List of top Mathematics Questions asked in JEE Main

The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:

The integral \[ \int_{1/4}^{3/4} \cos\left( 2 \cot^{-1} \sqrt{\frac{1 - x}{1 + x}} \right) \, dx \] is equal to:

Let $ a, ar, ar^2, \dots $ be an infinite G.P. If $$ \sum_{n=0}^\infty ar^n = 57 \quad \text{and} \quad \sum_{n=0}^\infty a^3 r^{3n} = 9747, $$ then $ a + 18r $ is equal to:

If $ \log_e y = 3 \sin^{-1}x $, then $ (1 - x)^2 y'' - xy' $ at $ x = \frac{1}{2} $ is equal to:

Let $$ B = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix} $$ and $A$ be a $2 \times 2$ matrix such that $$ AB^{-1} = A^{-1}. $$ If $BCB^{-1} = A$ and $$ C^4 + \alpha C^2 + \beta I = O, $$ then $2\beta - \alpha$ is equal to:

The sum of the coefficients of $ x^{2/3} $ and $ x^{-2/5} $ in the binomial expansion of $$ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 $$ is:

\[ \lim_{x \to \frac{\pi}{2}} \frac{\int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \left( \sin\left(2t^{1/3}\right) + \cos\left(t^{1/3}\right) \right) \, dt}{\left( x - \frac{\pi}{2} \right)^2} \] is equal to:

Between the following two statements: {Statement-I:} Let \[ \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \quad \text{and} \quad \vec{b} = 2\hat{i} + \hat{j} - \hat{k}. \] Then the vector $ \vec{r} $ satisfying \[ \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \quad \text{and} \quad \vec{a} \cdot \vec{r} = 0 \] is of magnitude $ \sqrt{10} $. {Statement-II:} In a triangle $ \triangle ABC $, \[ \cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}. \]

Let the range of the function \[ f(x) = \frac{1}{2 + \sin 3x + \cos 3x}, \, x \in \mathbb{R} \, \text{be } [a, b]. \] If $ \alpha $ and $ \beta $ are respectively the arithmetic mean (A.M.) and the geometric mean (G.M.) of $ a $ and $ b $, then $ \frac{\alpha}{\beta} $ is equal to:

Two vertices of a triangle $ \triangle ABC $ are $ A(3, -1) $ and $ B(-2, 3) $, and its orthocentre is $ P(1, 1) $. If the coordinates of the point $ C $ are $ (\alpha, \beta) $ and the centre of the circle circumscribing the triangle $ \triangle PAB $ is $ (h, k) $, then the value of \[ (\alpha + \beta) + 2(h + k) \] equals:

If the variance of the frequency distribution is 160, then the value of $ c \in \mathbb{N} $ is: \[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & c & 2c & 3c & 4c & 5c & 6c \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array} \]

The area (in square units) of the region enclosed by the ellipse $$ x^2 + 3y^2 = 18 $$ in the first quadrant below the line $ y = x $ is:

Let the foci of a hyperbola $ H $ coincide with the foci of the ellipse $ E : \frac{(x - 1)^2}{100} + \frac{(y - 1)^2}{75} = 1 $ and the eccentricity of the hyperbola $ H $ be the reciprocal of the eccentricity of the ellipse $ E $. If the length of the transverse axis of $ H $ is $ \alpha $ and the length of its conjugate axis is $ \beta $, then $ 3\alpha^2 + 2\beta^2 $ is equal to:

Let $ z $ be a complex number such that the real part of \[ \frac{z - 2i}{z + 2i} \] is zero. Then, the maximum value of $ |z - (6 + 8i)| $ is equal to:

If $\alpha$ satisfies the equation $x^2 + x + 1 = 0$ and $(1 + \alpha)^7 = A + B \alpha + C \alpha^2$, $A, B, C \geq 0$, then $5(3A - 2B - C)$ is equal to ______.

$\lim_{x \to 0} \frac{e - (1 + 2x)^{\frac{1}{2x}}}{x} \quad \text{is equal to:}$

Let \[ \int_{0}^{x} \sqrt{1 - (y'(t))^2} \, dt = \int_{0}^{x} y(t) \, dt, \quad 0 \leq x \leq 3, \, y \geq 0, \, y(0) = 0. \] Then, at $ x = 2 $, $ y'' + y + 1 $ is equal to:

Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$. The distance of the point $$ \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) $$ from the line $L$ along the line $$ \frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2} $$ is equal to:

Let $ f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) $, $ x \in \mathbb{R} $. Then $ f'(10) $ is equal to ______.

Let the set of all $ a \in \mathbb{R} $ such that the equation $\cos 2x + a \sin x = 2a - 7$ has a solution be $[p, q]$ and $ r = \tan 9^\circ - \tan 27^\circ - \frac{1}{\cot 63^\circ + \tan 81^\circ} $, then $ pqr $ is equal to ______.

Let the area of the region $\{(x, y) : x - 2y + 4 \geq 0, x + 2y^2 \geq 0, x + 4y^2 \leq 8, y \geq 0\}$ be $\frac{m}{n}$, where $ m $ and $ n $ are coprime numbers. Then $ m + n $ is equal to ______.

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required and let $a=P(X=3), b=P(X≥3)$, and $( c = P(X \geq 6|X>3).$ Then $\frac{b+c}{a}$ is equals to ____.

If $ 8 = 3 + \frac{1}{4}(3 + p) + \frac{1}{4^2}(3 + 2p) + \frac{1}{4^3}(3 + 3p) + \ldots \infty $, then the value of $ p $ is ______.

Let for a differentiable function $f : (0, \infty) \rightarrow \mathbb{R}$, $f(x) - f(y) \geq \log_e \left( \frac{x}{y} \right) + x - y, \quad \forall \; x, y \in (0, \infty).$
Then $\sum_{n=1}^{20} f'\left(\frac{1}{n^2}\right)$ is equal to ____.

If the solution of the differential equation $(2x + 3y - 2) \, dx + (4x + 6y - 7) \, dy = 0, \quad y(0) = 3,$ is $\alpha x + \beta y + 3 \log_e |2x + 3y - \gamma| = 6,$ then $\alpha + 2\beta + 3\gamma$ is equal to ____.