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

Let \( (1+x+x^{2})^{10}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{20}x^{20} \). If \( (a_{1}+a_{3}+a_{5}+\ldots+a_{19})-11a_{2}=121k \), then \( k \) is equal to __________.

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \( x \) denote the number of defective oranges, then the variance of \( x \) is:

The product of the last two digits of \( (1919)^{1919} \) is ______.

If \( \alpha = 1 + \displaystyle \sum_{r=1}^{6}(-3)^{\,r-1} \, {}^{12}C_{2r-1}, \) then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}\, y + 1 = 0 \) is ______.

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

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

Let \( p \) be the number of all triangles that can be formed by joining the vertices of a regular polygon \( P \) of \( n \) sides and \( q \) be the number of all quadrilaterals that can be formed by joining the vertices of \( P \). If \( p+q=126 \), then the eccentricity of the ellipse \( \dfrac{x^2}{16}+\dfrac{y^2}{n}=1 \).

The sum of the series \( 2\times 1\times {}^{20}C_{4}-3\times 2\times {}^{20}C_{5}+4\times 3\times {}^{20}C_{6}-5\times 4\times {}^{20}C_{7}+\ldots +18\times 17\times {}^{20}C_{20} \) equal to _________.

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

Let \( {}^nC_{r-1} = 28 \), \( {}^nC_r = 56 \) and \( {}^nC_{r+1} = 70 \).
Let \( A(4\cos t, 4\sin t) \), \( B(2\sin t, -2\cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( r \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals:

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

The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( \left(\sqrt[3]{7}+\sqrt[4]{11}\right)^n \) is \( 183 \), is:

If \( \displaystyle \sum_{r=0}^{5}\dfrac{{^{11}C_{\,2r+1}}}{2r+2}=\dfrac{m}{n},\, \gcd(m,n)=1 \), then \( m-n \) is equal to ______.

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 G.P., and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( \left(\sqrt[4]{3}+\sqrt[3]{4}\right)^{12} \). Then \( p+q \) is equal to:

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

If \( 1^2 \cdot ({^{15}C_1} )+ 2^2 \cdot ({^{15}C_2} )+ 3^2 \cdot ({^{15}C_3}) + \cdots + 15^2 \cdot ({^{15}C_{15}}) = 2^m \cdot 3^n \cdot 5^k \), where \( m,n,k \in \mathbb{N} \), then \( m + n + k \) is equal to:

Let \( \alpha, \beta, \gamma \) and \( \delta \) be the coefficients of \( x^{7}, x^{5}, x^{3} \) and \( x \) respectively in the expansion of \( \left(x + \sqrt{x^{3} - 1}\right)^{5} + \left(x - \sqrt{x^{3} - 1}\right)^{5} ,\; x > 1 \). If \( u \) and \( v \) satisfy the equations \( \alpha u + \beta v = 18, \gamma u + \delta v = 20 \), then \( u + v \) equals:

If \( \displaystyle \sum_{r=1}^{9} \left( \frac{r+3}{2^{r}} \right) \, {}^{.9}C_{r} = \alpha \left( \frac{3}{2} \right)^{9} - \beta \), \( \alpha , \beta \in \) \(\mathbb{R}\) , then \( (\alpha + \beta)^{2} \) is equal to:

For some \( n \neq 10 \), let the coefficients of the \( 5^{\text{th}} \), \( 6^{\text{th}} \) and \( 7^{\text{th}} \) terms in the binomial expansion of \( (1 + x)^{n+4} \) be in A.P. Then the largest coefficient in the expansion of \( (1 + x)^{n+4} \) is:

The term independent of \( x \) in the expansion of \( \left( \dfrac{x+1}{(x^{2/3}+1-x^{1/3})} - \dfrac{x-1}{(x-x^{1/2})} \right)^{10} , \; x>1, \) is:

If in the expansion of \( (1 + x)^{p} (1 - x)^{q} \), the coefficients of \( x \) and \( x^{2} \) are \( 1 \) and \( -2 \), respectively, then \( p^{2} + q^{2} \) is equal to:

Suppose \( A \) and \( B \) are the coefficients of \( 30^{\text{th}} \) and \( 12^{\text{th}} \) terms respectively in the binomial expansion of \( (1 + x)^{2n-1} \). If \( 2A = 5B \), then \( n \) is equal to:

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JEE Main 2025 28th Jan Morning Shift

If \( \alpha = 1 + \displaystyle \sum_{r=1}^{6}(-3)^{\,r-1} \, {}^{12}C_{2r-1}, \) then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}\, y + 1 = 0 \) is ______.