List of top Questions asked in JEE Main- 2025

Suppose that the number of terms in an A.P. is $ 2k, k \in \mathbb{N} $. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then $ k $ is equal to:

Let $ ABCD $ be a tetrahedron such that the edges $ AB $, $ AC $, and $ AD $ are mutually perpendicular. Let the areas of the triangles $ ABC $, $ ACD $, and $ ADB $ be 5, 6, and 7 square units respectively. Then the area (in square units) of the $ \triangle BCD $ is equal to:

Let the line $ x + y = 1 $ meet the circle $ x^2 + y^2 = 4 $ at the points A and B. If the line perpendicular to AB and passing through the midpoint of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to:

The sum of the squares of the roots of $ |x - 2|^2 + |x - 2| - 2 = 0 $ and the squares of the roots of $ x^2 |x - 3| - 5 = 0 $, is:

What is the sum of the infinite series $ S = \sum_{n=0}^{\infty} \frac{1}{3^n} $?

The system of equations is given as: \[ (\lambda + 1)x + (\lambda + 2)y + (\lambda - 1)z = 0 \] \[ \lambda x + (\lambda - 1)y + (\lambda + 1)z = 0 \] \[ (\lambda - 1)x + (\lambda + 1)y + (\lambda + 2)z = 0 \] If the above system of equations has infinite solutions, then $ \lambda^2 + \lambda $ is:

Let $(\alpha,\beta,\gamma)$ be the foot of the perpendicular from the point $(25,2,41)$ on the line \[ \frac{x-4}{3}=\frac{y+1}{7}=\frac{z-2}{3}. \] Then $\alpha+\beta+\gamma$ is equal to:

Let $ \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} $ and $ \vec{c} $ be three vectors such that $ \vec{c} $ is coplanar with $ \vec{a} $ and $ \vec{b} $. If the vector $ \vec{c} $ is perpendicular to $ \vec{b} $ and $ \vec{a} \cdot \vec{c} = 5 $, then $ |\vec{c}| $ is equal to:

If two vectors $ \mathbf{a} $ and $ \mathbf{b} $ satisfy the equation:

\[ \frac{|\mathbf{a} + \mathbf{b}| + |\mathbf{a} - \mathbf{b}|}{|\mathbf{a} + \mathbf{b}| - |\mathbf{a} - \mathbf{b}|} = \sqrt{2} + 1, \]

then the value of

\[ \frac{|\mathbf{a} + \mathbf{b}|}{|\mathbf{a} - \mathbf{b}|} \]

is equal to:

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

Let $ \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k} $, $ \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} $, and $ \vec{c} $ be a vector such that $ \vec{a} \times \vec{c} = \vec{a} \times \vec{b} $ and $ (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168 $. Then the maximum value of $ |\vec{c}|^2 $ is:

Let $ A = [a_{ij}] $ be a 2 $\times$ 2 matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$. Let the random variable X denote the possible values of the determinant of the matrix A. Then, the variance of X is:

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 :

The area of the region $ \{(x, y): |x - y| \le y \le 4\sqrt{x}\} $ is

Let $ A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} $ and $ P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta > 0. $ If $ B = P A P^T $, $ C = P^T B P $, and the sum of the diagonal elements of $ C $ is $ \frac{m}{n} $, where gcd(m, n) = 1, then $ m + n $ is:

Two parabolas have the same focus $(4, 3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:

Let $ \vec{a} $ and $ \vec{b} $ be the vectors of the same magnitude such that $ \frac{| \vec{a} + \vec{b} | + | \vec{a} - \vec{b} |}{| \vec{a} + \vec{b} | - | \vec{a} - \vec{b} |} = \sqrt{2} + 1. \quad \text{Then } \frac{| \vec{a} + \vec{b} |^2}{| \vec{a} |^2} \text{ is:} $

Let $ M $ and $ m $ respectively be the maximum and the minimum values of $ f(x) = \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ \sin^2x & 1 + \cos^2x & 4\sin4x \\ \sin^2x & \cos^2x & 1 + 4\sin4x \end{vmatrix}, \quad x \in \mathbb{R} $ for $ x \in \mathbb{R} $. Then $ M^4 - m^4 $ is equal to:

If $ \vec{a} $ is a non-zero vector such that its projections on the vectors $ 2\hat{i} - \hat{j} + 2\hat{k},\ \hat{i} + 2\hat{j} - 2\hat{k} $, and $ \hat{k} $ are equal, then a unit vector along $ \vec{a} $ 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:

Let a random variable X take values 0, 1, 2, 3 with $ P(X = 0) = P(X = 1) = p, \, P(X = 2) = P(X = 3), \, \text{and} \, F(X^2) = 2F(X). $ Then the value of $ 8p - 1 $ is:

The interior angles of a polygon with $ n $ sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then $ n $ is equal to:

The length of the chord of the ellipse: \[ \frac{x^2}{4} + \frac{y^2}{2} = 1, \] whose mid-point is $ \left( 1, \frac{1}{2} \right) $, is:

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:

Each of the angles $ \beta $ and $ \gamma $ that a given line makes with the positive y- and z-axes, respectively, is half the angle that this line makes with the positive x-axis. Then the sum of all possible values of the angle $ \beta $ is