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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:

Two parabolas have the same focus \( (4,3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersects at the points \( A \) and \( B \), then \( (AB)^2 \) is equal to:

Let the line \( x+y=1 \) meet the circle \( x^2+y^2=4 \) at the points \( A \) and \( B \). If the line perpendicular to \( AB \) and passing through the mid point of the chord \( AB \) intersects the circle at \( C \) and \( D \), then the area of the quadrilateral \( ADBC \) is equal to:

Consider an A.P. of positive integers, whose sum of the first three terms is \( 54 \) and the sum of the first twenty terms lies between \( 1600 \) and \( 1800 \). Then its \( 11^{\text{th}} \) term is:

Let \( \vec a=\hat i+2\hat j+\hat k \) and \( \vec b=2\hat i+7\hat j+3\hat k \). Let \( L_1:\ \vec r=(-\hat i+2\hat j+\hat k)+\lambda \vec a,\, \lambda\in \mathbb{R} \) and \( L_2:\ \vec r=(\hat j+\hat k)+\mu \vec b,\, \mu\in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2 \), and is parallel to \( \vec a+\vec b \), then \( L_3 \) passes through the point:

Let \( \vec a=2\hat i-\hat j+3\hat k \), \( \vec b=3\hat i-5\hat j+\hat k \) and \( \vec c \) be a vector such that \( \vec a\times \vec c=\vec c\times \vec b \) and \( (\vec a+\vec c)\cdot(\vec b+\vec c)=168 \). Then the maximum value of \( |\vec c|^2 \) is:

Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \displaystyle \sum_{i=1}^{10} (x_i-2)=30 \), \( \displaystyle \sum_{i=1}^{10} (x_i-\beta)^2=98 \), \( \beta>2 \), and their variance is \( \frac{4}{5} \). If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1-1)+4\beta \), \( 2(x_2-1)+4\beta \), \( \ldots \), \( 2(x_{10}-1)+4\beta \), then \( \frac{\beta \mu}{\sigma^2} \) is equal to:

The integral \( 80 \int_{0}^{\pi/4} (\frac{\sin \theta+\cos \theta}{9+16\sin 2\theta})\, d\theta \)is equal to:

Let \( \triangle ABC \) be a triangle formed by the lines \( 7x-6y+3=0 \), \( x+2y-31=0 \) and \( 9x-2y-19=0 \). Let the point \( (h,k) \) be the image of the centroid of \( \triangle ABC \) in the line \( 3x+6y-53=0 \). Then \( h^2+k^2+hk \) is equal to:

The value of\( \displaystyle \lim_{n \to \infty}\left( \sum_{k=1}^{n} \frac{k^3+6k^2+11k+5}{(k+3)!} \right) \) is:

Let \( y=y(x) \) be the solution of the differential equation \( \cos x(\log_e(\cos x))^2\,dy+(\sin x-3y\sin x\log_e(\cos x))\,dx=0 \), \( x\in\left(0,\frac{\pi}{2}\right) \). If \( y\left(\frac{\pi}{4}\right)=-\frac{1}{\log_e 2} \), then \( y\left(\frac{\pi}{6}\right) \) is equal to:

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If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij}= \displaystyle \sum_{k=1}^{2} a_{ik}A_{jk} \), \( 1\le i,j\le 2 \), and \( C=[C_{ij}] \), then \( 8|C| \) is equal to:

Let \( |z_1 - 8 - 2i| \le 1 \) and \( |z_2 - 2 + 6i| \le 2 \), \( z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is:

Let \( P \) be the set of seven digit numbers with sum of their digit equal to \( 11 \). If the numbers in \( P \) are formed by using the digits \( 1 \), \( 2 \) and \( 3 \) only, then the number of elements in the set \( P \) is:

Let the area of region \( \{(x, y) : 2y \le x^2 + 3,\; y + |x| \le 3,\; y \ge0| x - 1|\} \) be \( A \). Then \( 6A \) is equal to:

The number of solutions of the equation \( \left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right)\left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is:

Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1,\, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be \( 4 \). If \( E_1 \) and \( E_2 \) meet at \( A \), \( B \), \( C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals:

Let \( M \) and \( m \) respectively be the maximum and the minimum values of

Then \( M^4 - m^4 \) is equal to:

Define a relation \( R \) on the interval \( \left[0, \frac{\pi}{2}\right) \) by \( x \,R\, y \) if and only if \( \sec^2 x - \tan^2 y = 1 \). Then \( R \) is:

Let \( L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2} \) and \( L_2 : \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \( (\alpha, \beta, \gamma) \) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |\,5\alpha - 11\beta - 8\gamma\,| \) equals:

Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which

$\lim_{x \to 0^+} \left( x \left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \dots + \left[\frac{p}{x}\right] \right) - x^2 \left( \left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \dots + \left[\frac{9^2}{x^2}\right] \right) \right) \geq 1$

is equal to ______.

Let \( f : (0,\infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \ne 0 \), \( \displaystyle \int_{0}^{1} f(\lambda x)\, d\lambda = a f(x) \) and \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \) then \( 16 - f'\!\left(\frac{1}{16}\right) \) is equal to ______.

The number of \( 6 \)-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.

Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum\limits_{x \in S} (2x - 1)^2 \) is equal to ________.

Let \( S = \{ m \in \mathbb{Z} : A^{m^2} + A^m = 3I - A^{-6} \} \), where. Then \( n(S) \) is equal to ________.

JEE Main 2025 29th Jan Morning Shift

Mathematics Physics Chemistry

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Two parabolas have the same focus \( (4,3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersects at the points \( A \) and \( B \), then \( (AB)^2 \) is equal to: