JEE Main 2025 Apr 4 Shift 2 — Mathematics

Let the domains of the functions $f(x)=\log_4\log_3\log_7(8-\log_2(x^2+4x+5))$ and $g(x)=\sin^{-1}\left(\dfrac{7x+10}{x-2}\right)$ be $(\alpha,\beta)$ and $[\gamma,\delta]$, respectively. Then $\alpha^2+\beta^2+\gamma^2+\delta^2$ is equal to :

Let $A=\{-3,-2,-1,0,1,2,3\}$ and $R$ be a relation on $A$ defined by $xRy$ if and only if $2x-y\in\{0,1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations, respectively. Then $l+m+n$ is equal to :

Let the product of $\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta$ and $\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta$ be $\alpha+i\beta$, $i=\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha+\beta$ respectively. Then $p+q$ is equal to :

Let the matrix $A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$ satisfy $A^n=A^{n-2}+A^2-I$ for $n\geq 3$. Then the sum of all the elements of $A^{50}$ is :

If the sum of the first 20 terms of the series $$\frac{4\cdot 1}{4+3\cdot 1^2+1^4}+\frac{4\cdot 2}{4+3\cdot 2^2+2^4}+\frac{4\cdot 3}{4+3\cdot 3^2+3^4}+\frac{4\cdot 4}{4+3\cdot 4^2+4^4}+\ldots$$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to :

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and $p$ respectively and the sum and the product of the elements of B be 36 and $q$ respectively. Let $d$ and $D$ be the common differences of AP's in A and B respectively such that $D=d+3$, $d>0$. If $\dfrac{p+q}{p-q}=\dfrac{19}{5}$, then $p-q$ is equal to

If $1^2\cdot\left({}^{15}\mathrm{C}_1\right)+2^2\cdot\left({}^{15}\mathrm{C}_2\right)+3^2\cdot\left({}^{15}\mathrm{C}_3\right)+\ldots+15^2\cdot\left({}^{15}\mathrm{C}_{15}\right)=2^{\mathrm{m}}\cdot 3^{\mathrm{n}}\cdot 5^{\mathrm{k}}$, where $\mathrm{m},\mathrm{n},\mathrm{k}\in\mathbf{N}$, then $\mathrm{m}+\mathrm{n}+\mathrm{k}$ is equal to :

Let the mean and the standard deviation of the observation $2,3,3,4,5,7,a,b$ be $4$ and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is :

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2\sqrt{2}$ units from the origin, respectively. If the point $(1,\mathrm{k})$ lies on the parabola, then a possible value of $\mathrm{k}$ is :

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $\mathrm{E}:\dfrac{x^2}{4^2}+\dfrac{y^2}{3^2}=1$ at the points A and B. Let the line $y=x$ intersect E at the points C and D. Then the area of the quadrilateral ABCD is equal to :

The centre of a circle C is at the centre of the ellipse $\mathrm{E}:\dfrac{x^2}{\mathrm{a}^2}+\dfrac{y^2}{\mathrm{b}^2}=1$, $\mathrm{a}>\mathrm{b}$. Let C pass through the foci $\mathrm{F}_1$ and $\mathrm{F}_2$ of E such that the circle C and the ellipse E intersect at four points. Let P be one of these four points. If the area of the triangle $\mathrm{PF}_1\mathrm{F}_2$ is 30 and the length of the major axis of E is 17, then the distance between the foci of E is :

Let the sum of the focal distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}:\dfrac{x^2}{\mathrm{a}^2}-\dfrac{y^2}{\mathrm{b}^2}=1$ be $8\sqrt{\dfrac{5}{3}}$. If for H, the length of the latus rectum is $l$ and the product of the focal distances of the point P is $\mathrm{m}$, then $9l^2+6\mathrm{m}$ is equal to :

The sum of the infinite series $\cot^{-1}\left(\dfrac{7}{4}\right)+\cot^{-1}\left(\dfrac{19}{4}\right)+\cot^{-1}\left(\dfrac{39}{4}\right)+\cot^{-1}\left(\dfrac{67}{4}\right)+\ldots$ is :

Let the values of $p$, for which the shortest distance between the lines $\dfrac{x+1}{3}=\dfrac{y}{4}=\dfrac{z}{5}$ and $\vec{r}=\left(p\hat{i}+2\hat{j}+\hat{k}\right)+\lambda\left(2\hat{i}+3\hat{j}+4\hat{k}\right)$ is $\dfrac{1}{\sqrt{6}}$, be $a, b$, $(a<b)$. Then the length of the latus rectum of the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ is :

Let A be the point of intersection of the lines $\mathrm{L}_1:\dfrac{x-7}{1}=\dfrac{y-5}{0}=\dfrac{z-3}{-1}$ and $\mathrm{L}_2:\dfrac{x-1}{3}=\dfrac{y+3}{4}=\dfrac{z+7}{5}$. Let B and C be the points on the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ respectively such that $\mathrm{AB}=\mathrm{AC}=\sqrt{15}$. Then the square of the area of the triangle ABC is :

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1, f'(2)=4$. Let $\lim\limits_{x\to 0}\left(f(2+x)\right)^{3/x}=\mathrm{e}^{\alpha}$. Then the number of times the curve $y=4x^3-4x^2-4(\alpha-7)x-\alpha$ meets $x$-axis is :

Let $\mathrm{a}>0$. If the function $f(x)=6x^3-45\mathrm{a}x^2+108\mathrm{a}^2x+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1x_2=54$, then $\mathrm{a}+x_1+x_2$ is equal to :

Let $f(x)+2f\left(\dfrac{1}{x}\right)=x^2+5$ and $2g(x)-3g\left(\dfrac{1}{2}\right)=x$, $x>0$. If $\alpha=\displaystyle\int_1^2 f(x)\,\mathrm{d}x$, and $\beta=\displaystyle\int_1^2 g(x)\,\mathrm{d}x$, then the value of $9\alpha+\beta$ is :

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}:y^2=x-2$ at the point B in the first quadrant. The area, of the region bounded by the line AB, parabola P and the $x$-axis, is :

If a curve $y=y(x)$ passes through the point $\left(1,\dfrac{\pi}{2}\right)$ and satisfies the differential equation $\left(7x^4\cot y-\mathrm{e}^x\operatorname{cosec} y\right)\dfrac{\mathrm{d}x}{\mathrm{d}y}=x^5$, $x\geq 1$, then at $x=2$, the value of $\cos y$ is :

If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\displaystyle\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\dfrac{1}{\alpha^{\mathrm{k}}}\right)^2=20$, then $\mathrm{n}$ is equal to __________.

Let $\mathrm{m}$ and $\mathrm{n}$, $(\mathrm{m}<\mathrm{n})$, be two 2-digit numbers. Then the total numbers of pairs $(\mathrm{m},\mathrm{n})$, such that $\gcd(\mathrm{m},\mathrm{n})=6$, is __________.

A card from a pack of 52 cards is lost. From the remaining 51 cards, $\mathrm{n}$ cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\dfrac{11}{50}$, then $\mathrm{n}$ is equal to __________.

Let the three sides of a triangle ABC be given by the vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let G be the centroid of the triangle ABC. Then $6\left(|\overrightarrow{\mathrm{AG}}|^2+|\overrightarrow{\mathrm{BG}}|^2+|\overrightarrow{\mathrm{CG}}|^2\right)$ is equal to __________.

If $\displaystyle\int\dfrac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9}\,\mathrm{d}x=\dfrac{1}{\mathrm{m}}\left(\left(\sqrt{1+x^2}+x\right)^{\mathrm{n}}\left(\mathrm{n}\sqrt{1+x^2}-x\right)\right)+\mathrm{C}$ where C is the constant of integration and $\mathrm{m},\mathrm{n}\in\mathbf{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to __________.