JEE Main 2025 Jan 22 Shift 1 — Mathematics

Let $A=\{1,2,3,\ldots,10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m<n \text{ and } \gcd(m, n)=1\right\}$. Then $n(B)$ is equal to :

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

The product of all solutions of the equation $e^{5(\log_e x)^2+3}=x^8, x>0$, is :

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=\frac{-\pi}{4}$, $\arg(z_2)=0$ and $\arg(z_3)=\frac{\pi}{4}$. If $\left|z_1\bar{z}_2+z_2\bar{z}_3+z_3\bar{z}_1\right|^2=\alpha+\beta\sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2+\beta^2$ is :

If $\sum_{r=1}^{n} T_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\lim_{n \to \infty} \sum_{r=1}^{n}\left(\frac{1}{T_r}\right)$ is equal to :

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to :

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is :

A coin is tossed three times. Let X denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of X, then the value of $64(\mu+\sigma^2)$ is :

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n)=1$, then $m+n$ is equal to :

Let the triangle PQR be the image of the triangle with vertices $(1,3)$, $(3,1)$ and $(2,4)$ in the line $x+2y=2$. If the centroid of $\Delta$PQR is the point $(\alpha, \beta)$, then $15(\alpha-\beta)$ is equal to :

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle C at exactly two points. If the set of all possible values of r is the interval $(\alpha, \beta)$, then $3\beta-2\alpha$ is equal to :

Let the parabola $y=x^2+px-3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(-1,-1)$ passes through the points P, Q and R, then the area of $\Delta$PQR is :

Let the foci of a hyperbola be $(1,14)$ and $(1,-12)$. If it passes through the point $(1,6)$, then the length of its latus-rectum is :

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left((\sec^{-1}x)^2+(\operatorname{cosec}^{-1}x)^2\right)$ is :

Let $L_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $L_2: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$ ?

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(x+y)=f(x)f(y)$ for all $x, y \in \mathbb{R}$. If $f'(0)=4a$ and $f$ satisfies $f''(x)-3af'(x)-f(x)=0$, $a>0$, then the area of the region $R=\{(x, y) \mid 0 \le y \le f(ax), 0 \le x \le 2\}$ is :

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_e f(n)$ is equal to :

Let for $f(x)=7\tan^8 x+7\tan^6 x-3\tan^4 x-3\tan^2 x$, $I_1=\int_0^{\pi/4} f(x)\,dx$ and $I_2=\int_0^{\pi/4} x f(x)\,dx$. Then $7I_1+12I_2$ is equal to :

The area of the region, inside the circle $(x-2\sqrt{3})^2+y^2=12$ and outside the parabola $y^2=2\sqrt{3}\,x$ is :

Let $x=x(y)$ be the solution of the differential equation $y^2 dx+\left(x-\frac{1}{y}\right)dy=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

Let A be a square matrix of order 3 such that $\det(A)=-2$ and $\det(3\,\mathrm{adj}(-6\,\mathrm{adj}(3A)))=2^{m+n} \cdot 3^{mn}$, $m>n$. Then $4m+2n$ is equal to ________.

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

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda\hat{i}+4\hat{k}$, $\lambda>0$, on the vector $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.

Let $L_1: \frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $L_2: \frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}$, $\alpha \in \mathbb{R}$, be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A$(1,1,-1)$ on $L_2$, then the value of $26\,\alpha(PB)^2$ is ________.

Let the function, $f(x)=\begin{cases}-3ax^2-2, & x<1 \\ a^2+bx, & x \geq 1\end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a>1$, $b \in \mathbb{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta\sqrt{3}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha+\beta$ is ________.