List of top Mathematics Questions asked in JEE Main

Let $ ABC $ be the triangle such that the equations of lines $ AB $ and $ AC $ are: $ 3y - x = 2 \quad \text{and} \quad x + y = 2, $ respectively, and the points $ B $ and $ C $ lie on the x-axis. If $ P $ is the orthocentre of the triangle $ ABC $, then the area of the triangle $ PBC $ is equal to:
Let $ x_1, x_2, x_3, x_4 $ be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from $ x_1, x_2, x_3, x_4 $, then the resulting numbers are in an arithmetic progression. Then the value of $ \frac{1}{24} (x_1 x_2 x_3 x_4) $ is:
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 $ C_1 $ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let $ C_2 $ be the circle with center $ (1, 3) $ that touches $ C_1 $ externally at the point $ (\alpha, \beta) $. If $ (\beta - \alpha)^2 = \frac{m}{n} $, and $ \gcd(m, n) = 1 $, then $ m + n $ is equal to:
The integral $ \int_0^\pi \frac{(x + 3) \sin x}{1 + 3 \cos^2 x} \, dx $ is equal to:
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:

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

The sum of the infinite series $ \cot^{-1} \left( \frac{7}{4} \right) + \cot^{-1} \left( \frac{19}{4} \right) + \cot^{-1} \left( \frac{39}{4} \right) + \cot^{-1} \left( \frac{67}{4} \right) + ... $ is:
If $ \alpha $ is a root of the equation $ x^2 + x + 1 = 0 $ and $$ \sum_{k=1}^{n} \left( \alpha^k + \frac{1}{\alpha^k} \right)^2 = 20$$, then n is equal to _______
Let m and n, $ m<n $ be two 2-digit numbers. Then the total number of pairs (m, n) such that $ \gcd(m, n) = 6 $, is _______

If $$ \int \frac{\left( \sqrt{1 + x^2} + x \right)^{10}}{\left( \sqrt{1 + x^2} - x \right)^9} \, dx = \frac{1}{m} \left( \left( \sqrt{1 + x^2} + x \right)^n \left( n\sqrt{1 + x^2} - x \right) \right) + C, $$ $\text{where } m, n \in \mathbb{N} \text{ and }$ $C \text{ is the constant of integration, then } m + n$ $\text{ is equal to:}$

Let $ f(x) + 2f\left( \frac{1}{x} \right) = x^2 + 5 $ and $ 2g(x) - 3g\left( \frac{1}{2} \right) = x, \, x>0. \, \text{If} \, \alpha = \int_{1}^{2} f(x) \, dx, \, \beta = \int_{1}^{2} g(x) \, dx, \text{ then the value of } 9\alpha + \beta \text{ is:}$

The center of a circle $ C $ is at the center of the ellipse $ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, where $ a>b $. Let $ C $ pass through the foci $ F_1 $ and $ F_2 $ of $ E $ such that the circle $ C $ and the ellipse $ E $ intersect at four points. Let $ P $ be one of these four points. If the area of the triangle $ PF_1F_2 $ is 30 and the length of the major axis of $ E $ is 17, then the distance between the foci of $ E $ is:

If a curve $ y = y(x) $ passes through the point $ \left(1, \frac{\pi}{2}\right) $ and satisfies the differential equation $$ (7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5, \quad x \geq 1, \text{ then at } x = 2, \text{ the value of } \cos y \text{ is:} $$
Consider two sets $ A $ and $ B $, each containing three numbers in A.P. Let the sum and the product of the elements of $ A $ be 36 and $ p $, respectively, and the sum and the product of the elements of $ B $ be 36 and $ q $, respectively. Let $ d $ and $ D $ be the common differences of A.P's in $ A $ and $ B $, respectively, such that $ D = d + 3 $, $ d>0 $. If $ \frac{p+q}{p-q} = \frac{19}{5} $, then $ p - q $ is equal to:
Let for two distinct values of $ p $, the lines $ y = x + p $ touch the ellipse $ E: \frac{x^2}{4} + \frac{y^2}{9} = 1 $ at the points $ A $ and $ B $. Let the line $ y = x $ intersect $ E $ at the points $ C $ and $ D $. Then the area of the quadrilateral $ ABCD $ is equal to:

If
$ 2^m 3^n 5^k, \text{ where } m, n, k \in \mathbb{N}, \text{ then } m + n + k \text{ is equal to:} $

Let the domains of the functions $ f(x) = \log_4 \log_3 \log_7 (8 - \log_2(x^2 + 4x + 5)) $ and $ g(x) = \sin^{-1} \left( \frac{7x + 10}{x - 2} \right)$ be $ (\alpha, \beta) $ and $ [\gamma, \delta] $, respectively. Then $ \alpha^2 + \beta^2 + \gamma^2 + \delta^2 $ is equal to:
The axis of a parabola is the line $ y = x $ and its vertex and focus are in the first quadrant at distances $ \sqrt{2} $ and $ 2\sqrt{2} $ units from the origin, respectively. If the point $ (1, k) $ lies on the parabola, then a possible value of $ k $ is:
Let the values of $ p $, for which the shortest distance between the lines $ \frac{x + 1}{3} = \frac{y}{4} = \frac{z}{5} $ and $ \vec{r} = (p \hat{i} + 2 \hat{j} + \hat{k}) + \lambda (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) $ is $ \frac{1}{\sqrt{6}} $, be $ a, b $, where $ a<b $. Then the length of the latus rectum of the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ is:
Let $ a>0 $. If the function $ f(x) = 6x^3 - 45ax^2 + 108a^2x + 1 $ attains its local maximum and minimum values at the points $ x_1 $ and $ x_2 $ respectively such that $ x_1x_2 = 54 $, then $ a + x_1 + x_2 $ is equal to:

Let the matrix $ A = \begin{pmatrix} 1 & 0 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \end{pmatrix} $ satisfy $ A^n = A^{n-2} + A^2 - I $ for $ n \geq 3 $. Then the sum of all the elements of $ A^{50} $ is:

Let the three sides of a triangle ABC be given by the vectors $$ 2\hat{i} - \hat{j} + \hat{k}, \quad \hat{i} - 3\hat{j} - 5\hat{k}, \quad \text{and} \quad 3\hat{i} - 4\hat{j} - 4\hat{k}. $$ Let G be the centroid of the triangle ABC. Then $$ 6 \left( |\vec{AG}|^2 + |\vec{BG}|^2 + |\vec{CG}|^2 \right) \text{ is equal to}$$ _______
A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\frac{11}{50}, \text{ then } n \text{ is equal to}$ _______