Q JEE Main 2024

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the...

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Q Jee Main 2024

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

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Q JEE MAIN 2024

Let $\alpha=\sum_{\mathrm{k}=0}^{\mathrm{n}}\left(\frac{\left({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}\right)^2}{\mathrm{k}+1}\right)$ and $\beta=\sum_{\mathrm{k}=0}^{\mathrm{n}-1}\left(\frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}+1}}{\mathrm{k}+2}\right)$. If $5 \alpha=6 \beta$, then $n$ equals

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Q JEE-Main 2024

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

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Q JEE MAIN 2024

Suppose $2-p, p, 2-\alpha, \alpha$ are the coefficient of four consecutive terms in the expansion of $(1+x)^n$. Then the value...

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Q Jee Main 2024

Let the coefficient of $x^r$ in the expansion of $$ \begin{aligned} & (x+3)^{n-1}+(x+3)^{n-2}(x+2)+ \\ & (x+3)^{n-3}(x+2)^2+\cdots \cdots+(x+2)^{n-1} \end{aligned} $$ be...

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Q JEE MAIN 2024

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

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Q JEE MAIN 2026

Given below are two statements : Statement I: $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7 . Statement II: The integral part of...

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Q JEE MAIN 2026

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\ldots+x^{1000}$ is :

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Q JEE MAIN 2026

Let $\mathrm{S}=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\ldots$ up to 13 terms. If $13 \mathrm{~S}=\frac{2^k}{n!}, k \in \mathrm{~N}$, then $n+k$ is equal to

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Q JEE MAIN_2026_

The sum of all possible values of $\mathrm{n} \in \mathbf{N}$, so that the coefficients of $x_2 x^2$ and $x^3$ in...

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Q JEE MAIN_2026_

The value of $\frac{{ }^{100} C_{50}}{51}+\frac{{ }^{100} C_{51}}{52}+\ldots .+\frac{{ }^{100} C_{100}}{101}$ is:

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Q JEE MAIN_2026_

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^2+3(1+x)^3+\ldots+100(1+x)^{100}$ is equal to

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Q JEE MAIN 2026

Let $\mathrm{C}_{\mathrm{r}}$ denote the coefficient of $x^{\mathrm{r}}$ in the binomial expansion of $...

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Q JEE MAIN 2026

If $...

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Q JEE MAIN 2026

The largest $\mathrm{n} \in \mathbf{N}$, for which $7^{\mathrm{n}}$ divides $101!$, is :

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Q JEE MAIN 2026

If the coefficient of x in the expansion of $\left(a x^2+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $x^2$...

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Q JEE MAIN_2019

If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $...

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Q JEE MAIN 2019

If the fourth term in the Binomial expansion of $\left(\frac{2}{x}+x^{\log _8 x}\right)^6$ (x > 0) is 20 $\times 8^7$ ,...

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Q JEE MAIN 2019

The term independent of $x$ in the expansion of $\left(\frac{1}{60}-\frac{x^8}{81}\right) \cdot\left(2 x^2-\frac{3}{x^2}\right)^6$ is equal to :

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Q JEE MAIN 2020

For a positive integer $n,\left(1+\frac{1}{x}\right)^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are...

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Q JEE MAIN 2019

If ${ }^{20} \mathrm{C}_1+\left(2^2\right)^{20} \mathrm{C}_2+\left(3^2\right)^{20} \mathrm{C}_3+\ldots \ldots . .+\left(20^2\right)^{20} \mathrm{C}_{20}=\mathrm{A}\left(2^\beta\right)$, then the ordered pair $(\mathrm{A}, \beta)$ is equal to:

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Q JEE Main 2020

Let $\alpha>0, \beta>0$ be such that $\alpha^3+\beta^2=4$. If the maximum value of the term independent of $x$ in the binomial...

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Q JEE-MAIN 2019

A ratio of the 5 th term from the beginning to the $5^{\text {th }}$ term from the end in...

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Q JEE MAIN 2019

The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is

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Q JEE MAIN 2020

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

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Q JEE MAIN 2020

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 ,...

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Q JEE MAIN 2019

The total number of irrational terms in the binomial expansion of $\left(7^{1 / 5}-3^{1 / 10}\right)^{60}$ is :

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Q JEE MAIN 2019

If ${ }^{\mathrm{n}} \mathrm{C}_4,{ }^{\mathrm{n}} \mathrm{C}_5$ and ${ }^{\mathrm{n}} \mathrm{C}_6$ are in A.P., then n can be :

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Q JEE MAIN 2020

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

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Q JEE MAIN 2020

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

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Q JEE MAIN 2019

The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is

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Q JEE MAIN 2020

The coef ficient of $x^4$ in the expansion of $\left(1+x+x^2+x^3\right)^6$ in powers of $x$, is $\_\_\_\_$ ...

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Q JEE MAIN 2019

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^3}{3}+\frac{3}{x}\right)^8$ equals...

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Q JEE MAIN 2020

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

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Q JEE MAIN 2019

The value of r for which $...

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Q JEE MAIN 2019

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 $...

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Q JEE MAIN 2020

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

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Q JEE Main 2019

If the third term in the binomial expansion of $\left(1+x^{\log _2 x}\right)^5$ equals 2560 , then a possible value of...

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Q JEE MAIN 2020

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

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Q JEE Main 2019

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is

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Q JEE Main 2019

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

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Q JEE MAIN 2020

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

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Q JEE MAIN 2020

If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in...

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Q JEE MAIN 2019

Let $S_n=1+q+q^2+\ldots+q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+\left(\frac{q+1}{2}\right)^2+\ldots .+\left(\frac{q+1}{2}\right)^n$ where $q$ is a real number and $q \neq 1$. If $...

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Q JEE MAIN 2019

Let $(x+10)^{50}+(x-10)^{50} =a_0+a_1 x+a_2 x^2+\ldots+a_{50} x^{50}$, for all $x \in R$ : then $\frac{a_2}{a_0}$ is equal to :

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Q JEE MAIN 2020

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

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Q JEE-MAIN 2020

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

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Q JEE MAIN 2020

If the sum of the coefficients of all even powers of x in the product $\left(1+x+x^2+\ldots \ldots+x^{2 n}\right)\left(1-x+x^2-x^3+\ldots .+x^{2 n}\right)$...

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Q JEE MAIN 2019

If some three consecutive coefficients in the binomial expansion of $(x+1)^n$ in powers of $x$ are in the ratio $...

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Q JEE MAIN 2021

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

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Q JEE Main 2021

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

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Q JEE MAIN 2019

If $\sum_{\mathrm{r}=0}^{25}\left\{{ }^{50} \mathrm{C}_{\mathrm{r}} \cdot{ }^{50-\mathrm{r}} \mathrm{C}_{25-\mathrm{r}}\right\}=\mathrm{K}\left({ }^{50} \mathrm{C}_{25}\right)$, then K is equal to:

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Q JEE MAIN 2019

The positive value of $\lambda$ for which the co-efficient of $x^2$ in the expression $x^2\left(\sqrt{x}+\frac{\pi}{x^2}\right)^{10}$ is 720 , is

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Q JEE MAIN 2021

$\sum_{\mathrm{k}=0}^{20}{ }^{20} \mathrm{C}_{\mathrm{k}} \cdot{ }^{20} \mathrm{C}_{20-\mathrm{k}}$ is equal to :

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Q JEE Main 2021

If ${ }^{20} \mathrm{C}_{\mathrm{r}}$ is the co-efficient of $\mathrm{x}^{\mathrm{r}}$ in the expansion of $(1+\mathrm{x})^{20}$, then the value of $...

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Q JEE MAIN 2021

Let $\binom{n}{k}$ denotes ${ }^n C_k$ and $...

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Q JEE MAIN_2021_

$3 \times 7^{22}+2 \times 10^{22}-44$ when divided by 18 leaves the remainder $\_\_\_\_$

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Q JEE MAIN 2019

The sum of the series $...

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Q JEE MAIN 2019

The sum of the co-efficients 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...

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Q JEE Main 2019

If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to

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Q JEE MAIN 2021

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

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Q JEE MAIN_2021

The term independent of ' $x$ ' in the expansion of $\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}$, where $...

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Q JEE MAIN_2021

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

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Q JEE MAIN_2021

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers...

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Q JEE MAIN 2021 S2

A possible value of ' $x$ ', for which the ninth term in the expansion of $...

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Q JEE MAIN 2021 S2

If the co-efficient 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...

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Q JEE MAIN 2021

Let $n \in N$ and $[x]$ denote the greatest integer less than or equal to $x$. If the sum of...

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Q JEE MAIN 2019

If the fourth term in the binomial expansion of $\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^6$ is equal to 200 , and $x>1$, then...

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Q JEE MAIN 2021

If the coefficients of $x^7$ in $\left(x^2+\frac{1}{b x}\right)^{11}$ and $x^{-7}$ in $\left(x-\frac{1}{b x^2}\right)^{11}, b \neq 0$, are equal, then the...

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Q JEE MAIN 2021

The lowest integer which is greater than $\left(1+\frac{1}{10^{100}}\right)^{10^{100}}$ is

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Q JEE MAIN 2021

If the greatest value of the term independent of ' $x$ ' in the expansion of $...

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$

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Q JEE MAIN 2021

The coefficien of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^2+x+1\right)^{100}$ is :

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Q JEE-MAIN 2021

Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^2}\right)^n, x \neq 0$ be in the...

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Q JEE MAIN 2021

If the remainder when x is divided by 4 is 3 , then the remainder when $(2020+\mathrm{x})^{2022}$ is divided by...

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Q JEE MAIN 2021

The total number of two digit numbers ' n ', such that $3^n+7^n$ is a multiple of 10 , is...

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Q JEE MAIN_2021_

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

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Q JEE MAIN 2021

Let ${ }^n C_r$ denote the binomial coefficient of $\underline{X^r}$ in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$. If $...

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Q JEE MAIN 2021

The term independent of x in the expansion of $\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1$ is...

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Q JEE MAIN 2021

If $\sum_{r=1}^{10} r!\left(r^3+6 r^2+2 r+5\right)=\alpha(11!)$ then the value of $\alpha$ is equal to $\_\_\_\_$ .

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Q JEE-MAIN 2021

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 :

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Q JEE MAIN_2021_

If the fourth term in the expansion of $\left(x+x^{\log _2 x}\right)^7$ is 4480 , then the value of $x$ where...

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

For integers $n$ and $r$, let $...

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Q JEE MAIN 2021

If $n \geq 2$ is a positive integer, then the sum of the series $...

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Q JEE MAIN 2021

Let $\left(1+x+2 x^2\right) 20=a_0+a_1 x+a_2 x_2+\ldots+a_{40} x^{40}$. then $\mathbf{a}_1+\mathbf{a}_3+\mathbf{a}_5+\ldots+\mathbf{a}_{37}$ is equal to

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Q JEE MAIN 2021

Let $[x]$ denote greatest integer less than or equal to $x$. If for $n \in \mathbb{N},\left(1-x+x^3\right)^n=\sum_{j=0}^{3 n} a_j x^j m$...

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

Let $m, n \in N$ and $\operatorname{gcd}(2, n)=1$, If $$ 30\binom{30}{0}+29\binom{30}{1}+\ldots+2\binom{30}{28}+1\binom{30}{29}=n \cdot 2^m $$ then $n+m$ is equal to (Here...

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Q JEE MAIN 2021

The value of ${ }^{-15} C_1+2 .{ }^{15} C_2-3 .{ }^{15} C_3+\ldots \ldots$. $...

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Q JEE MAIN 2021

The maximum value of the term independent of ' t ' in the expansion of $\left(t^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{t}\right)^{10}$ where $x \in (0,1)$...

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Q JEE MAIN_2022

1`$x$ and $x^2$ in the expansion of $(1+x)^p(1-x)^q, p, q \leq 15$, are -3 and -5 respectively, then the coefficient...

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Q JEE MAIN 2022

If the maximum value of the term independent of $t$ in the expansion of $\left(t^2 x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{t}\right)^{15}, x \geq 0$ is...

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Q JEE MAIN 2022

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left(x^n+\frac{2}{x^5}\right)^7$ is...

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Q JEE MAIN 2022

If $\left({ }^{40} \mathrm{C}_0\right)+\left({ }^{41} \mathrm{C}_1\right)+\left({ }^{42} \mathrm{C}_2\right)+\ldots .+\left({ }^{60} \mathrm{C}_{20}\right)=\frac{m}{n}{ }^{60} \mathrm{C}_{20}, m$ and $n$ are coprime, then $m+n$...

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Q JEE MAIN 2022

Let $n \geq 5$ be an integer. If $9^n-8 n-1=64 \alpha$ and $6^n-5 n-1=25 \beta$, then $\alpha-\beta$ is equal to...

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Q JEE MAIN 2022

The remainder on dividing $1+3+3^2+3^3+\ldots+3^{2021}$ by 50 is $\_\_\_\_$

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Q JEE MAIN 2022

If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $...

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Q JEE MAIN 2022

The term independent of $x$ in the expression of $\left(1-x^2+3 x^3\right)\left(\frac{5}{2} x^3-\frac{1}{5 x^2}\right)^{11}, x \neq 0$ is

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Q JEE MAIN 2022

If the constant term in the expansion of $\left(3 x^3-2 x^2+\frac{5}{x^5}\right)^{10}$ is $2^k . l$, where $l$ is an odd...

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Q JEE MAIN 2022

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\ldots . . x^{500} \ldots>0$, is

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Q JEE MAIN 2022

The number of positive integers $k$ such that the constant term in the binomial expansion of $...

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Q JEE MAIN 2022

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}$ is $5^k l$, where $l, k \in N$ and...

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Q JEE MAIN 2022

If $\sum_{k=1}^{31}\left({ }^{31} C_k\right)\left({ }^{31} C_{k-1}\right)-\sum_{k=1}^{30}\left({ }^{30} C_k\right)\left({ }^{30} C_{k-1}\right)=\frac{\alpha(60!)}{(30!)(31!)}$, Where $\alpha \in R$, then the value of $16 \alpha$...

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Q JEE MAIN 2022

Let $C_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x)^{10}$. If $...

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Q JEE MAIN 2022

The remainder when $(2021)^{2023}$ is divided by 7 is :

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Q JEE MAIN 2022

If $\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^2 \cdot 3^9}+\ldots \cdot \frac{1}{2^{10} \cdot 3}=\frac{\mathrm{K}}{2^{10} \cdot 3^{10}}$, then the remainder when K is divided by...

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Q JEE MAIN 2022

The remainder when $3^{2022}$ is divided by 5 is

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Q JEE MAIN_2023

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

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Q JEE MAIN 2023

The remainder, when $7^{103}$ is divided by 17 is $\_\_\_\_$ $-$

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Q JEE MAIN 2025

The product of the last two digits of $(1919)^{1919}$ is $\_\_\_\_$

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Q JEE MAIN 2025

The number of integral terms in the expansion of $\left(5^{\frac{1}{2}}+7^{\frac{1}{8}}\right)^{1016}$ is

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Q JEE MAIN 2023

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

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Q JEE MAIN 2023

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

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Q JEE MAIN 2023

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

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Q JEE-Main 2023

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

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Q JEE MAIN 2023

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

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Q JEE MAIN_2025

The sum of the series $...

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Q JEE MAIN 2023

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

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Q JEE MAIN 2025

Let $\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots+a_{20} x^{20}$. If $\left(a_1+a_3+a_5+\ldots+a_{19}\right)-11 a_2=121 k$, then $k$ is equal to $\_\_\_\_$

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Q JEE MAIN 2025

If $\sum_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^r}\right) \cdot{ }^{11} C_{r+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to :

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Q JEE MAIN 2025

If $1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k$, where $...

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Q JEE MAIN 2023

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)^n, n \leq 15$. If the sum of the...

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Q JEE MAIN 2023

The number of elements in the set $\left\{n \in N: 10 \leq n \leq 100\right.$ and $3^n-3$ is a multiple...

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Q JEE MAIN 2023

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

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Q JEE MAIN 2023

Let $\left(a+b x+c x^2\right)^{10}=\sum_{i=0}^{20} p_i x^i, a, b, c \in \mathbb{N}$. If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to

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Q JEE MAIN 2023

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

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Q JEE-Main

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion...

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Q JEE MAIN 2023

The mean of the coefficients of $x, x^2$ $\_\_\_\_$ $x^7$ in the binomial expansion of $(2+x)^9$ is $\_\_\_\_$ .

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Q JEE MAIN 2023

The largest natural number $n$ such that $3^n$ divides $66!$ is $\_\_\_\_$

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Q JEE MAIN 2023

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

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Q JEE MAIN 2023

If $\frac{1}{n+1}{ }^n C_n+\frac{1}{n}{ }^n C_{n-1}+\ldots . .+\frac{1}{2}{ }^n C_1+{ }^n C_0=\frac{1023}{10}$ then $n$ is equal to-

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Q JEE MAIN 2023

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

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Q JEE MAIN 2023

If the coefficients of the three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1: 5: 20$,...

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Q JEE MAIN 2023

The coefficient of $x^7$ in $\left(1-x+2 x^3\right)^{10}$ is $\_\_\_\_$

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Q JEE-Main 2023

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

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Q JEE MAIN 2023

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

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Q JEE-Main 2023

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion...

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Q JEE MAIN

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

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Q JEE-Main 2024

If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135,30 and $\frac{10}{3}$, respectively, then $6\left(n^3+x^2+y\right)$ is...

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Q JEE MAIN 2024

Let a $$ \begin{aligned} t a=1+\frac{{ }^2 C_2}{3!}+\frac{{ }^3 C_2}{4!}+\frac{{ }^4 C_2}{5!}+\cdots, & \\ & \quad b=1+\frac{{ }^1 C_0+{ }^1...

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Q JEE MAIN 2024

Let $\alpha=\sum_{\mathrm{r}=0}^{\mathrm{n}}\left(4 \mathrm{r}^2+2 \mathrm{r}+1\right)^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$ and $\beta=\left(\sum_{r=0}^n \frac{{ }^n C_r}{r+1}\right)+\frac{1}{n+1}$. If $140

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Q JEE MAIN 2024

If the constant term in the expansion of (1+2x-3x^3 ) (3/2 x^2-1/3x)^9 is p, then 108p is equal to

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Q JEE MAIN

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

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Q JEE MAIN 2024

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

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Q JEE MAIN 2024

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

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Q JEE-Main 2024

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

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Q JEE-Main 2024

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

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Q JEE MAIN

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

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Q JEE MAIN 2024

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

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Q JEE MAIN 2024

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

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Q JEE MAIN 2025

The remainder, when $7^{103}$ is divided by 23 , is equal to :

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Q JEE MAIN 2025

Let the coefficients of three consecutive terms $T_{r}, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a...

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Q JEE MAIN 2025

The remainder when ${\left( {{{(64)}^{(64)}}} \right)^{(64)}}$ is divided by 7 is equal to

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Q JEE MAIN 2025

In the expansion of ${\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^n},n \in \;{\rm{N}}$ , if the ratio of ${15^{{\rm{th }}}}$ term from...

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Q JEE MAIN 2025

For an integer $n \ge 2$, if the arithmetic mean of all coefficients in the binomial expansion of $...

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Q JEE MAIN 2025

The sum of all rational terms in the expansion of ${(2 + \sqrt 3 )^8}$ is

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Q JEE MAIN 2025

If $...

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Q JEE MAIN 2025

Suppose $A$ and $B$ are the coefficients of $30^{\text {th }}$ and $12^{\text {th }}$ terms respectively in the binomial...

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Q JEE MAIN 2025

The term independent of x in the expansion of $...

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Q JEE MAIN 2025

If in the expansion of $(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}$, the coefficients of x and $x^{2}$ are 1 and -2 , respectively, then $p^{2}+q^{2}$...

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Q JEE MAIN 2025

The least value of n for which the number of integral terms in the Binomial expansion of ${(\sqrt[3]{7} + \sqrt[{12}]{{11}})^n}$...

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Q JEE MAIN 2025

If $\alpha = 1 + \sum\limits_{r = 1}^6 {{{( - 3)}^{r - 1}}} {\quad ^{12}}{{\rm{C}}_{2r - 1}}$, then the distance...

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Q JEE MAIN 2025

Let $^{\rm{n}}{{\rm{C}}_{{\rm{r}} - 1}} = 28{,^{\rm{n}}}{{\rm{C}}_{\rm{r}}} = 56$ and $^{\rm{n}}{{\rm{C}}_{{\rm{r}} + 1}} = 70$. Let $...

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Q JEE MAIN 2025

For some ${\rm{n}} \ne 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of...

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Q JEE MAIN 2025

The sum of all rational terms in the expansion of ${\left( {1 + {2^{1/3}} + {3^{1/2}}} \right)^6}$ is equal to...

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Q JEE MAIN 2025

If $\sum_{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \times 2^{29}$, then $\alpha$ is equal to

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Q JEE MAIN 2025

If $\sum_{r=0}^5 \frac{{ }^{11} C_{2 r+1}}{2 r+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to ..........

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