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If the constant term in the expansion of \( \left(\dfrac{\sqrt[5]{3}}{x}+\dfrac{2x}{\sqrt[3]{5}}\right)^{12} , \; x\neq 0 \), is \(\alpha \times 2^{8} \times \sqrt[5]{3} \), then \( 25 \alpha \) is equal to:

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 maximum value of \( n \) is:

Let \( a \) be the sum of all coefficients in the expansion of \( (1-2x+2x^{2})^{2023}(3-4x^{2}+2x^{3})^{2024} \) and \( b=\ \displaystyle \lim_{x\to 0}\left(\frac{\displaystyle \int_{0}^{x}\frac{\log(1+t)}{t^{2024}+1}\,dt}{x^{2}}\right) \). If the equations \( cx^{2}+dx+e=0 \) and \( 2bx^{2}+ax+4=0 \) have a common root, where \( c,d,e\in \mathbb{R} \), then \( d:c:e \) equals:

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

Let the coefficient of \( x^{r} \) in the expansion of \( (x+3)^{n-1}+ (x+3)^{n-2}(x+2)+ (x+3)^{n-3}(x+2)^{2}+\cdots +(x+2)^{n-1} \) be \( \alpha_{\gamma} \). If \(\displaystyle \sum_{r=0}^{n}a_{\gamma}=\beta ^{n}-\gamma ^{n},\ \beta ,\gamma \in \mathbb{N} \), then the value of \( \beta ^{2}+\gamma ^{2} \) equals ________.

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

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

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 _________.

Remainder when \( 64^{\,32^{32}} \) is divided by \( 9 \) is equal to ________.

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

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

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

Let \( a = 1 + \frac{{}^{2}C_{2}}{3!} + \frac{{}^{3}C_{2}}{4!} - \frac{{}^{4}C_{2}}{5!} \ldots \), \( b = 1 + \frac{{}^{1}C_{0} + {}^{1}C_{1}}{1!} + \frac{{}^{2}C_{0} + {}^{2}C_{1} + {}^{2}C_{2}}{2!} + \frac{{}^{3}C_{0} + {}^{3}C_{1} + {}^{3}C_{2} + {}^{3}C_{3}}{3!} + \ldots \). Then \( \dfrac{2b}{a^{2}} \) is equal to ________.

If for some \( m ,n\) : \( {^6}C_m + 2({^6}C_{m+1}) + {^6}C_{m+2} > {^8}C_3 \) and \( {^{\,n-1}}P_3 : {^{\,n}}P_4 = 1:8 \) then \( {^{\,n}}P_{m+1} + {^{\,n+1}}C_m \) is equal to:

Suppose \( 2-p, \ p, \ 2-\alpha, \ \alpha \) are the coefficients of four consecutive terms in the expansion of \( (1+x)^{n} \). Then the value of \( p^{2} - \alpha^{2} + 6\alpha + 2p \) equals.

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

If the constant term in the expansion of \( (1 + 2x - 3x^{3}) \left( \dfrac{3}{2}x^{2} - \dfrac{1}{3x} \right)^{9} \) is \( p \), then \( 108p \) is equal to \( \_\_\_\_\_\_\_\_\_\_\_ \).

If \( \frac{{}^{11}C_{1}}{2}+\frac{{}^{11}C_{2}}{3}+\ldots+\frac{{}^{11}C_{9}}{10}=\frac{n}{m} \) with \( \gcd(n,m)=1 \), then \( n+m \) is equal to __________.

The remainder when \( 428^{2024} \) is divided by \( 21 \) is ________.

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

Let \( 0 \le r \le n \). If \( \ ^{n+1}C_{r+1} : \ ^{n}C_{r} : \ ^{n-1}C_{r-1} = 55 : 35 : 21 \), then \( 2n + 5r \) is equal to:

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

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 coefficients of \( x^3 \) and \( x^{-13} \) is equal to ______

Let \( \alpha = \displaystyle \sum_{r=0}^{n} (4r^{2} + 2r + 1)\, { }^{n}C_{r} \), and \( \beta = \left( \displaystyle \sum_{r=0}^{n} \frac{{ }^{n}C_{r}}{r+1} \right) + \frac{1}{n+1} \). If \( 140 < \frac{2\alpha}{\beta} < 281 \), then the value of \( n \) is _______.

Let \( \alpha = \displaystyle\sum_{k=0}^{n} \left( \dfrac{({}^{n}C_{k})}{k+1} \right)^{2} \) and \( \beta =\displaystyle \sum_{k=0}^{n-1} \left( \dfrac{{}^{n}C_{k}\,{}^{n}C_{k+1}}{k+2} \right) \). If \( 5\alpha = 6\beta \), then \( n \) equals _______.

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JEE Main 2024 1st Feb Morning Shift

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