List of top Mathematics Questions asked in JEE Main

If $ \alpha + i\beta $ and $ \gamma + i\delta $ are the roots of the equation $ x^2 - (3-2i)x - (2i-2) = 0 $, $ i = \sqrt{-1} $, then $ \alpha\gamma + \beta\delta $ 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:

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

The square of the distance of the point $\left( \frac{15}{7}, \frac{32}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\mathbf{i} + 4\mathbf{j} + 7\mathbf{k}$ is:

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

Let the position vectors of three vertices of a triangle be $ \overrightarrow{p} = 4\hat{i} + \hat{j} - 3\hat{k} $, $ \overrightarrow{q} = -5\hat{i} + 2\hat{j} + 3\hat{k} $, and $ \overrightarrow{r} = -5\hat{i} + 3\hat{j} + 2\hat{k} $. Then $ \alpha + 2\beta + 5\gamma $ is equal to:

For some $ a, b $, let $ f(x) = \left| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} \right| $, where $ x \neq 0 $, $ \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b $.
Then $ (\lambda + \mu + \nu)^2 $ is equal to:

If $ 7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \frac{1}{7^3}(5 + 3\alpha) + \cdots $, then the value of $ \alpha $ is:

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

Let $ \mathbf{a} = 3\hat{i} - \hat{j} + 2\hat{k} $, $ \mathbf{b} = \mathbf{a} \times (\hat{i} - 2\hat{k}) $ and $ \mathbf{c} = \mathbf{b} \times \hat{k} $. Then the projection of $ \mathbf{c} - 2\hat{j} $ on $ \mathbf{a} $ is:

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

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:

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 $ f: (0, \infty) \to \mathbb{R} $ be a function which is differentiable at all points of its domain and satisfies the condition $ x^2 f'(x) = 2f(x) + 3 $, with $ f(1) = 4 $. Then $ 2f(2) $ is equal to:

If $ \alpha>\beta>\gamma>0 $, then the expression \[ \cot^{-1} \beta + \left( \frac{1 + \beta^2}{\alpha - \beta} \right) + \cot^{-1} \gamma + \left( \frac{1 + \gamma^2}{\beta - \gamma} \right) + \cot^{-1} \alpha + \left( \frac{1 + \alpha^2}{\gamma - \alpha} \right) \] is equal to:

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

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:

Let \[ S = \left\{ m \in \mathbb{Z} : A m^2 + A^n = 31 - A^6 \right\}, \quad \text{where} \quad A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \] Then $ n(S) $ is equal to:

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word "MATHS" such that any letter that appears in the word must appear at least twice is: