List of top Mathematics Questions asked in JEE Main

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:

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:}

Among the statements: (S1): The set $ \{ z \in \mathbb{C} - \{-i\} : |z| = 1 \text{ and } \frac{z - i}{z + i} \text{ is purely real} \} $ contains exactly two elements.
(S2): The set $ \{ z \in \mathbb{C} - \{-1\} : |z| = 1 \text{ and } \frac{z - 1}{z + 1} \text{ is purely imaginary} \} $ contains infinitely many elements. Then, which of the following is correct?

Evaluate the limit: \[ \lim_{x \to 0} \csc{x} \left( \sqrt{2 \cos^2{x} + 3 \cos{x}} - \sqrt{\cos^2{x} + \sin{x} + 4} \right) \] 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:

Let $ C_{t-1} = 28, C_t = 56 $ and $ C_{t+1} = 70 $. Let $ A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \text{ and } C(3r - n_1, r^2 - n - 1) $ be the vertices of a triangle ABC, where $ t $ is a parameter. If $ (3x - 1)^2 + (3y)^2 = \alpha $ is the locus of the centroid of triangle ABC, then $ \alpha $ equals:

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:

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_1, a_2, a_3, \dots $ be a G.P. of increasing positive terms. If $ a_1 a_5 = 28 $ and $ a_2 + a_4 = 29 $, then the value of $ a_6 $ is equal to:

If the function \[ f(x) = \begin{cases} \frac{2}{x} \left( \sin(k_1 + 1)x + \sin(k_2 -1)x \right), & x<0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e \left( \frac{2 + k_1 x}{2 + k_2 x} \right), & x>0 \end{cases} \] is continuous at $ x = 0 $, then $ k_1^2 + k_2^2 $ is equal to:

Let $ f(x) $ be a real differentiable function such that $ f(0) = 1 $ and $ f(x + y) = f(x)f'(y) + f'(x)f(y) $ for all $ x, y \in \mathbb{R} $. Then $ \sum_{n=1}^{100} \log_e f(n) $ is equal to :

Let the triangle PQR be the image of the triangle with vertices $ (1, 3), (3, 1) $ and $ (2, 4) $ in the line $ x + 2y = 2 $. If the centroid of $ \Delta PQR $ is the point $ (\alpha, \beta) $, then $ 15(\alpha - \beta) $ is equal to :

Let integers $ a, b \in [-3, 3] $ be such that $ a + b \neq 0 $. $\text{Then the number of all possible ordered pairs}$ $ (a, b) $, $\text{for which}$ \[ \left| \frac{z - a}{z + b} \right| = 1 \quad \text{and} \quad \left| \begin{matrix} z + 1 & \omega & \omega^2 \\ \omega^2 & 1 & z + \omega \\ \omega^2 & 1 & z + \omega \end{matrix} \right| = 1, \] $\text{is equal to:}$

Let $ A(6,8) $, $ B(10\cos\alpha, -10\sin\alpha) $, and $ C(-10\sin\alpha, 10\cos\alpha) $ be the vertices of a triangle. If $ L(a,9) $ and $ G(h,k) $ be its orthocenter and centroid respectively, then $ 5a - 3h + 6k + 100\sin2\alpha $ is equal to _________.

The variance of the numbers 8, 21, 34, 47, \dots, 320, is:

The integral \[ \int_0^\pi \frac{8x}{4\cos^2 x + \sin^2 x} \, dx \text{ is equal to:} \]

For positive integers $ n $, if $ 4 a_n = \frac{n^2 + 5n + 6}{4} $ and \[ S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right), \text{ then the value of } 507 S_{2025} \text{ is:} \]

1 + 3 + $5^2$ + 7 + $9^2$ + $\ldots$ upto 40 terms is equal to

If \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96x^2 \cos^2 x}{1 + e^x} dx = \pi(a\pi^2 + \beta), \quad a, \beta \in \mathbb{Z}, \] then $ (a + \beta)^2 $ equals:

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is:

Let the equation of the circle, which touches x-axis at the point $ (a, 0) $ and cuts off an intercept of length $ b $ on y-axis be $ x^2 + y^2 - cx + dy + e = 0 $. If the circle lies below x-axis, then the ordered pair $ (2a, b^2) $ is equal to:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all $ x, y \in \mathbb{R}. $

If $ f'(0) = \frac{1}{2} $, then the value of $ 24f''\left( \frac{5\pi}{3} \right) $ is:

Let $ b_1 = 3, b_2, b_3, \ldots $ be a geometric progression of increasing positive numbers. Let \[ \sum_{n=1}^{20} b_{3n} = 4\sum_{n=1}^{20} b_{3n-2}. \] Then, the sum of the first ten terms of the G.P. is:

If the line $ \arg(z) = \frac{\pi}{3} $ intersects the curve $ |z - 2\sqrt{3}i| = 2 $, $ z \in \mathbb{C} $, at two distinct points $A$ and $B$, then $AB$ equals:

If the system of linear equations : \[ x+y+z=4,\quad x+2y+3z=6,\quad 4x+5y+\lambda z=\mu \] has more than one solution, then the value of $ \lambda+\mu $ is equal to: