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

Question 4 :
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 5 :
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 6 :
Find the coefficient of the term independent of x in the expansion of $\displaystyle\left(6x^3-\frac{5}{x^6}\right)^{12}$.

Question 7 :
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 8 :
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 9 :
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/>