List of top Mathematics Questions asked in JEE Main

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 ______.

Let $ H_1: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ and $ H_2: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 $ be two hyperbolas having lengths of latus rectums $ 15\sqrt{2} $ and $ 12\sqrt{5} $ respectively. Let their eccentricities be $ e_1 = \frac{5}{\sqrt{2}} $ and $ e_2 $ respectively. If the product of the lengths of their transverse axes is $ 100\sqrt{10} $, then $ 25e_2^2 $ is equal to:

If \[ \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left( \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| \right) + C, \] where $ C $ is the constant of integration, then $ \alpha + 2\beta $ is equal to ________________

Number of functions $ f: \{1, 2, \dots, 100\} \to \{0, 1\} $, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:

Let $ P $ be the image of the point $ Q(7, -2, 5) $ in the line $ L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} $, and let $ R(5, p, q) $ be a point on $ L $. Then the square of the area of $ \triangle PQR $ is:

Let $ y = y(x) $ be the solution of the differential equation \[ 2\cos x \frac{dy}{dx} = \sin 2x - 4y \sin x, \quad x \in \left( 0, \frac{\pi}{2} \right). \]
If $ y\left( \frac{\pi}{3} \right) = 0 $, then $ y\left( \frac{\pi}{4} \right) + y\left( \frac{\pi}{4} \right) $ is equal to ________.

If the area of the larger portion bounded between the curves $x^2 + y^2 = 25$ and $y = |x - 1|$ is $ \frac{1}{4} (b\pi + c) $, where $b, c \in \mathbb{N}$, then $ b + c $ is equal .

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

Let the circle C touch the line $x - y + 1 = 0$, have the center on the positive x-axis, and cut off a chord of length $ \frac{4}{\sqrt{13}} $ along the line $ -3x + 2y = 1 $. Let H be the hyperbola $ \frac{x^2}{\alpha^2} - \frac{y^2}{\beta^2} = 1 $, whose one of the foci is the center of C and the length of the transverse axis is the diameter of C. Then $ 2\alpha^2 + 3\beta^2 $ is equal to:

If the set of all values of $ a $, for which the equation $ 5x^3 - 15x - a = 0 $ has three distinct real roots, is the interval $ (\alpha, \beta) $, then $ \beta - 2\alpha $ 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 .

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

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:

If \[ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6, { then } f(1) { is equal to:} \]

Let $ f: [0, 3] \to A $ be defined by $ f(x) = 2x^3 - 15x^2 + 36x + 7 $ and $ g: [0, \infty) \to B $ be defined by $ g(x) = \frac{x{x^{2025} + 1}. $ If both functions are onto and $ S = \{ x \in \mathbb{Z} : x \in A { or } x \in B \} $, then $ n(S) $ is equal to:}

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 $ S $ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $ S $, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

Let $ f $ be a real-valued continuous function defined on the positive real axis such that $ g(x) = \int_0^x t f(t) \, dt $. If $ g(x^3) = x^6 + x^7 $, then the value of $ \sum_{r=1}^{15} f(r^3) $ is:

If $ A $ and $ B $ are the points of intersection of the circle $ x^2 + y^2 - 8x = 0 $ and the hyperbola $ \frac{x^2}{9} - \frac{y^2}{4} = 1 $, and a point $ P $ moves on the line $ 2x - 3y + 4 = 0 $, then the centroid of $ \triangle PAB $ lies on the line:

Two equal sides of an isosceles triangle are along $ -x + 2y = 4 $ and $ x + y = 4 $. If $ m $ is the slope of its third side, then the sum of all possible distinct values of $ m $ is:

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: