List of top Mathematics Questions asked in JEE Main

Let $ (2, 3) $ be the largest open interval in which the function $ f(x) = 2 \log_e (x - 2) - x^2 + ax + 1 $ is strictly increasing, and $ (b, c) $ be the largest open interval, in which the function $ g(x) = (x - 1)^3 (x + 2 - a)^2 $ is strictly decreasing. Then $ 100(a + b - c) $ is equal to:

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

Let $ |z_1 - 8 - 2i| \leq 1 $ and $ |z_2 - 2 + 6i| \leq 2 $, where $ z_1, z_2 \in \mathbb{C} $. Then the minimum value of $ |z_1 - z_2| $ is:

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

The function $ f: (-\infty, \infty) \to (-\infty, 1) $, defined by \[ f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}, \] is:

Let $ A = \left\{ x \in (0, \pi) \mid - \log\left(\frac{2}{\pi}\right)\sin x + \log\left(\frac{2}{\pi}\right)\cos x = 2 \right\} $ and

\[ B = \left\{ x \geq 0 : \sqrt{x}(\sqrt{x - 4}) - 3\sqrt{x - 2} + 6 = 0 \right\}. \]

Then $ n(A \cup B) $ is equal to:

Let $ A = [a_{ij}] $ be a square matrix of order 2 with entries either 0 or 1. Let $ E $ be the event that $ A $ is an invertible matrix. Then the probability $ P(E) $ is:

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

Let $A$ be the area of the region
$\left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}$.
Then $540 A$ is equal to ______.

The sum of all rational terms in the expansion of $ \left( 1 + 2^{1/3} + 3^{1/2} \right)^6 $ 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 .

For positive integers $ n $, if $ 4 a_n = \frac{n^2 + 5n + 6}{4} $ and \[ S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right), \text{ then the value of } 507 S_{2025} \text{ is:} \]

Let A, B, C be three points in the xy-plane, whose position vectors are given by $ \sqrt{3} \hat{i} + \hat{j} $, $ \hat{i} + \sqrt{3} \hat{j} $, and $ a\hat{i} + (1-a) \hat{j }$ respectively with respect to the origin $ O $. If the distance of the point C from the line bisecting the angle between the vectors $ \overrightarrow{OA} $ and $ \overrightarrow{OB} $ is $ \frac{9}{\sqrt{2}} $, then the sum of all possible values of $ a $ is:

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:

Let $ f : \mathbb{R} \setminus \{0\} \to (-\infty, 1) $ be a polynomial of degree 2, satisfying $ f(x)f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right) $. If $ f(K) = -2K $, then the sum of squares of all possible values of $ K $ is:

Let $ f $ be a differentiable function such that \[ 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, \] for $ x \geq 0 $. Then $ f(2) $ is equal to:

The product of all the rational roots of the equation \[ (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3, \] is equal to:

N>40000, where N is divisible by 5. How many such 5 digit numbers can be formed using 0,1,3,5,7,9 without repetition.

The sum of all local minimum values of the function $ f(x) $ as defined below is:

\[ f(x) = \left\{ \begin{array}{ll} 1 - 2x & \text{if } x < -1 \\ \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2 \\ \frac{11}{18} (x-4)(x-5) & \text{if } x > 2 \end{array} \right. \]

Let $\vec{a} = i + j + k$, $\vec{b} = 2i + 2j + k$ and $\vec{d} = \vec{a} \times \vec{b}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $\|\vec{c} - 2\vec{d}\| = 8$ and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$, then $\left|10 - 3\vec{b} \cdot \vec{c} + |\vec{d}|\right|^2$ is equal to:

If $ 7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \cdots \infty \text{ terms}, $ then $ \alpha $ 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 $ \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 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:

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is _____.