List of top Questions asked in JEE Main- 2025

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) - 3a f'(x) - f(x) = 0 $, where $ a > 0 $, then the area of the region R = {(x, y) | 0 $\leq$ y $\leq$ f(ax), 0 $\leq$ x $\leq$ 2 is :

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 ______.

Let $ f(x) = x - 1 $ and $ g(x) = e^x $ for $ x \in \mathbb{R} $. If $\frac{dy}{dx} = \left( e^{-2\sqrt{x}} g\left(f\left(f(x)\right)\right) - \frac{y}{\sqrt{x}} \right), \, y(0) = 0,$ then $ y(1) $ is

If $ f(x) = x - 1 $ and $ g(x) = e^x $, and the differential equation is: $ \frac{dy}{dx} = \left( e^{-2\sqrt{x}} g(f(f(f(x)))) - \frac{y}{\sqrt{x}} \right) $ with the initial condition $ y(0) = 0 $, then $ y(1) $ equals:

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

Let g be a differentiable function such that $ \int_0^x g(t) dt = x - \int_0^x tg(t) dt $, $ x \ge 0 $ and let $ y = y(x) $ satisfy the differential equation $ \frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x) $, $ x \in \left[ 0, \frac{\pi}{2} \right) $. If $ y(0) = 0 $, then $ y\left( \frac{\pi}{3} \right) $ is equal to

Let $ [x] $ denote the greatest integer function, and let $ m $ and $ n $ respectively be the numbers of the points, where the function $ f(x) = [x] + |x - 2| $, $ -2<x<3 $, is not continuous and not differentiable. Then $ m + n $ is equal to:

Let for some function $ y = f(x) $, $\int_0^x t f(t) \, dt = x^2 f(x), x>0$ and $ f(2) = 3 $. Then $ f(6) $ is equal to:

Let $ y = f(x) $ be the solution of the differential equation\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\] such that $ f(0) = 0 $. If \[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\] then $ \alpha^2 $ is equal to __________.

If $ y = y(x) $ is the solution of the differential equation, \[ \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right), \] where $ -2 \leq x \leq 2 $, and $ y(2) = \frac{\pi^2 - 8}{4} $, then $ y^2(0) $ is equal to:

Let $ y = y(x) $ be the solution of the differential equation \[ \cos(x \log(\cos x))^2 \, dy + (\sin x - 3 \sin x \log(\cos x)) \, dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right) \] If $ y\left( \frac{\pi}{4} \right) = -1 $, then $ y\left( \frac{\pi}{6} \right) $ is equal to:

If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and

and $ f(0) = \frac{5}{4} $, then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to:

Let $ y = f(x) $ be the solution of the differential equation\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\] such that $ f(0) = 0 $. If \[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\] then $ \alpha^2 $ is equal to ______.

If $ x = f(y) $ is the solution of the differential equation \[ (1 + y^2) + (x - 2e^{\tan^{-1}y}) \frac{dy}{dx} = 0, \quad y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right), \] with $ f(0) = 1 $, then $ f\left( \frac{1}{\sqrt{3}} \right) $ is equal to:

Let $ y = f(x) $ be the solution of the differential equation

\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]

such that $ f(0) = \frac{e^3}{3} + 1 $, then $ f\left( \frac{\pi}{4} \right) $ is equal to:

If a curve $ y = y(x) $ passes through the point $ \left(1, \frac{\pi}{2}\right) $ and satisfies the differential equation $$ (7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5, \quad x \geq 1, \text{ then at } x = 2, \text{ the value of } \cos y \text{ is:} $$

The shortest distance between the curves $ y^2 = 8x $ and $ x^2 + y^2 + 12y + 35 = 0 $ is:

Let C be the circle of minimum area enclosing the ellipse E: $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ with eccentricity $ \frac{1}{2} $ and foci $ (\pm 2, 0) $. Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 29 is parallel to the major axis and contains the point of intersection of E with the negative y-axis. Then the maximum area of the triangle PQR is:

Let the product of the focal distances of the point P(4, $ 2\sqrt{3} $) on the hyperbola H: $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then $ p^2 + q^2 $ is equal to .....

A line passing through the point P$(\sqrt{5}, \sqrt{5})$ intersects the ellipse $ \frac{x^2}{36} + \frac{y^2}{25} = 1 $ at A and B such that (PA).(PB) is maximum. Then 5(PA$^2$ + PB$^2$) is equal to :

Let the distance between two parallel lines be 5 units and a point $ P $ lies between the lines at a unit distance from one of them. An equilateral triangle $ POR $ is formed such that $ Q $ lies on one of the parallel lines, while $ R $ lies on the other. Then $ (QR)^2 $ is equal to _______.

Consider the lines $ x(3\lambda + 1) + y(7\lambda + 2) = 17\lambda + 5 $. If P is the point through which all these lines pass and the distance of L from the point $ Q(3, 6) $ is $ d $, then the distance of L from the point $ (3, 6) $ is $ d $, then the value of $ d^2 $ is

If the four distinct points $ (4, 6) $, $ (-1, 5) $, $ (0, 0) $ and $ (k, 3k) $ lie on a circle of radius $ r $, then $ 10k + r^2 $ is equal to

Let the equation $ x(x+2) * (12-k) = 2 $ have equal roots. The distance of the point $ \left(k, \frac{k}{2}\right) $ from the line $ 3x + 4y + 5 = 0 $ is

Let the line $x + y = 1$ meet the axes of x and y at A and B, respectively. A right-angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is $\frac{4}{9}$ of the area of the triangle OAB and $AN : NB = \lambda : 1$, then the sum of all possible values of $\lambda$ is: