Embibe Experts Solutions for Chapter: Binomial Theorem, Exercise 1: Exercise 1

If $n$ be a positive integer and $(1+x{)}^n=a_0+a_{1}x+a_2{x}^2+\dots +a_n{x}^n,$ then what is $a_0+a_1+a_2+\dots +a_n$ equal to?

The sum of all those terms which are rational numbers in the expansion of ${\left(2^{\frac{1}{3}}+3^{\frac{1}{4}}\right)}^{12}$ is:

The number of irrational terms in the expansion of ${\left(5^{\frac{1}{6}}+2^{\frac{1}{8}}\right)}^{100}$ is

For the natural numbers $m, n,$ if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+\dots .+a_{m+n}{y}^{m+n}$ and $a_1=a_2=10,$ then the value of $m, n,$ is equal to:

If the coefficients of ${\left(2r+4\right)}^{\mathrm{th}},{\left(r-2\right)}^{\mathrm{th}}$ terms in the expansion of ${\left(1+x\right)}^{18}$ are equal, then the value of $r$ is

If the term independent of $x$ in the expansion of ${\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]}^{10}, x\ne 1$, is equal to $k^{th}$ term, then the value of $k$ is 

The value of the numerically greatest term in the expansion of $(2x+3y{)}^{11}$ when $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is

In the expansion of ${\left(\sqrt{x}+\frac{1}{3x^2}\right)}^{10}$ the value of constant term (independent of $x$ ) is