JEE Main 2025 (22 Jan Shift 1)

67 Questions • 75 Minutes • 2025

Take This Paper Online

All Questions — Click to View Solution

Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive terms. If $a_1a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to:

View Answer & Solution

Let $x=x(y)$ be the solution of the differential equation $y^2\,dx+\left(x-\frac1y\right)dy=0$. If $x(1)=1$, then $x\left(\frac12\right)$ is:

View Answer & Solution

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 $\frac mn$, where $\gcd(m,n)=1$, then $m+n$ is equal to :

View Answer & Solution

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

View Answer & Solution

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 $\triangle PQR$ is the point $(\alpha,\beta)$, then $15(\alpha-\beta)$ is equal to :

View Answer & Solution

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

View Answer & Solution

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 $\triangle PQR$ is :

View Answer & Solution

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 $L_2$ ?

View Answer & Solution

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 $x,y\in\mathbf R$. Then $\sum_{n=1}^{100}\log_ef(n)$ is equal to :

View Answer & Solution

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 :

View Answer & Solution

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 :

View Answer & Solution

Let $f:\mathbf R\to\mathbf R$ be a twice differentiable function such that $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbf R$. If $f'(0)=4a$ and $f$ satisfies $f''(x)-3af'(x)-f(x)=0$, $a>0$, then the area of the region $R=\{(x,y)\mid0\le y\le f(ax),\ 0\le x\le2\}$ is:

View Answer & Solution

The area of the region, inside the circle $(x-2\sqrt3)^2+y^2=12$ and outside the parabola $y^2=2\sqrt3x$ is :

View Answer & Solution

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 :

View Answer & Solution

If $\sum_{r=1}^nT_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\lim_{n\to\infty}\sum_{r=1}^n\left(\frac1{T_r}\right)$ is equal to :

View Answer & Solution

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 :

View Answer & Solution

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

View Answer & Solution

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 :

View Answer & Solution

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

View Answer & Solution

Let $z_1,z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=-\frac\pi4$, $\arg(z_2)=0$ and $\arg(z_3)=\frac\pi4$. If $|z_1\bar z_2+z_2\bar z_3+z_3\bar z_1|^2=\alpha+\beta\sqrt2$, $\alpha,\beta\in\mathbf Z$, then the value of $\alpha^2+\beta^2$ is :

View Answer & Solution

Paper Info

Exam: JEE
Year: 2025
Questions: 67
Duration: 75 mins

Take Online All Papers