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Given three indentical bags each containing \( 10 \) balls, whose colours are as follows :

A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is \( p \) and if the ball is Green, the probability that it is from bag III is \( q \), then the value of \( \left( \frac{1}{p} + \frac{1}{q} \right) \) is:

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:

Let \( A = \{ 1, 2, 3, \ldots , 100 \} \) and \( R \) be a relation on \( A \) such that \( R = \{ (a,b) : a = 2b + 1 \} \). Let \( (a_{1}, a_{2}), (a_{2}, a_{3}), (a_{3}, a_{4}), \ldots , (a_{k}, a_{k+1}) \) be a sequence of \( k \) elements of \( R \) such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer \( k \), for which such a sequence exists, is equal to:

If \( \displaystyle \sum_{r=0}^{10} \left[ \frac{10^{\, r+1} - 1}{10^{\, r}} \right] {^{11}\!C_{r+1}} = \frac{\alpha^{11} - 1^{11}}{10^{10}} \), then \( \alpha \) is equal to:

\( 4 \displaystyle\int_{0}^{1} \left( \dfrac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}} \right) dx - 3 \log_{e}(\sqrt{3})\) is equal to:

If \( \theta\in \left[ -\dfrac{7\pi}{6}, \dfrac{4\pi}{3} \right] \), then the number of solutions of \( \sqrt{3}\ {cosec^{2}}\theta - 2(\sqrt{{3}-1)}\,cosec\theta - 4 = 0 \), is equal to:

Let \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \vec{c} \) be such that \( (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \) and \( \vec{a} \cdot \vec{c} = 3 \). If \( \vec{b} \times \vec{c} = \vec{d} \), then \( \left| \vec{a} \cdot \vec{d} \right| \) is equal to:

The number of terms of an A.P. is even; the sum of all the odd terms is \( 24 \), the sum of all the even terms is \( 30 \) and the last term exceeds the first by \( \frac{21}{2} \). Then the number of terms which are integers in the A.P. is:

Let the area of the triangle formed by a straight line \( L: x + by + c = 0 \) with co-ordinate axes be \( 48 \) square units. If the perpendicular drawn from the origin to the line \( L \) makes an angle of \( 45^\circ \) with the positive \( x \)-axis, then the value of \( b^2 + c^2 \) is:

The line \( L_{1} \) is parallel to the vector \( \vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k} \) and passes through the point \( (7, 6, 2) \) and the line \( L_{2} \) is parallel to the vector \( \vec{b} = 2\hat{i} + \hat{j} + 3\hat{k} \) and passes through the point \( (5, 3, 4) \). The shortest distance between the lines \( L_{1} \) and \( L_{2} \) is:

Let \( (a,b) \) be the point of intersection of the curve \( x^{2} = 2y \) and the straight line \( y - 2x - 6 = 0 \) in the second quadrant. Then the integral \( I = \int_{a}^{b} \dfrac{9x^{2}}{1 + 5^{x}} \, dx \) is equal to:

If the image of the point \( P(1, 0, 3) \) in the line joining the points \( A(4, 7, 1) \) and \( B(3, 5, 3) \) is Q\((\alpha, \beta, \gamma)\), then \( \alpha + \beta + \gamma \) is equal to:

Let the point \( P \) of the focal chord \( PQ \) of the parabola \( y^{2}=16x \) be \( (1,-4) \). If the focus of the parabola divides the chord \( PQ \) in the ratio \( m:n \), \( \gcd(m,n)=1 \), then \( m^{2}+n^{2} \) is equal to:

If the mean and the variance of \( 6,4,a,8,b,12,10,13 \) are \( 9 \) and \( 9.25 \) respectively, then \( a+b+ab \) is equal to:

Let \( f:[1,\infty)\to \ [2,\infty) \) be a differentiable function. If \( 10\int_{1}^{x} f(t)\,dt=5x f(x)-x^{5}-9 \) for all \( x\ge 1 \), then the value of \( f(3) \) is:

The number of ways, in which the letters \( A,B,C,D,E \) can be placed in the \( 8 \) boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:

If the domain of the function \( f(x)=\dfrac{1}{\sqrt{10+3x-x^{2}}}+\dfrac{1}{\sqrt{x+|x|}} \) is \( (a,b) \), then \( (1+a^{2})+b^{2} \) is equal to:

Let \( A \) be a \( 3\times 3 \) real matrix such that \( A^{2}(A-2I)-4(A-I)=0 \), where \( I \) and \( O \) are the identity and null matrices, respectively. If \( A^{5}=\alpha A^{2}+\beta A+\gamma I \), where \( \alpha,\beta \) and \( \gamma \) are real constants, then \( \alpha+\beta+\gamma \) is equal to:

If the system of equations
\( 2x + \lambda y + 3z = 5 \)
\( 3x + 2y - z = 7 \)
\( 4x + 5y + \mu z = 9 \)
has infinitely many solutions, then \( \left(\lambda^{2} + \mu^{2}\right) \) is equal to :

If \( \lim_{x \to 0} \dfrac{\cos(2x) + a \cos(4x) - b}{x^{4}} \) is finite, then \( (a + b) \) is equal to:

Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} + 2y \sec^{2} x = 2 \sec^{2} x + 3 \tan x \cdot \sec^{2} x \) such that \( y(0) = \dfrac{5}{4} \). Then \( 12 \left( y\!\left(\dfrac{\pi}{4}\right) - e^{-2} \right) \) is equal to __________________.

If the set of all \( a \in \mathbb{R} - \{1\} \), for which the roots of the equation \( (1 - a)x^{2} + 2(a - 3)x + 9 = 0 \) are positive is \( (-\infty, -\alpha] \cup [\beta, \gamma) \), then \( 2\alpha + \beta + \gamma \) is equal to ________________.

If the sum of the first \( 10 \) terms of the series \( \dfrac{4 \cdot 1}{1 + 4 \cdot 1^{4}} + \dfrac{4 \cdot 2}{1 + 4 \cdot 2^{4}} + \dfrac{4 \cdot 3}{1 + 4 \cdot 3^{4}} + \ldots \) is \( \dfrac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is equal to __________.

\( y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right) \), then \( (x - y)^2 + 3y^2 \) is equal to ____________.

Let \( A(4,-2) \), \( B(1,1) \) and \( C(9,-3) \) be the vertices of a triangle \( ABC \). Then the maximum area of the parallelogram \( AFDE \), formed with vertices \( D \), \( E \) and \( F \) on the sides \( BC \), \( CA \) and \( AB \) of the triangle \( ABC \) respectively, is __________________.

JEE Main 2025 2nd April Evening Shift

Mathematics Physics Chemistry

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\( y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right) \), then \( (x - y)^2 + 3y^2 \) is equal to ____________.