\Big)^2 \cdot$ $\frac{1}{3^4}$ $=$ $\frac{6 \cdot 5}{2 \cdot 1}$ $\times$ $\frac{3^2}{2^2}$ $\times$ $\frac{1}{3^4}$ $=$ $\frac{5}{12}$

$\\$

Question 17: If the coefficients of $( 2r + 4)^{th}$ and $(r-2)^{th}$ terms in the expansion of $( 1 + x)^{18}$ are equal, find $r$.

$[ \text{Note: When } ^n C_x = ^n C_y \Rightarrow x+y = n ]$

Given expression: $( 1 + x)^{18}$

$T_{2r+4} = ^{18} C_{2r+3} \cdot (x)^{2r+3}$

$T_{r-2} = ^{18} C_{r-3} \cdot (x)^{r-3}$

Given $^{18} C_{2r+3} = ^{18} C_{r-3}$

$\Rightarrow 2r+3 + r-3 = 18 \Rightarrow 3r = 18 \Rightarrow r = 6$

$\\$

Question 18: If the coefficient of $(2r+1)^{th}$  and $(r + 2)^{th}$ term in the expansion of $( 1 + x)^{43}$ are equal, find $r$.

$[ \text{Note: When} ^n C_x = ^n C_y \Rightarrow x+y = n ]$

Given expression: $( 1 + x)^{43}$

$T_{2r+1} = ^{43} C_{2r} \cdot (x)^{2r}$

$T_{r+2} = ^{43} C_{r+1} \cdot (x)^{r+1}$

Given $^{43} C_{2r} = ^{43} C_{r+1}$

$\Rightarrow 2r+r+1 = 43 \Rightarrow 3r = 42 \Rightarrow r = 14$

$\\$

Question 19: Prove 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 $(r+1)^{th}$ terms in the expansion of $( 1+x)^n$.

For expression: $( 1+x)^{n+1}$

$T_{r+1} = ^{n+1} C_r (x)^r$

For expression: $(1+x)^n$

$T_{r+1} = ^n C_r ( x)^r$

$T_r = ^n C_{r-1} (x)^{r-1}$

Sum of the coefficients $= ^n C_r + ^n C_{r-1} = ^{n+1} C_r$

Hence proven.

$\\$

Question 20: Prove that the term independent of $x$ in the expansion of

$\Big( x+$ $\frac{1}{x}$ $\Big)^{2n}$ is $\frac{1\cdot 3\cdot 5 \ldots (2n-1)}{n!}$ $\cdot 2^n$.

Given expression: $\Big( x+$ $\frac{1}{x}$ $\Big)^{2n}$

$T_{r+1} = ^{2n} C_r (x)^{2n-r} \Big($ $\frac{1}{x}$ $\Big)^r$

$= ^{2n} C_r x^{2n-2r}$

If the term is independent of  $x$,  then

$2n-2r = 0 \Rightarrow r = n$

$\therefore T_{r+1} = ^{2n} C_n =$ $\frac{(2n)!}{ n! n!}$

$=$ $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \ldots (2n-2) \cdot (2n-1) \cdot (2n)}{n! n!}$

$=$ $\frac{[ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n-1)][ 2 \cdot 4 \cdot 6 \cdot 8 \ldots (2n-2) \cdot (2n)}{n! n!}$

$=$ $\frac{[ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n-1)] 2^n [ 1 \cdot 2 \cdot 3 \cdot 4 \ldots (n-1) \cdot (n)]}{n! n!}$

$=$ $\frac{[ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n-1)] 2^n}{n!}$

Therefore the term independent of $x$ is $\frac{[ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n-1)] 2^n}{n!}$

$\\$

Question 21: The coefficient of the $5^{th}, 6^{th}$ and $7^{th}$ terms in the expansion of  $(1+x)^n$ are in A.P. , find $n$.

Given expression: $(1+x)^n$

Coefficient of $5^{th} \text{term} = ^n C_{5-1} = ^n C_4$

Coefficient of $6^{th} \text{term} = ^n C_{6-1} = ^n C_5$

Coefficient of $7^{th} \text{term} = ^n C_{7-1} = ^n C_6$

Since they are in AP

$2 \cdot ^n C_5 = ^n C_4 + ^n C_6$

$\Rightarrow 2 \cdot$ $\frac{n!}{5! (n-5)!}$ $=$ $\frac{n!}{4! (n-4)!}$ $+$ $\frac{n!}{6! (n-6)!}$

$\Rightarrow$ $\frac{2}{5 \cdot 4! (n-5) (n-6)!}$ $=$ $\frac{1}{4! (n-4)(n-5) (n-6)!}$ $+$ $\frac{1}{6 \cdot 5 \cdot 4! (n-6)!}$

$\Rightarrow$ $\frac{1}{(n-1)} \Big[$ $\frac{2}{5}$ $-$ $\frac{1}{(n-4)}$ $\Big] =$ $\frac{1}{30}$

$\Rightarrow$ $\frac{1}{(n-1)}$ $\Big[$ $\frac{2n - 8 - 5}{5 (n-4)}$ $\Big] =$ $\frac{1}{30}$

$\Rightarrow 6(2n-13) = (n-4)(n-5)$

$\Rightarrow 12n - 78 = n^2 - 9n +20$

$\Rightarrow n^2 - 21 n + 108 = 0$

$\Rightarrow (n-7)(n-14) = 0$

$\Rightarrow n = 7 \ or \ n = 14$

$\\$

Question 22: If the coefficient of $2^{nd}, 3^{rd}$ and $4^{th}$ terms in the expansion of $(1+x)^{2n}$ are in A.P., show that $2n^2 - 9n + 7 = 0$.

Given expression: $(1+x)^{2n}$

$T_{1+1} = ^{2n} C_1 \cdot x^1$

$T_{2+1} = ^{2n} C_2 \cdot x^2$

$T_{3+1} = ^{2n} C_3 \cdot x^3$

Therefore  $2 ( ^{2n} C_2 ) = ^{2n} C_1 + ^{2n} C_3$

$\Rightarrow 2 \cdot$ $\frac{(2n)!}{2! (2n-2)!}$ $=$ $\frac{(2n)!}{1! (2n-1)!}$ $+$ $\frac{(2n)!}{3! (2n-3)!}$

$\Rightarrow$ $\frac{2}{2 (2n-2)(2n-3)!}$ $=$ $\frac{1}{(2n-1)(2n-2)(2n-3)!}$ $+$ $\frac{1}{3 \cdot 2 \cdot 1 ( 2n-3)!}$

$\Rightarrow$ $\frac{1}{2n-2}$ $=$ $\frac{1}{(2n-1)(2n-2)}$ $+$ $\frac{1}{6}$

$\Rightarrow$ $\frac{1}{(2n-2)}$ $\Big[ 1 -$ $\frac{1}{2n-1}$