List of top Mathematics Questions asked in JEE Main

If $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + ... \infty = \frac{\pi^4}{90}, $ $ \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + ... \infty = \alpha, $ $ \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + ... \infty = \beta, $ then $ \frac{\alpha}{\beta} $ is equal to:

Find the length of the chord whose midpoint is $ \left( \frac{3}{2}, 0 \right) $ of the ellipse \[ \frac{x^2}{2} + \frac{y^2}{4} = 1. \]

Let $ L_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} $ and $ L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5} $ be two lines. Then which of the following points lies on the line of the shortest distance between $ L_1 $ and $ L_2 $?

The perpendicular distance of the line $ \frac{x - 1}{2} = \frac{y + 2}{-1} = \frac{z + 3}{2} $ from the point $ P(2, -10, 1) $ is:

The value of \[ \int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left( (\log_e x)^2 +1 \right)^{-1}}}{e^{\left( (\log_e x)^2 +1 \right)^{-1}} + e^{\left( (6-\log_e x)^2 +1 \right)^{-1}}} \right) dx \] is:

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $ r $ be the radius of a circle that has centre at the point $ (2, 5) $ and intersects the circle C at exactly two points. If the set of all possible values of $ r $ is the interval $ (\alpha, \beta) $, then $ 3\beta - 2\alpha $ is equal to:

If the system of equations \[ 2x - y + z = 4, \] \[ 5x + \lambda y + 3z = 12, \] \[ 100x - 47y + \mu z = 212, \] has infinitely many solutions, then $ \mu - 2\lambda $ is equal to:

The sum of all values of $ \theta \in [0, 2\pi] $ satisfying $ 2\sin^2\theta = \cos 2\theta $ and $ 2\cos^2\theta = 3\sin\theta $ is:

Let $ \alpha_1 $ and $ \beta_1 $ be the distinct roots of $ 2x^2 + (\cos\theta)x - 1 = 0, \ \theta \in (0, 2\pi) $. If $ m $ and $ M $ are the minimum and the maximum values of $ \alpha_1 + \beta_1 $, then $ 16(M + m) $ equals:

If the system of equations \[ x + 2y - 3z = 2, \quad 2x + \lambda y + 5z = 5, \quad 14x + 3y + \mu z = 33 \]

has infinitely many solutions, then $ \lambda + \mu $ 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:

If $ \sin x + \sin^2 x = 1 $, $ x \in \left(0, \frac{\pi}{2} \right) $, then the expression \[ (\cos^2 x + \tan^2 x) + 3(\cos^4 x + \tan^4 x + \cos^4 x + \tan^4 x) + (\cos^6 x + \tan^6 x) \] is equal to:

The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is:

Let $ y^2 = 12x $ be the parabola and $ S $ its focus. Let $ PQ $ be a focal chord of the parabola such that $ (SP)(SQ) = \frac{147}{4} $. Let $ C $ be the circle described by taking $ PQ $ as a diameter. If the equation of the circle $ C $ is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then $ \beta - \alpha $ is equal to:

For a $ 3 \times 3 $ matrix $ M $, let trace(M) denote the sum of all the diagonal elements of $ M $. Let $ A $ be a $ 3 \times 3 $ matrix such that $ |A| = \frac{1}{2} $ and $ \text{trace}(A) = 3 $. If $ B = \text{adj}(\text{adj}(2A)) $, then the value of $ |B| + \text{trace}(B) $ equals:

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 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:

Let $ A = \{1, 2, 3, \dots, 10\} $ and $ B = \left\{ \frac{m}{n} : m, n \in A, m <n \text{ and } \gcd(m, n) = 1 \right\} $. Then $ n(B) $ is equal to :

Let $ \hat{a} $ be a unit vector perpendicular to the vectors \[ \mathbf{b} = \hat{i} - 2\hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{c} = 2\hat{i} + 3\hat{j} - \hat{k}, \] $\text{and makes an angle of}$ $ \cos\left( -\frac{1}{3} \right) $ $\text{with the vector}$ $ \hat{i} + \alpha \hat{j} + \hat{k} $. $\text{If}$ $ \hat{a} $ $\text{makes an angle of}$ $ \frac{\pi}{3} $ $\text{with the vector}$ $ \hat{i} + \alpha \hat{j} + \hat{k} $, then the value of $ \alpha $ $\text{is:}$

Let the function $f(x) = (x^2 - 1)|x^2 - ax + 2| + \cos|x| $ be not differentiable at the two points $ x = \alpha = 2 $ and $ x = \beta $. Then the distance of the point $(\alpha, \beta)$ from the line $12x + 5y + 10 = 0$ is equal to:

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:

Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is:

Let $ \langle a_n \rangle $ be a sequence such that $ a_0 = 0 $, $ a_1 = \frac{1}{2} $, and $ 2a_{n+2} = 5a_{n+1} - 3a_n $.n= 0,1,2,3.... Then $ \sum_{k=1}^{100} a_k $ is equal to:

Let $ A $ be a matrix of order $ 3 \times 3 $ and $ |A| = 5 $. If $ |2 \, \text{adj}(3A \, \text{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N} $ then $ \alpha + \beta + \gamma $ is equal to

Let the matrix $ A = \begin{pmatrix} 1 & 0 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \end{pmatrix} $ satisfy $ A^n = A^{n-2} + A^2 - I $ for $ n \geq 3 $. Then the sum of all the elements of $ A^{50} $ is: