List of top Questions asked in JEE Main- 2025

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

Three distinct numbers are selected randomly from the set $ \{1, 2, 3, \dots, 40\} $. If the probability, that the selected numbers are in an increasing G.P. is $ \frac{m}{n} $, where $ \gcd(m, n) = 1 $, then $ m + n $ is equal to:

Let $ S $ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $ S $, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order 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:

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability that the ball drawn is white is $\frac{29}{45}$, then n is equal to:

If $ A $ and $ B $ are two events such that $ P(A) = 0.7 $, $ P(B) = 0.4 $ and $ P\left( A \cap \overline{B} \right) = 0.5 $, where $\overline{B}$ denotes the complement of $ B $, then $ P\left( B | \left( A \cup \overline{B} \right) \right) $ is equal to

If A and B are two events such that $ P(A \cap B) = 0.1 $, and $ P(A|B) $ and $ P(B|A) $ are the roots of the equation $ 12x^2 - 7x + 1 = 0 $, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$

A board has 16 squares as shown in the figure. Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:

If A and B are two events such that $ P(A \cap B) = 0.1 $, and $ P(A|B) $ and $ P(B|A) $ are the roots of the equation $ 12x^2 - 7x + 1 = 0 $, then the value of $ \frac{P(A \cup B)}{P(A \cap B)} $ is:

A conducting bar moves on two conducting rails as shown in the figure. A constant magnetic field $ B $ exists into the page. The bar starts to move from the vertex at time $ t = 0 $ with a constant velocity. If the induced EMF is $ E \propto t^n $, then the value of $ n $ is _____.

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 $ [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:

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

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 $ 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 __________.

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

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:

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 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: