List of top Questions asked in JEE Main

A metal string A is suspended from a rigid support and its free end is attached to a block of mass M. Second block having mass 2M is suspended at the bottom of the first block using a string B. The area of cross sections of strings A and B are same. The ratio of lengths of strings of A to B is 2 and the ratio of their Young's moduli ($Y_A / Y_B$) is 0.5. The ratio of elongations in A to B is _______.

The height in terms of radius of the earth ($R$), at which the acceleration due to gravity becomes $g/9$, where $g$ is acceleration due to gravity on earth's surface, is

At $t = 0$, a body of mass 100 g starts moving under the influence of a force $(5\hat{i} + 10\hat{j})$ N. After 2 s its position is $(2x\hat{i} + 5y\hat{j})$ m. The ratio $x : y$ is______.

Two cars A and B are moving in the same direction along a straight line with speeds 100 km/h and 80 km/h, respectively such that car A is moving ahead of car B. A person in car B throws a stone with a speed $v$ so that it hits the car A with a speed of 5 m/s. The value of $v$ is ______ km/h.

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

Let $f$ be a twice differentiable function such that $f(x) = \int_0^x \tan(t-x) dt - \int_0^x f(t) \tan t dt, x \in (-\frac{\pi}{2}, \frac{\pi}{2})$. Then $f''(\frac{\pi}{6}) + 12 f'(-\frac{\pi}{6}) + f(\frac{\pi}{6})$ is equal to ________.

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2 = 16x$. Let the vertex B containing the right angle be $(4, 8)$ and the locus of the centroid of $\Delta ABC$ be a conic $C_0$. Then three times the length of latus rectum of $C_0$ is _______.

Let A, B be points on the two half-lines $x - \sqrt{3}|y| = \alpha, \alpha>0$ at a distance of $\alpha$ from their point of intersection P. The line segment AB meets the angle bisector of the given half-lines at the point Q. If $PQ = \frac{9}{2}$ and R is the radius of the circumcircle of $\Delta PAB$, then $\frac{\alpha^2}{R}$ is equal to ________

The integral $\int_{0}^{1} \cot^{-1}(1 + x + x^2) dx$ is equal to:

Let $ y = y(x) $ be the solution of the differential equation:
\[ \frac{dy}{dx} + \left( \frac{6x^2 + (3x^2 + 2x^3 + 4)e^{-2x}}{(x^3 + 2)(2 + e^{-2x})} \right)y = 2 + e^{-2x}, \quad x \in (-1, 2) \]
satisfying $ y(0) = \frac{3}{2} $.
If $ y(1) = \alpha \left(2 + e^{-2}\right) $, then the value of $ \alpha $ is ________.

The area of the region bounded by the curves $x + 3y^2 = 0$ and $x + 4y^2 = 1$ is equal to:

Let $A = \{ (a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22 \}$. Then $n(A)$ is equal to:

Let for some $\alpha \in \mathbb{R}$, $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $f(x + y) = f(x) + 2y^2 + y + \alpha xy$ for all $x, y \in \mathbb{R}$. If $f(0) = -1$ and $f(1) = 2$, then the value of $\sum_{n=1}^{5} (\alpha + f(n))$ is:

Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}$. If $\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} \times \hat{v})$, then $\lambda$ is equal to:

If $(2\alpha + 1, \alpha^2 - 3\alpha, \frac{\alpha - 1}{2})$ is the image of $(\alpha, 2\alpha, 1)$ in the line $\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z}{1}$, then the possible value(s) of $\alpha$ is (are):

$\max_{0 \leq x \leq \pi} \left( 16 \sin\left(\frac{x}{2}\right) \cos^3\left(\frac{x}{2}\right) \right)$ is equal to:

Let $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\frac{8}{3}$. If the line $x = \alpha$ intersects the hyperbola H at the points A and B such that the area of the triangle AOB is $4\sqrt{15}$, where O is the origin, then $a^2$ equals:

Let $P(3\cos\alpha, 2\sin\alpha), \alpha \neq 0$, be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. $Q$ be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$ and $R$ be a point on the line $x + y = 5$ such that the centroid of the triangle $PQR$ is $\left( 2 + \cos\alpha, 3 + \frac{2}{3}\sin\alpha \right)$. Then the sum of the ordinates of all possible points $R$ is:

In the expansion of $\left( 9x - \frac{1}{3\sqrt{x}} \right)^{18}, x>0$, if the term independent of $x$ is $(221)k$, then $k$ is equal to:

For 10 observations $x_1, x_2, \dots, x_{10}$, if $\sum_{i=1}^{10} (x_i + 2)^2 = 180$ and $\sum_{i=1}^{10} (x_i - 1)^2 = 90$, then their standard deviation is:

Let \[ \alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \infty \] and \[ \beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty. \]
Then the value of \[ (0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_{5}(\beta)} \] is equal to: ________

Let $A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $\det(A - \alpha I) = 0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x - p)^2 + (y - 2p)^2 = 320$, intersects the co-ordinate axes at:

If the quadratic equation $(\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \lambda \neq -2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{-1}$ are two roots of the equation
$x^3 + ax^2 + bx + c = 0, a, b, c \in \mathbb{R}$, then $\int_{-1}^{1} (x^3 + ax^2 + bx + c) dx$ is equal to:

Let $S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}$. Then $\sum_{z \in S} |z + \sqrt{3}i|^2$ is equal to: