Binomial Theorem — JEE Main PYQs

Let $\mathrm{m},\mathrm{n}\in \mathrm{N}$ and $\gcd (2,\mathrm{n})=1$. If $30\left(\genfrac{}{}{0ex}{}{30}{0}\right)+29\left(\genfrac{}{}{0ex}{}{30}{1}\right)+\dots .+2\left(\genfrac{}{}{0ex}{}{30}{28}\right)+1\left(\genfrac{}{}{0ex}{}{30}{29}\right)=\mathrm{n}.2^{\mathrm{m}}$, then $\mathrm{n}+\mathrm{m}$ is equal to ______ …

${\sum }_{\mathrm{k}=0}^{20}{\left({}^{20}{\mathrm{C}}_{\mathrm{k}}\right)}^2$ is equal to

If the coefficient of $a^7{b}^8$ in the expansion of $(a+2b+4ab{)}^{10}$ is $K\cdot 2^{16}$, then $K$ is equal to ____.

If $\left(\frac{3^6}{4^4}\right)\mathrm{k}$ is the term, independent of x , in the binomial expansion of ${\left(\frac{\mathrm{x}}{4}-\frac{12}{{\mathrm{x}}^2}\right)}^{12}$, then k is equal to ____.

If the coefficients of $x^7$ in ${\left(x^2+\frac{1}{bx}\right)}^{11}$ and $x^{-7}$ in ${\left(x-\frac{1}{b{x}^2}\right)}^{11},b\ne 0$ are equal, then the value of $b$ is equal to :

If the fourth term in the expansion of ${\left(x+x^{{\log }_{2}x}\right)}^7$ is $4480$, then the value of $x$ where $x\in \mathbb{N}$ is equal to:

If $(2021{)}^{3762}$ is divided by 17, then the remainder is ____

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity
$\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\dots +\frac{1}{a-nb}=\alpha n+\beta n^2+\gamma n^3$,

the term independent of ' $x$ ' in the expression of ${\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)}^{10}$, where $x\ne 0,1$ is equal to ____

The ratio of the coefficient of the middle term in the expansion of $(1+x{)}^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x{)}^{19}$ is

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 :

If the greatest value of the term independent of ' $x$ ' in the expansion of ${\left(x\sin \alpha +a\frac{\cos \alpha }{x}\right)}^{10}$ is $\frac{10!}{(5!{)}^2}$, then the value of ' $a$ ' is equal to :

If the co-efficients of $x^7$ and $x^8$ in the expansion of ${\left(2+\frac{x}{3}\right)}^n$ are equal, then the value of $n$ is equal to ____

Let $n\in N$ and $[x]$ denote the greatest integer less than or equal to $x$. If the sum of ( $n+1$ ) terms ${}^{\mathrm{n}}{\mathrm{C}}_0,3.{}^{\mathrm{n}}{\mathrm{C}}_1,5.{}^{\mathrm{n}}{\mathrm{C}}_2,7.{}^{\mathrm{n}}{\mathrm{C}}_3,\dots .$. is equal …

A possible value of '$x$', for which the ninth term in the expansion of ${\left\{3^{{\log }_3\sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right){\log }_3\left(5^{x-1}+1\right)}\right\}}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right){\log }_3\left(5^{x-1}+1\right)}$ …

If the sum of the coefficients in the expansion of $(x+y{)}^n$ is 4096 , then the greatest coefficient in the expansion is

If $n$ is the number or irrational terms in the expansion of ${\left(3^{1/4}+5^{1/8}\right)}^{60}$, then ( $n-1$ ) is divisible by:

Let $[x]$ denote greatest integer less than or equal to $x$. If for $n\in N,{\left(1-x+x^3\right)}^n={\sum }_{j=0}^{3n}{a}_j{x}^j$, then …

Let $n$ be a positive integer. Let $A={\sum }_{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})\left[{\left(\frac{1}{2}\right)}^k+{\left(\frac{3}{4}\right)}^k+{\left(\frac{7}{8}\right)}^k+{\left(\frac{15}{16}\right)}^k+{\left(\frac{31}{32}\right)}^k\right]$. If $63A=1-\frac{1}{2^{30}}$, then …

Let ${}^n{C}_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x{)}^n$. If ${\sum }_{k=0}^{10}{\left(2^2+3k\right)}^n{C}_k=\alpha \cdot 3^{10}+\beta \cdot 2^{10},\alpha ,\beta \in R$, then $\alpha +\beta$ is equal to

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.....

The value of ${\sum }_{\mathrm{r}=0}^6\left({}^6{\mathrm{C}}_{\mathrm{r}}\cdot {}^6{\mathrm{C}}_{6-\mathrm{r}}\right)$ is equal to:

Let the coefficients of third, fourth and fifth terms in the expansion of ${\left(x+\frac{a}{x^2}\right)}^n$, $x\ne 0$, be in the ratio 12:8:3. Then the term independent of x in the expansion is equal to ____.

${\left(1+x+2x^2\right)}^{20}=a_0+a_{1}x+a_2{x}^2+\dots .+a_{40}{x}^{40}$. Then, $a_1+a_3+a_5+\dots .+a_{37}$ is equal to :

If the sum of the coefficients of all even powers of $x$ in the product $\left(1+x+x^2+\dots \dots +x^{2n}\right)\left(1-x+x^2-x^3+\dots ..+x^{2n}\right)$ is 61 , then $n$ is equal to

The number of ordered pairs $(r,k)$ for which $6.{}^{35}{C}_r=\left(k^2-3\right)\cdot {}^{36}{C}_{r+1}$, where $k$ is an integer, is:

The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$ is:

If $\mathrm{a},\mathrm{b}$ and c are the greatest values of ${}^{19}{\mathrm{C}}_{\mathrm{p}},{}^{20}{\mathrm{C}}_{\mathrm{q}}$ and ${}^{21}{\mathrm{C}}_{\mathrm{r}}$ respectively, then:

If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of ${\left(x+\sqrt{x^2-1}\right)}^6+{\left(x-\sqrt{x^2-1}\right)}^6$, then:

In the expansion of ${\left(\frac{x}{\cos \theta }+\frac{1}{x\sin \theta }\right)}^{16}$, if $I_1$ is the least value of the term independent of $x$ when $\frac{\pi }{16}\le \theta \le \frac{\pi }{8}$, and $I_2$ is the least value of the term independent of $x$ when …

If ${\mathrm{C}}_{\mathrm{r}}={}^{25}{\mathrm{C}}_{\mathrm{r}}$ and ${\mathrm{C}}_0+5\cdot {\mathrm{C}}_1+9\cdot {\mathrm{C}}_2+\dots \dots \dots +(101)\cdot {\mathrm{C}}_{25}=2{}^{25}\cdot \mathrm{k}$, then k is equal to ______.

The coefficient of $x^4$ in the expansion of ${\left(1+x+x^2\right)}^{10}$ is ______.

The coefficient of $x^4$ in the expansion of ${\left(1+x+x^2+x^3\right)}^6$ in powers of $x$, is ____.

If the number of integral terms in the expansion of ${\left(3^{\frac{1}{2}}+5^{\frac{1}{8}}\right)}^n$ is exactly 33, then the least value of '$n$' is

If the term independent of $x$ in the expansion of ${\left(\frac{3}{2}{x}^2-\frac{1}{3x}\right)}^9$ is $k$, then $18k$ is equal to:

If the constant term in the binomial expansion of ${\left(\sqrt{x}-\frac{k}{x^2}\right)}^{10}$ is 405, then $|k|$ equals:

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of ${\left(x^m+\frac{1}{x^2}\right)}^{22}$ is 1540, is ______

If $\{\mathrm{p}\}$ denotes the fractional part of the number p , then $\left\{\frac{3^{200}}{8}\right\}$, is equal to

The value of ${\sum }_{\mathrm{r}=0}^{20}{}^{50-\mathrm{r}}{\mathrm{C}}_6$ is equal to:

Let ${\left(2x^2+3x+4\right)}^{10}={\sum }_{r=0}^{20}{a}_r{x}^r$. Then $\frac{a_7}{a_{13}}$ is equal to

If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of $(1+x{)}^{N+5}$ are in the ratio $5:10:14$, then the largest coefficient in the expansion is:

$2\cdot {}^{20}{\mathrm{C}}_0+5\cdot {}^{20}{\mathrm{C}}_1+8\cdot {}^{20}{\mathrm{C}}_2+11\cdot {}^{20}{\mathrm{C}}_3+\dots \cdot +62\cdot {}^{20}{\mathrm{C}}_{20}$ is equal to

The sum of the co-efficient of all even degree terms in $x$ in the expansion of ${\left(x+\sqrt{x^3-1}\right)}^6+{\left(x-\sqrt{x^3-1}\right)}^6,(x>1)$ is equal to:

If the fourth term in the Binomial expansion of ${\left(\frac{2}{x}+x^{{\log }_{8}x}\right)}^6(x>0)$ is $20\times 8^7$, then a value of $x$ is:

If ${\sum }_{\mathrm{i}=1}^{20}{\left(\frac{{}^{20}{\mathrm{C}}_{\mathrm{i}-1}}{{}^{20}{\mathrm{C}}_{\mathrm{i}}+{}^{20}{\mathrm{C}}_{\mathrm{i}-1}}\right)}^3=\frac{\mathrm{k}}{21}$, then k equals:

If the third term in the binomial expansion of ${\left(1+x^{{\log }_{2}x}\right)}^5$ equals 2560 , then a possible value of $x$ is:

Let $S_n=1+q+q^2+\dots +q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+{\left(\frac{q+1}{2}\right)}^2+\dots +{\left(\frac{q+1}{2}\right)}^n$ where $q$ is a real number and $q\ne 1$. If ${}^{101}{C}_1{+}^{101}{C}_2\cdot S_1+\dots {+}^{101}{C}_{101}\cdot S_{100}=\alpha T_{100}$ …

Let $(x+10{)}^{50}+(x-10{)}^{50}=a_0+a_{1}x+a_2{x}^2+\dots +a_{50}{x}^{50}$, for $x\in R$; then $\frac{a_2}{a_0}$ is equal to:

The coefficient of $x^{18}$ in the product $(1+x)(1-x{)}^{10}{\left(1+x+x^2\right)}^9$ is:

The smallest natural number n , such that the coefficient of x in the expansion of ${\left(x^2+\frac{1}{x^3}\right)}^n$ is ${}^n{\mathrm{C}}_{23}$ is