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

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

If the last term in ${\left(\sqrt[3]{2}+\frac{1}{\sqrt{2}}\right)}^n$ is ${\left[\frac{1}{3(9{)}^{1/3}}\right]}^{{\log }_{3}8},$ then the fifth term is

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 in the expansion of ${\left(x^3-\frac{1}{x^2}\right)}^n$, the sum of coefficients of $x^5$ and $x^{10}$ is $0$, then the coefficient of the third term is

If $a_i\left(i=0, 1, 2,\dots , 16\right)$ be real constants such that for every real value of $x$, ${\left(1+x+x^2\right)}^8=a_0+a_{1}x+a_2{x}^2+\dots +a_{16}{x}^{16}$, then $a_5$ is equal to

Find the ratio of the coefficient of $x^{10}$ in ${(1 - x^2)}^{10}$ and the term independent of $x$ in ${\left(x-\frac{2}{x}\right)}^{10}$ 

The value of the sum $\sum _{j=0}^8\left[\frac{1}{\left(j+1\right)\left(j+2\right)}\left({}^{8}C_j\right)\right]$ is

If in the expansion of ${\left(1+x\right)}^n; C_0, C_1, C_2, \dots C_n$ denote the binomial coefficients, then the value of $\frac{C_o}{1}+\frac{C_2}{3}+\frac{C_4}{5}+\frac{C_6}{7}+\dots$