List of top Mathematics Questions asked in JEE Main

If the image of the point $ (4, 4, 3) $ in the line $ \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} $ is $ (a, \beta, \gamma) $, then $ a + \beta + \gamma $ is equal to:

If the set of all $ a \in \mathbb{R} \setminus \{1\} $, for which the roots of the equation $ (1 - a)x^2 + 2(a - 3)x + 9 = 0 $ are positive is $ (-\infty, -\alpha] \cup [\beta, \gamma] $, then $ 2\alpha + \beta + \gamma $ is equal to ...........

The sum of the squares of all the roots of the equation $ x^2 + [2x - 3] - 4 = 0 $ is:

Let for some function $ y = f(x) $, $\int_0^x t f(t) \, dt = x^2 f(x), x>0$ and $ f(2) = 3 $. Then $ f(6) $ is equal to:

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is:

Let $ y = y(x) $ be the solution curve of the differential equation $ x(x^2 + e^x) \, dy + \left( e^x(x - 2) y - x^3 \right) \, dx = 0, \quad x>0, $ passing through the point $ (1, 0) $. Then $ y(2) $ is equal to:

Let P be the parabola, whose focus is $ (-2, 1) $ and directrix is $ 2x + y + 2 = 0 $. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:

The remainder when $ \left( (64)^{64} \right)^{64} $ is divided by 7 is equal to:

Let $ x = -1 $ and $ x = 2 $ be the critical points of the function $ f(x) = x^3 + ax^2 + b \log|x| + 1 $, where $ x \neq 0 $. Let $ m $ and $ M $ be the absolute minimum and maximum values of $ f $ in the interval $ \left[-2, -\frac{1}{2}\right] $. Then, $ |M + m| $ is equal to:

If the shortest distance between the lines $ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} $ and $ \frac{x}{1} = \frac{y}{\alpha} = \frac{z-5}{1} $ is $ \frac{5}{\sqrt{6}} $, then the sum of all possible values of $ \alpha $ is:

If the sum of the first 10 terms of the series \[ \frac{4 \cdot 1}{1 + 4 \cdot 1^4} + \frac{4 \cdot 2}{1 + 4 \cdot 2^4} + \frac{4 \cdot 3}{1 + 4 \cdot 3^4} + \ldots \] is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to _____.

Let $ A(4, -2), B(1, 1) $ and $ C(9, -3) $ be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.

Evaluate the following limit: $ \lim_{x \to 0^+} \frac{\tan\left(5x^{\frac{1}{3}}\right) \log\left(1 + 3x^2\right)}{\left(\tan^{-1}\left(3\sqrt{x}\right)\right)^2 \left(e^{5x^{\frac{4}{3}}} - 1\right)} $

If $ y = \cos \left( \frac{\pi}{3} + \cos^{-1} \frac{x}{2} \right) $, then $ (x - y)^2 + 3y^2 $ is equal to ________.

Let $ y = y(x) $ be the solution of the differential equation \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] such that $ y(0) = \frac{5}{4} $. Then \[ 12 \left( y\left( \frac{\pi}{4} \right) - e^{-2} \right) \] is equal to _____.

Let $ A $ be a $ 3 \times 3 $ real matrix such that \[ A^{2}(A - 2I) - 4(A - I) = O, \] where $ I $ and $ O $ are the identity and null matrices, respectively.
If \[ A^{5} = \alpha A^{2} + \beta A + \gamma I, \] where $ \alpha, \beta, \gamma $ are real constants, then $ \alpha + \beta + \gamma $ is equal to:

Let $ \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} $, $ \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} $ and a vector $ \vec{c} $ be such that \[ (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \] and \[ \vec{a} \cdot \vec{c} = 3. \] If $ \vec{b} \times \vec{c} = \vec{d} $, then find $ |\vec{a} \cdot \vec{d}| $.

Let the area of the triangle formed by a straight line $ L: x + by + c = 0 $ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line $ L $ makes an angle of $ 45^\circ $ with the positive x-axis, then the value of $ b^2 + c^2 $ is:

Let the point $ P $ of the focal chord $ PQ $ of the parabola $ y^2 = 16x $ be $ (1, -4) $. If the focus of the parabola divides the chord $ PQ $ in the ratio $ m : n $, gcd($m, n$) = 1, then $ m^2 + n^2 $ is equal to:

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:

If \[ \sum_{r=0}^{10} \left( \frac{10^{r+1} - 1}{10^r} \right) \cdot {^{11}C_{r+1}} = \frac{\alpha^{11} - 11^{11}}{10^{10}}, \] then $ \alpha $ is equal to:

$$ \int \frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) $$ is equal to:

If $ \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} $ is finite, then $ (a + b) $ is equal to:

If the domain of the function $ f(x) = \dfrac{1}{\sqrt{10 + 3x - x^2}} + \dfrac{1}{\sqrt{x + |x|}} $ is $ (a, b) $, then $(1 + a)^2 + b^2$ 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: