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Let $\alpha > 0$, be the smallest number such that the expansion of $\left( x^{\frac{2}{3}} + \frac{2}{x^3} \right)^{30}$ has a term $\beta x^{-\alpha}, \beta \in N$.

Then $\alpha$ is equal to _____.

Let \( x = (8\sqrt{3} + 13)^{13} \) and \( y = (7\sqrt{2} + 9)^{9} \). If \( [t] \) denotes the greatest integer \( \le t \), then:

If \( \dfrac{1}{n+1} \, {}^{n}C_{n} + \dfrac{1}{n} \, {}^{n}C_{n-1} + \ldots + \dfrac{1}{2} \, {}^{n}C_{1} + {}^{n}C_{0} = \dfrac{1023}{10} \) then \( n \) is equal to:

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 \( n \in \mathbb{N} \)

The value of \( \dfrac{1}{1!50!}+\dfrac{1}{3!48!}+\dfrac{1}{5!46!}+\ldots+\dfrac{1}{49!2!}+\dfrac{1}{51!1!} \) is:

The mean of the coefficients of \( x, x^{2}, \ldots, x^{7} \) in the binomial expansion of \( (2+x)^{9} \) is __________ .

The remainder on dividing \( 5^{99} \) by \( 11 \) is ______.

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

If the coefficients of 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:

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

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

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

If $\left({}^{30}C_1\right)^2 + 2\left({}^{30}C_2\right)^2 + 3\left({}^{30}C_3\right)^2 + \ldots + 30\left({}^{30}C_{30}\right)^2 = \frac{\alpha 60!}{(30!)^2}$ then $\alpha$ is equal to:

If \( a_r \) is the coefficient of \( x^{10-r} \) in the Binomial expansion of \( (1+x)^{10} \), then \( \displaystyle \sum_{r=1}^{10} r^3 \left( \frac{a_r}{a_{r-1}} \right)^2 \) is equal to:

The coefficient of \( x^{-6} \), in the expansion of \( \left(\dfrac{4x}{5}+\dfrac{5}{2x^{2}}\right)^{9} \), is ______.

If the coefficient of \( x^{15} \) in the expansion of \( \left(ax^{3}+\frac{1}{bx^{1/3}}\right)^{15} \) is equal to the coefficient of \( x^{-15} \) in the expansion of \( \left(ax^{1/3}-\frac{1}{bx^{3}}\right)^{15} \), where \( a \) and \( b \) are positive real numbers, then for each such ordered pair \( (a,b) \),

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

Let the sixth term in the binomial expansion of \( \left( \sqrt{2^{\log_{2}(10-3^{x})}} + \sqrt[5]{2^{(x-2)\log_{2} 3}} \right)^{m} \), 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 expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of \( x \) is _______.

Let the sum of the coefficients of the first three terms in the expansion of \( \left(x-\dfrac{3}{x^{2}}\right)^{n} , \ x \neq 0, \ n \in \mathbb{N}, \) be \( 376 \). Then the coefficient of \( x^{4} \) is __________.

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

Fractional part of the number \( \dfrac{4^{2022}}{15} \) is equal to:

Let \( K \) be the sum of the coefficients of the odd powers of \( x \) in the expansion of \( (1+x)^{99} \). Let \( a \) be the middle term in the expansion of \( \left( 2+\dfrac{1}{\sqrt{2}} \right)^{200} \). If \( \dfrac{{}^{200}C_{99}K}{a}=\dfrac{2^{\ell}m}{n} \), where \( m \) and \( n \) are odd numbers, then the ordered pair \( (\ell , n) \) is equal to:

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

The remainder when \( (2023)^{2023} \) is divided by \( 35 \) is ______.

Let \( x \) and \( y \) be distinct integers where \( 1 \le x \le 25 \) and \( 1 \le y \le 125 \). Then, the number of ways of choosing \( x \) and \( y \), such that \( x + y \) is divisible by \( 5 \), is __________.

The constant term in the expansion of \( \left(2x+\frac{1}{x^7}+3x^2\right)^5 \) is _______.

\( 50^{\text{th}} \) root of a number \( x \) is \( 12 \) and \( 50^{\text{th}} \) root of another number \( y \) is \( 18 \). Then the remainder obtained on dividing \( (x + y) \) by \( 25 \) is __________.

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:

If the constant term in the binomial expansion of \( \left(\dfrac{x^{5/2}}{2}-\dfrac{4}{x^{l}}\right)^{9} \) is \( -84 \) and coefficient of \( x^{-3l} \) is \( 2^{\alpha}\beta \), where \( \beta<0 \) is an odd number, then \( |\alpha l-\beta| \) is equal to ________.

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

The remainder, when \( 7^{103} \) is divided by \( 17 \), is __________.

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

\( \displaystyle \sum_{k=0}^{6} {}^{\,51-k}\!C_{3} \) is equal to:

The remainder, when \( 19^{200} + 23^{200} \) is divided by \( 49 \), is __________.

\( 25^{190} - 19^{190} - 8^{190} + 2^{190} \) is divisible by:

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

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

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

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

Let \( (a+bx+cx^{2})^{10}=\sum\limits_{i=0}^{20} p_{i}x^{i},\; a,b,c\in \mathbb{N} \). If \( p_{1}=20 \) and \( p_{2}=210 \), then \( 2(a+b+c) \) is equal to:

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

Suppose \( \displaystyle \sum_{r=0}^{2023} r^{2} \, {^{2023}\!C_{r}} = 2023 \times \alpha \times 2^{2022} \). Then the value of \( \alpha \) is _______.

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

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JEE Main 2023 31st Jan Morning Shift

The remainder on dividing \( 5^{99} \) by \( 11 \) is ______.