Binomial Theorem — Class 12 Mathematics PYQs

What is the coefficient of $x^{10}$ in the expansion of $(1-x^2{)}^{20}{\left(2-x^2-\frac{1}{x^2}\right)}^{-5}$?

In a binomial expansion of
$(x+y{)}^{2n+1}(x-y{)}^{2n+1}$,
the sum of middle terms is zero. What is the value of $\left(\frac{x^2}{y^2}\right)$?

Direction : Consider the following for the two (02) items that follow :

Let $(8+3\sqrt{7}{)}^{20}=U+V$ and $(8-3\sqrt{7}{)}^{20}=W$, where $U$ is an integer and $0<V<1$.

What is $V+W$ equal to?

A set $S$ contains $(2n+1)$ elements. There are $4096$ subsets of $S$ which contain at most $n$ elements. What is $n$ equal to ?

If the sixth term in the binomial expansion of ${\left(x^{-\frac{8}{3}}+x^2{\log }_{10}x\right)}^8$ is $5600$, then what is the value of $x$?

How many terms are there in the expansion of $(3x-y{)}^4(x+3y{)}^4$?

Consider the following for the next three (03) items that follow :

Let $(1+x{)}^n=1+T_{1}x+T_2{x}^2+T_3{x}^3+\dots +T_n{x}^n$.

What is $T_1+2T_2+3T_3+\dots +n{T}_n$ equal to ?

Consider the following for the next three (03) items that follow :

Let $(1+x{)}^n=1+T_{1}x+T_2{x}^2+T_3{x}^3+\dots +T_n{x}^n$.

What is $1-T_1+2T_2-3T_3+\dots +(-1{)}^{n}n{T}_n$ equal to ?

Consider the following for the next three (03) items that follow :

Let $(1+x{)}^n=1+T_{1}x+T_2{x}^2+T_3{x}^3+\dots +T_n{x}^n$.

What is $T_1+T_2+T_3+\dots +T_n$ equal to ?

What is ${\sum }_{r=0}^n{2}^{r}C(n,r)$ equal to ?