Binomial Theorem — JEE Main PYQs

If the term independent of x in the expansion of ${\left(\sqrt{\mathrm{a}}{\mathrm{x}}^2+\frac{1}{2{\mathrm{x}}^3}\right)}^{10}$ is 105 , then ${\mathrm{a}}^2$ is equal to :

The coefficient of $x^{70}$ in $x^2(1+x{)}^{98}+x^3(1+x{)}^{97}+{\mathrm{x}}^4(1+\mathrm{x}{)}^{96}+\dots \dots ..+{\mathrm{x}}^{54}(1+\mathrm{x}{)}^{46}$ is ${}^{99}{\mathrm{C}}_{\mathrm{p}}-{}^{46}{\mathrm{C}}_{\mathrm{q}}$.

Then a possible value to $\mathrm{p}+\mathrm{q}$ …

The remainder when $428^{2024}$ is divided by 21 is

The sum of the coefficient of $x^{2/3}$ and $x^{-2/5}$ in the binomial expansion of ${\left(x^{2/3}+\frac{1}{2}{x}^{-2/5}\right)}^9$ is :

Remainder when $64^{32^{32}}$ is divided by 9 is equal to ______.

Let the coefficient of ${\mathrm{x}}^{\mathrm{r}}$ in the expansion of
…

Number of integral terms in the expansion of ${\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}}^{824}$ is equal to ______.

The coefficient of $x^{2012}$ in the expansion of $(1-x{)}^{2008}{\left(1+x+x^2\right)}^{2007}$ is equal to

${}^{n-1}{C}_r=\left(k^2-8\right){}^n{C}_{r+1}$ if and only if :

If A denotes the sum of all the coefficients in the expansion of ${\left(1-3x+10x^2\right)}^n$ and $B$ denotes the sum of all the coefficients in the expansion of ${\left(1+x^2\right)}^n$, then :

Let m and n be the coefficients of seventh and thirteenth terms respectively in the expansion of ${\left(\frac{1}{3}{\mathrm{x}}^{\frac{1}{3}}+\frac{1}{2{\mathrm{x}}^{\frac{2}{3}}}\right)}^{18}$. Then ${\left(\frac{\mathrm{n}}{\mathrm{m}}\right)}^{\frac{1}{3}}$ is :

In the expansion of $(1+x)(1-x^2){\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)}^5,x\ne 0$, the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to ____

If the Coefficient of $x^{30}$ in the expansion of ${\left(1+\frac{1}{x}\right)}^6{\left(1+x^2\right)}^7{\left(1-x^3\right)}^8;x\ne 0$ is $\alpha$, then $|\alpha |$ equals ____.

If $\frac{{}^{11}{C}_1}{2}+\frac{{}^{11}{C}_2}{3}+\dots +\frac{{}^{11}{C}_9}{10}=\frac{n}{m}$ with $\gcd (n,m)=1$, then $n+m$ is equal to

Let $\alpha ={\sum }_{\mathrm{k}=0}^{\mathrm{n}}\left(\frac{{\left({}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{k}}\right)}^2}{\mathrm{k}+1}\right)$ and …

Among the statements:
(S1): $2023^{2022}-1999^{2022}$ is divisible by 8.
(S2): $13(13{)}^n-11n-13$ is divisible by 144 for infinitely many $n\in \mathbb{N}$.

If the coefficients of $x^7$ in ${\left(a{x}^2+\frac{1}{2bx}\right)}^{11}$ and $x^{-7}$ in ${\left(ax-\frac{1}{3b{x}^2}\right)}^{11}$ are equal, then

If $(20{)}^{19}+2(21)(20{)}^{18}+3(21{)}^2(20{)}^{17}+\dots +20(21{)}^{19}=\mathrm{k}(20{)}^{19}$, then k is equal to

If the coefficients of the three consecutive terms in the expansion of $(1+x{)}^n$ are in the ratio $1:5:20$, then the coefficient of the fourth term is

Let $[\mathrm{t}]$ denotes the greatest integer $\le \mathrm{t}$. If the constant term in the expansion of ${\left(3x^2-\frac{1}{2x^5}\right)}^7$ is $\alpha$, then $[\alpha ]$ is equal to

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of ${\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)}^{\mathrm{n}}$ is $\sqrt{6}:1$, then the third term from the beginning is:

The coefficient of $x^{18}$ in the expansion of ${\left(x^4-\frac{1}{x^3}\right)}^{15}$ is

The absolute difference of the coefficients of $x^{10}$ and $x^7$ in the expansion of ${\left(2x^2+\frac{1}{2x}\right)}^{11}$ is equal to

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by

If the coefficients of $x$ and $x^2$ in $(1+x{)}^p(1-x{)}^q$ are 4 and -5 respectively, then $2p+3q$ is equal to

Let the number $(22{)}^{2022}+(2022{)}^{22}$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7. Then $({\alpha }^2+{\beta }^2)$ is equal to

If the coefficient of $x^7$ in ${\left(ax-\frac{1}{b{x}^2}\right)}^{13}$ and the coefficient of $x^{-5}$ in ${\left(ax+\frac{1}{b{x}^2}\right)}^{13}$ are equal, then $a^4{b}^4$ is equal to :

The coefficient of $x^7$ in $(1-x+2x^3{)}^{10}$ is ______.

The sum of the coefficients of the first 50 terms in the binomial expansion of $(1-x{)}^{100}$, is equal to

If $\frac{1}{n+1}{}^n{C}_n+\frac{1}{n}{}^n{C}_{n-1}+\dots +\frac{1}{2}{}^{\mathrm{n}}{\mathrm{C}}_1+{}^{\mathrm{n}}{\mathrm{C}}_0=\frac{1023}{10}$ then n is equal to

The coefficient of $x^5$ in the expansion of ${\left(2x^3-\frac{1}{3x^2}\right)}^5$ is

The remainder, when $7^{103}$ is divided by 17 is ______.

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

Let $\alpha$ be the constant term in the binomial expansion of ${\left(\sqrt{x}-\frac{6}{x^{3/2}}\right)}^n$, $n\le 15$. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ …

Let ${\left(a+bx+c{x}^2\right)}^{10}={\sum }_{i=0}^{20}{p}_i{x}^i,a,b,c\in \mathbb{N}$. If $p_1=20$ and ${\mathrm{p}}_2=210$, then $2(\mathrm{a}+\mathrm{b}+\mathrm{c})$ is equal to

The number of elements in the set $\{\mathrm{n}\in \mathbb{N}:10\le \mathrm{n}\le 100$ and $3^{\mathrm{n}}-3$ is a multiple of 7$\}$ is ____

If the $1011^{\text{th}}$ term from the beginning in the binomial expansion of ${\left(\frac{4\mathrm{x}}{5}-\frac{5}{2\mathrm{x}}\right)}^{2022}$ is 1024 times the $1011^{\text{th}}$ term from the end, then $|\mathrm{x}|$ is equal to

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x{)}^{n+2}$, which are in the ratio $1:3:5$, is equal to

The mean of the coefficients of $x,x^2,\dots ,x^7$ in the binomial expansion of $(2+\mathrm{x}{)}^9$ is

The number of integral terms in the expansion of ${\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)}^{680}$ is equal to

The value ${\sum }_{\mathrm{r}=0}^{22}{}^{22}{\mathrm{C}}_{\mathrm{r}}{}^{23}{\mathrm{C}}_{\mathrm{r}}$ is

Suppose ${\sum }_{\mathrm{r}=0}^{2023}{\mathrm{r}}^2{}^{2023}{\mathrm{C}}_{\mathrm{r}}=2023\times \alpha \times 2^{2022}$. Then the value of $\alpha$ is

If the coefficient of $x^{15}$ in the expansion of ${\left(a{x}^3+\frac{1}{b{x}^{\frac{1}{3}}}\right)}^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of ${\left(a{x}^{\frac{1}{3}}-\frac{1}{b{x}^3}\right)}^{15}$, where $a$ and $b$ are positive real numbers, th …

The coefficient of $x^{301}$ in $(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+\dots +x^{500}$ is:

Let the sixth term in the binomial expansion of ${\left(\sqrt{2^{{\log }_2\left(10-3^x\right)}+\sqrt[5]{2^{(x-2){\log }_{2}3}}}\right)}^m$, in the increasing powers of $2^{(x-2){\log }_{2}3}$, be 21. If the binomial coefficients of the second, third and fourth terms in the expa …

If the term without $x$ in the expansion of ${\left(x^{\frac{2}{3}}+\frac{\alpha }{x^3}\right)}^{22}$ is 7315, then $|\alpha |$ is equal to ______.

Let K be the sum of the coefficients of the odd powers of x in the expansion of $(1+\mathrm{x}{)}^{99}$. Let a be the middle term in the expansion of ${\left(2+\frac{1}{\sqrt{2}}\right)}^{200}$. If …

If the co-efficient of $x^9$ in ${\left(\alpha x^3+\frac{1}{\beta x}\right)}^{11}$ and the co-efficient of ${\mathrm{x}}^{-9}$ in ${\left(\alpha \mathrm{x}-\frac{1}{\beta {\mathrm{x}}^3}\right)}^{11}$ are equal, then $(\alpha \beta {)}^2$ is equal to ____.

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2\mathrm{x}{)}^{\mathrm{n}}$ be in the ratio $2:5:8$. Then the coefficient of the term, which is in the middle of these three terms, is ____.

${\sum }_{\mathrm{k}=0}^6{}^{51-\mathrm{k}}{\mathrm{C}}_3$ is equal to