List of top Questions asked in JEE Main- 2025

Let $ a_1, a_2, a_3, \dots $ be an A.P. and $ \sum_{k=1}^{12} a_{2k-1} = -\frac{72}{5} a_1 $ and $ \sum_{k=1}^{n} a_k = 0 $. Then the value of $ n $ is:

Let $ S = \mathbb{N} \cup \{0\} $. Define a relation $ R $ from $ S $ to $ \mathbb{R} $ by: \[ R = \left\{ (x, y) : \log_e y = x \log_e \left(\frac{2}{5}\right), x \in S, y \in \mathbb{R} \right\}. \] Then, the sum of all the elements in the range of $ R $ is equal to:

Let the curve $($$ z(1 + i) + \overline{z(1 - i)} = 4$, $\, z \in \mathbb{C}$ $)$, divide the region $|z - 3| \leq 1$ into two parts of areas $ \alpha $ and $ \beta $. Then $ |\alpha - \beta| $ equals:

If the equation of a circle is $ 4x^2 + 4y^2 - 12x + 8y = 0 $, what is the radius of the circle?

Let $ D = \{a, b, c\} $. How many distinct ways can $ D $ be partitioned into non-empty subsets, representing equivalence relations?

Suppose A and B are the coefficients of the 30th and 12th terms respectively in the binomial expansion of $ (1 + x)^{2n - 1} $. If $ 2A = 5B $, then $ n $ is equal to:

The number of words that can be formed using all the letters of the word "DAUGHTER" such that all the vowels never come together, is:

Let $ x = x(y) $ be the solution of the differential equation: \[ y = \left( x - y \frac{dx}{dy} \right) \sin\left( \frac{x}{y} \right), \, y > 0 \, \text{and} \, x(1) = \frac{\pi}{2}. \] Then $ \cos(x(2)) $ 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:

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 $ 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:

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

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 $ 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:

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:

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:

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:

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 $ 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:

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:

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:

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:

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:

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: