Question 1 :
The coefficient of ${ x }^{ 9 }$ in the expansion of ${ \left( { x }^{ 3 }+\cfrac { 1 }{ { 2 }^{ t } } \right) }^{ 11 }$, where $t=\log _{ \sqrt { 2 } }{ { (x }^{ \tfrac 32 } } )$,<br/>

Question 2 :
Let {tex} S _ { n } ( x ) = \underset{ k = 0 }{ \stackrel { n } \sum}{ } ^ { n } C _ { k } \sin ( k x ) \cos [ ( n - k ) x ] , {/tex} then

Question 3 :
The coefficient of $x^{6}.y^{-2}$ in the in the expansion of $\left ( \displaystyle \frac{x^{2}}{y}-\frac{y}{x} \right )^{12}$ is<br>

Question 4 :
i. Three consecutive binomial coefficients cannot be in $G.P.$<br/>ii. Three consecutive binomial coefficients can be in $H.P.$<br/>Which of the above statement is correct

Question 6 :
If the coefficients of $2^{nd}, 3^{rd}$ and $4^{th}$ terms of the expansion of $(1+x)^{2n}$ are in $\mathrm{A}.\mathrm{P}$, then the value of $2n^2-9n+7$ is

Question 7 :
If (1 + x + x²)ⁿ = a₀ + a₁x + a₂ x² + ......+ a_2nxⁿ, then the value of a₀ + a₃ + a₆+.........is -

Question 8 :
The ratio of $(r + 1) ^{th}$ and $(r - 1)^ {th}$ terms in the expansion of $({a}-b)^{n}$ is<br/>

Question 9 :
If the second term of the expansion $\displaystyle \left [ a^{1/13}+\frac{a}{\sqrt{a^{-1}}} \right ]^{n}\: \: is\: \: 14a^{5/2}$, then the value of $\displaystyle \frac{^{n}{C}_{3}}{^{n}{C}_{2}}$ is

Question 10 :
The coefficient of ${ x }^{ m }$ in : ${ \left( 1+x \right) }^{ m }+{ \left( 1+x \right) }^{ m+1 }+....+{ \left( 1+x \right) }^{ n },\quad m\le n$ is<br>

Question 11 :
If the last term in the binomial expansion of <br>${ \left( { 2 }^{ 1/3 }-\cfrac { 1 }{ \sqrt { 2 } } \right) }^{ n }$ is ${ \left( \cfrac { 1 }{ { 3 }^{ 5/3 } } \right) }^{ \log _{ 3 }{ 8 } }$, then the 5th term from the beginning is<br>

Question 12 :
Coefficient of $x^{50}$ in the polynomial <br/>$\left(x+_{ }^{ 50 }{ { C }_{ 0 } }\right)\left(x+3._{ }^{ 50 }{ { C }_{ 1 } }\right)\left(x+5._{ }^{ 50 }{ { C }_{ 2 } }\right).....\left[x+(101)._{ }^{ 50 }{ { C }_{ 50 } }\right]$ is

Question 13 :
Arrange the values of $n$ in ascending order<br/>A : If the term independent of $x$ in the expansion of $\left(\displaystyle \sqrt{x}-\frac{n}{x^{2}}\right)^{10}$ is $405$<br/>B : If the fourth term in the expansion of $\left(\displaystyle \frac{1}{n}+n^{\log_{n}10}\right)^{5}$ is $1000$, <span>( $ n&lt; 10 $)<br/>C : In the binomial expansion of $(1+x)^{n}$ the coefficients of <span> $5^{\mathrm{t}\mathrm{h}},\ 6^{\mathrm{t}\mathrm{h}}$ and $7^{\mathrm{t}\mathrm{h}}$ terms are in A.P.</span></span><br/>

Question 14 :
Find the coefficient of ${ x }^{ 50 }$ in the expression:<br>${ \left( 1+x \right) }^{ 1000 }+2x{ \left( 1+x \right) }^{ 999 }+3{ x }^{ 2 }{ \left( 1+x \right) }^{ 998 }+....+1001{ x }^{ 1000 }$<br>

Question 15 :
Find the coefficient of the term independent of x in the expansion of $\displaystyle\left(6x^3-\frac{5}{x^6}\right)^{12}$.

Question 16 :
The coefficient ${x^n}$ in the expression of ${\left( {1 + x} \right)^{2n}}$ and ${\left( {1 + x} \right)^{2n - 1}}$ are in the ratio.

Question 18 :
If sum of the coefficients of ${x}^{7}$ and ${x}^{4}$ in the expansion of ${ \left( \cfrac { { x }^{ 2 } }{ a } -\cfrac { b }{ x } \right) }^{ 11 }$ is zero, then<br>

Question 19 :
Let $n$ be a positive integer such that ${ \left( 1+x+{ x }^{ 2 } \right) }^{ n }={ a }_{ 0 }+{ a }_{ 1 }x+{ a }_{ 2 }{ x }^{ 2 }+...+{ a }_{ 2n }{ x }^{ 2n },$ then ${a}_{r}=$

Question 20 :
If there is a term containing $x^{2r}$ in $\left( x + \dfrac{1}{x^2} \right )^{n - 3}$, then