JEE Main 2025 Apr 2 Shift 2 — Mathematics

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

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 :

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2I)-4(A-I)=O$, 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 $(\lambda^2+\mu^2)$ 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 :

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 :

an arrangement of 8 boxes in three rows — top row of 3 boxes, middle row of 3 boxes, bottom row of 2 boxes, with the rows offset from one another

If $\displaystyle\sum_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^r}\right) \cdot {}^{11}C_{r+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ 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 :

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

	Red	Blue	Green
Bag I	3	2	5
Bag II	4	3	3
Bag III	5	1	4

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 :

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 $45°$ with the positive $x$-axis, then the value of $b^2+c^2$ is :

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 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 :

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

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 $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}+2\hat{j}+5\hat{k}$ 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 :

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

$4\displaystyle\int_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)\mathrm{d}x-3\log_e\left(\sqrt{3}\right)$ is equal to :

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 $I=\displaystyle\int_a^b \frac{9x^2}{1+5^x}\,\mathrm{d}x$ is equal to :

Let $f: [1, \infty) \to [2, \infty)$ be a differentiable function. If $10\displaystyle\int_1^x f(\mathrm{t})\,\mathrm{dt}=5xf(x)-x^5-9$ for all $x \geqslant 1$, then the value of $f(3)$ is :

If the set of all $a \in \mathbf{R}-\{1\}$, 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 $\frac{4 \cdot 1}{1+4 \cdot 1^4}+\frac{4 \cdot 2}{1+4 \cdot 2^4}+\frac{4 \cdot 3}{1+4 \cdot 3^4}+\ldots$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\gcd(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to _________.

If $y=\cos\left(\frac{\pi}{3}+\cos^{-1}\frac{x}{2}\right)$, 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 _________.

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}+2y\sec^2 x=2\sec^2 x+3\tan x \cdot \sec^2 x$ such that $y(0)=\frac{5}{4}$. Then $12\left(y\left(\frac{\pi}{4}\right)-\mathrm{e}^{-2}\right)$ is equal to _________.