List of top Mathematics Questions asked in JEE Main

The number of real solution(s) of the equation $ x^2 + 3x + 2 = \min \left( |x - 3|, |x + 2| \right) $ is:

Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:

Let $ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} $ and $ \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k} $. Let $ L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in {R} $ and $ L_2 : \vec{r} = (\hat{j} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} $ be two lines. If the line $ L_3 $ passes through the point of intersection of $ L_1 $ and $ L_2 $, and is parallel to $ \vec{a} + \vec{b} $, then $ L_3 $ passes through the point:

For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:

In a group of 3 girls and 4 boys, there are two boys $ B_1 $ and $ B_2 $. The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $ B_1 $ and $ B_2 $ are not adjacent to each other, is:

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:

Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is:

Let $ A = [a_{ij}] $ be a matrix of order 3 $\times$ 3, with $a_{ij} = (\sqrt{2})^{i+j}$. If the sum of all the elements in the third row of $ A^2 $ is $ \alpha + \beta\sqrt{2} $, where $\alpha, \beta \in \mathbb{Z}$, then $\alpha + \beta$ 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 the triangle PQR be the image of the triangle with vertices $ (1, 3), (3, 1) $ and $ (2, 4) $ in the line $ x + 2y = 2 $. If the centroid of $ \Delta PQR $ is the point $ (\alpha, \beta) $, then $ 15(\alpha - \beta) $ is equal to :

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:

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 _____________.

If the equation $ a(b - c)x^2 + b(c - a)x + c(a - b) = 0 $ has equal roots, where $ a + c = 15 $ and $ b = \frac{36}{5} $, then $ a^2 + c^2 $ is equal to .

The value of $ (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) $ is:

The variance of the numbers 8, 21, 34, 47, \dots, 320, is:

Let A and B be the two points of intersection of the line $ y + 5 = 0 $ and the mirror image of the parabola $ y^2 = 4x $ with respect to the line $ x + y + 4 = 0 $. If $ d $ denotes the distance between A and B, and $ a $ denotes the area of $ \Delta SAB $, where $ S $ is the focus of the parabola $ y^2 = 4x $, then the value of $ (a + d) $ is:

Let $ x_1, x_2, x_3, x_4 $ be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from $ x_1, x_2, x_3, x_4 $, then the resulting numbers are in an arithmetic progression. Then the value of $ \frac{1}{24} (x_1 x_2 x_3 x_4) $ 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:

If \[ \sum_{r=1}^{13} \frac{1}{\sin \frac{\pi}{4} + (r-1) \frac{\pi}{6}} \sin \frac{\pi}{4} + \frac{\pi}{6} = a \sqrt{3} + b, \quad a, b \in \mathbb{Z}, { then } a^2 + b^2 { is equal to:} \]

If the equation of the parabola with vertex $\left(\frac{3}{2},\, 3\right)$ and the directrix $x + 2y = 0$ is
$$ ax^2 + by^2 - cxy - 30x - 60y + 225 = 0, $$ then $\alpha + \beta + \gamma$ is equal to:

Let $ A(6,8) $, $ B(10\cos\alpha, -10\sin\alpha) $, and $ C(-10\sin\alpha, 10\cos\alpha) $ be the vertices of a triangle. If $ L(a,9) $ and $ G(h,k) $ be its orthocenter and centroid respectively, then $ 5a - 3h + 6k + 100\sin2\alpha $ is equal to _________.

Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} $ at the point $ A(\alpha, \beta, \gamma) $ and the line $ L_2: x - 6 = y = -z + 4 $ at the point $ B(a, b, c) $. Then $ \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \text{ is equal to} $

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

The integral $ \int_{-1}^{\frac{3}{2}} \left( \pi^2 x \sin(\pi x) \right) dx $ is equal to:

Let $ A = \{0, 1, 2, 3, 4, 5\} $. Let $ R $ be a relation on $ A $ defined by $(x, y) \in R$ if and only if $\max\{x, y\} \in \{3, 4\}$. Then among the statements $ (S_1) : $ The number of elements in $ R $ is 18, and $ (S_2) : $ The relation $ R $ is symmetric but neither reflexive nor transitive Options 1. only $ (S_1) $ is true
2. both are true
3. only $ (S_2) $ is true
4. both are false