List of top Questions asked in JEE Main- 2025

The sum of all rational numbers in $ (2 + \sqrt{3})^8 $ is:

Let a curve $ y = f(x) $ pass through the points $ (0,5) $ and $ (\log 2, k) $. If the curve satisfies the differential equation: \[ 2(3+y)e^{2x}dx - (7+e^{2x})dy = 0, \] then $ k $ is equal to:

Evaluate the following limit: \[ \lim_{x \to \infty} \frac{(2x^2 - 3x + 5) \left( 3x - 1 \right)^{x/2}}{(3x^2 + 5x + 4) \sqrt{(3x + 2)^x}}. \] The value of the limit is:

Let the line passing through the points $ (-1, 2, 1) $ and parallel to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \] intersect the line \[ \frac{x + 2}{3} = \frac{y - 3}{2} = \frac{z - 4}{1} \] at the point P. Then the distance of P from the point Q(4, -5, 1) is:

A line passing through the point $ P(a, 0) $ makes an acute angle $ \alpha $ with the positive $ x $-axis. Let this line be rotated about the point $ P $ through an angle $ \frac{\alpha}{2} $ in the clock-wise direction. If in the new position, the slope of the line is $ 2 - \sqrt{3} $ and its distance from the origin is $ \frac{1}{\sqrt{2}} $, then the value of $ 3a^2 \tan^2 \alpha - 2\sqrt{3} $ is

Let $ ABC $ be the triangle such that the equations of lines $ AB $ and $ AC $ are: $ 3y - x = 2 \quad \text{and} \quad x + y = 2, $ respectively, and the points $ B $ and $ C $ lie on the x-axis. If $ P $ is the orthocentre of the triangle $ ABC $, then the area of the triangle $ PBC $ is equal to:

Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that
\[\sum_{i=1}^{10} (x_i - 2) = 30, \quad \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \quad \beta > 2\]
and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of
\[ 2(x_1 - 1) + 4\beta, 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta\]
then $\frac{\beta \mu}{\sigma^2}$ is equal to:

Let ABCD be a trapezium whose vertices lie on the parabola $ y^2 = 4x $. Let the sides AD and BC of the trapezium be parallel to the y-axis. If the diagonal AC is of length $ \frac{25}{4} $ and it passes through the point $ (1, 0) $, then the area of ABCD is:

Let R = {(1, 2), (2, 3), (3, 3)}} be a relation defined on the set $ \{1, 2, 3, 4\} $. Then the minimum number of elements needed to be added in $ R $ so that $ R $ becomes an equivalence relation, is:

If the system of equations \[ (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \] \[ \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \] \[ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \] has infinitely many solutions, then $ \lambda^2 + \lambda $ is equal to:

The integral $ \int_0^\pi \frac{(x + 3) \sin x}{1 + 3 \cos^2 x} \, dx $ is equal to:

Bag $ B_1 $ contains 6 white and 4 blue balls, Bag $ B_2 $ contains 4 white and 6 blue balls, and Bag $ B_3 $ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag $ B_2 $ is:

Let the area of the region $$ \{(x, y) : 2y \leq x^2 + 3, \quad y + |x| \leq 3, \quad y \geq |x-1|\} $$ be $ A $. Then $ 6A $ is equal to:

If $\int e^x \left( \frac{x \sin^{-1} x}{\sqrt{1-x^2}} + \frac{\sin^{-1} x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C$, where C is the constant of integration, then $g\left( \frac{1}{2} \right)$equals:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a twice-differentiable function such that $ f(2) = 1 $. If $ F(x) = x f(x) $ for all $ x \in \mathbb{R} $, and the integrals $ \int_0^2 x F'(x) \, dx = 6 $ and $ \int_0^2 x^2 F''(x) \, dx = 40 $, then $ F'(2) + \int_0^2 F(x) \, dx $ is equal to:

If $ \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] $, then the number of solutions of the equation \[ \sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0 \] is:

The equation of the chord of the ellipse $ \frac{x^2}{25} + \frac{y^2}{16} = 1 $, whose mid-point is $ (3, 1) $ is:

If \[ I = \int_0^{\frac{\pi}{2}} \frac{\sin^2 \frac{3}{2}x}{\sin^2 x + \cos^2 x} \, dx, \] then \[ \int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals:

Let $ A = [a_{ij}] $ be a $3 \times 3 $ matrix such that: $$ A = \begin{bmatrix} 0 & 0 & 4 \\ 1 & 0 & 0 \\ 0 & 1 & 3 \\ \end{bmatrix}, A^{-1} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & 0 \\ 2 & 1 & 0 \end{bmatrix}. $$ Then $ a_{23} $ equals:

The value of $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k+3)!}$ is:

The number of solutions of the equation $ 2x + 3\tan x = \pi $, $ x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} $ is

Let $ f : \mathbb{R} \rightarrow \mathbb{R} $ be a function defined by $ f(x) = ||x+2| - 2|x|| $. If m is the number of points of local maxima of f and n is the number of points of local minima of f, then m + n is

Let $ [x] $ denote the greatest integer less than or equal to $ x $. Then the domain of $ f(x) = \sec^{-1} (2[x] + 1) $ is:

Let the area of a triangle $ \triangle PQR $ with vertices $ P(5, 4) $, $ Q(-2, 4) $, and $ R(a, b) $ be 35 square units. If its orthocenter and centroid are $ O(2, \frac{14}{5}) $ and $ C(c, d) $ respectively, then $ c + 2d $ is equal to:

Let $ f: \mathbb{R} \to \mathbb{R} $ be a function defined by $ f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 $. If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of $ 28 \sum_{i=1}^5 f(i) $ is: