Embibe Experts Solutions for Chapter: Binomial Theorem, Exercise 1: TS EAMCET - 5 August 2021 Shift 2

If $f\left(n\right)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x{)}^n,$ then $f\left(2021\right)=$

If $p$ and $q$ are respectively the coefficient of $x^{-3}$ and $x^{-5}$ in the expansion of
${\left(x^{1/3}+\frac{1}{2x^{1/3}}\right)}^{21}, x>0,$ then $\frac{5p}{4q}=$

If $(\sqrt{3}+i{)}^8-(\sqrt{3}-i{)}^8=\alpha +i\beta ,$ then $\alpha -\frac{\sqrt{3}}{2}\beta =$

In the binomial expansion of $(1+x{)}^{2k},$ if its middle term is the only numerically greatest term, then $x$ lies in the interval

Sum of the coefficients of $x^r, (r=0, 1, 2, 3,..., 15)$ in the expansion of $(3x-1{)}^{15}$ is equal to the sum of the binomial coefficients of which of the following expansions?

$\left(a\right)$ $(1+x{)}^{15}$

$\left(b\right)$ $(1+x{)}^{16}+(1-x{)}^{16}$

$\left(c\right)$ $(1+x{)}^{16}-(1-x{)}^{16}$

For $x>0,$ if $p^{\text{th }}$term is the first negative term in the expansion of ${\left(1+\frac{3x}{5}\right)}^{22/3}$and in the expansion of ${\left(1-\frac{3x}{5}\right)}^{22/3}$ from $r^{th}$term onwards all the terms are positive, then the number of terms in the expansion of ${\left(px+\frac{r}{x}\right)}^{pr}$ is

For $n,p\in N-\left\{1\right\},$ the coefficient of $x^3$ in $\frac{(1-x{)}^{-1/p}}{(1-x{)}^n}=$

The ${13}^{\mathrm{th}}$ term in the expansion of $(1-4x{)}^{-4}$ is