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If the constant term in the expansion of \( \left( 3x^{3} - 2x^{2} + \dfrac{5}{x^{5}} \right)^{10} \) is \( 2^{k} \cdot l \), where \( l \) is an odd integer, then the value of \( k \) is equal to:

The term independent of \( x \) in the expansion of \( \left(1 - x^2 + 3x^3\right)\left(\dfrac{5}{2}x^3 - \dfrac{1}{5x^2}\right)^{11} , \; x \ne 0 \) is:

If the maximum value of the term independent of \( t \) in the expansion of \( \left(t^{2}x^{1/5}+\dfrac{(1-x)^{1/10}}{t}\right)^{15},\ x\ge 0, \) is \( K \), then \( 8K \) is equal to ______.

If \( \displaystyle \sum_{k=1}^{10} K^{2}\left({10}_{C_{K}}\right)^{2}=22000L \), then \( L \) is equal to ________.

Let for the \( 9^{\text{th}} \) term in the binomial expansion of \( (3+6x)^{n} \), in the increasing powers of \( 6x \), to be the greatest for \( x=\dfrac{3}{2} \), the least value of \( n \) is \( n_{0} \). If \( k \) is the ratio of the coefficient of \( x^{6} \) to the coefficient of \( x^{3} \), then \( k+n_{0} \) is equal to:

If the coefficients of \( x \) and \( x^{2} \) in the expansion of \( (1+x)^{p}(1-x)^{q} \), \( p,q\leq15 \), are \( -3 \) and \( -5 \) respectively, then the coefficient of \( x^{3} \) is equal to ______.

Let the coefficients of the middle terms in the expansion of \( \left(\dfrac{1}{\sqrt{6}} + \beta x\right)^4 \), \( (1 - 3 \beta x)^2 \) and \( \left(1 - \dfrac{\beta}{2} x\right)^6 \), \( \beta > 0 \) , respectively from the first three terms of an A.P. If \( d \) is the common difference of this A.P., then \( 50 -\dfrac{ 2d}{\beta^2} \) is equal to ____________.

If \( 1 + (2 + {}^{49}C_{1} + {}^{49}C_{2} + \ldots + {}^{49}C_{49})({}^{50}C_{2} + {}^{50}C_{4} + \ldots + {}^{50}C_{50}) \) is equal to \( 2^{n} \cdot m \), where \( m \) is odd, then \( n + m \) is equal to ______________.

If the sum of the coefficients of all the position powers of \( x \), in the Binomial expansion of \( \left(x^{n}+\frac{2}{x^{5}}\right)^{7} \) is \( 939 \), then the sum of all the possible values of \( n \) is __________.

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

The remainder when \( 7^{2022} + 3^{2022} \) is divided by \( 5 \) is:

The number of positive integers \( k \) such that the constant term in the binomial expansion of \( \left(2x^3 + \frac{3}{x^k}\right)^{12} , \, x \neq 0 \) is \( 2^{8} \).\(\ell\) where \( \ell \) is an odd integer is ________.

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

Let the coefficients of \( x^{-1} \) and \( x^{-3} \) in the expansion of \( \left(2x^{1/5}-\frac{1}{x^{1/5}}\right)^{15} \), \( x>0 \), be \( m \) and \( n \) respectively. If \( r \) is a positive integer such that \( mn^{2}=\,^{15}\!C_{r}\cdot 2^{r} \), then the value of \( r \) is equal to ______.

If $\sum_{k=1}^{31} \left({}^{31}C_k\right)\left({}^{31}C_{k-1}\right) - \displaystyle \sum_{k=1}^{30} \left({}^{30}C_k\right)\left({}^{30}C_{k-1}\right) = \frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha \in R$, then the value of $16\alpha$ is equal to:

The remainder when \( (11)^{1011}+(1011)^{11} \) is divided by \( 9 \) is:

If the coefficient of \( x^{10} \) in the binomial expansion of \( \left( \dfrac{\sqrt{x}}{5^{1/4}} + \dfrac{\sqrt{5}}{x^{1/3}} \right)^{60} \) is \( {5^{k}}{} .l\), where \( l, k \in \mathbb{N} \) and \( l \) is co-prime, to \( 5 \), then \( k \) is equal to _______.

The remainder when \( (2021)^{2022}+(2022)^{2021} \) is divided by \( 7 \) is:

\(\displaystyle \sum_{i,j=0}^{n} {}^{n}C_{i}\,{}^{n}C_{j} \) is equal to \( i\neq j \)

The remainder when \( 3^{2022} \) is divided by \( 5 \) is :

The remainder on dividing \( 1 + 3 + 3^{2} + 3^{3} + \ldots + 3^{2021} \) by \( 50 \) is ________ .

If $\left(^{40}C_{0}\right) + \left(^{41}C_{1}\right) + \left(^{42}C_{2}\right) + \dots + \left(^{60}C_{20}\right) = \frac{m}{n} {}^{60}C_{20}$, $m$ and $n$ are coprime, then $m+n$ is equal to _____________.

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

If the sum of the co-efficients of all the positive even powers of \( x \) in the binomial expansion of \( \left( 2x^3 + \frac{3}{x} \right)^{10} \) is \( 5^{10} - \beta \cdot 3^9 \), then \( \beta \) is equal to __________.

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JEE Main 2022 26th July Evening Shift

\(\displaystyle \sum_{i,j=0}^{n} {}^{n}C_{i}\,{}^{n}C_{j} \) is equal to \( i\neq j \)