List of top Mathematics Questions asked in JEE Main

Let $ \vec{c} $ be the projection vector of $ \mathbf{b} = \lambda \hat{i} + 4 \hat{k}, \lambda > 0 $, on the vector $ \vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k} $. If $ |\vec{a} + \vec{c}| = 7 $, then the area of the parallelogram formed by the vectors $ \vec{b} $ and $ \vec{c} $ is:

Let $ L_1 : \frac{x - 1}{3} = \frac{y - 1}{-1} = \frac{z + 1}{0} $ and $ L_2 : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{\alpha} $, where $ \alpha \in \mathbb{R} $, be two lines which intersect at the point $ B $. If $ P $ is the foot of the perpendicular from the point $ A(1, 1, -1) $ on $ L_2 $, then the value of $ 26 \alpha (PB)^2 $ is:

Three distinct numbers are selected randomly from the set $ \{1, 2, 3, \dots, 40\} $. If the probability, that the selected numbers are in an increasing G.P. is $ \frac{m}{n} $, where $ \gcd(m, n) = 1 $, then $ m + n $ is equal to:

Let $ A $ be a square matrix of order 3 such that $ \text{det}(A) = -2 $ and $ \text{det}(3 \cdot \text{adj}(-6 \cdot \text{adj}(3A))) = 2^{m+n} \cdot 3^{mn} $, where $ m > n $. Then $ 4m + 2n $ is equal to:

If $ \sum_{r=0}^5 \frac{{}^{11}C_{2r+1}}{2r+2} = \frac{m}{n} $, gcd(m, n) = 1, then $ m - n $ is equal to:

Let the function, $f(x)$ = $\begin{cases} -3ax^2 - 2, & x < 1 \\a^2 + bx, & x \geq 1 \end{cases}$ Be differentiable for all $ x \in \mathbb{R} $, where $ a > 1 $, $ b \in \mathbb{R} $. If the area of the region enclosed by $ y = f(x) $ and the line $ y = -20 $ is $ \alpha + \beta\sqrt{3} $, where $ \alpha, \beta \in \mathbb{Z} $, then the value of $ \alpha + \beta $ is:

Let the foci of a hyperbola be $ (1, 14) $ and $ (1, -12) $. If it passes through the point $ (1, 6) $, then the length of its latus-rectum is:

If the area of the region \[ \{(x, y) : |4 - x^2| \leq y \leq x^2, y \leq 4, x \geq 0\} \] is $ \frac{80\sqrt{2}}{\alpha - \beta} $, where $ \alpha, \beta \in \mathbb{N} $, then $ \alpha + \beta $ is equal to:

The area of the region, inside the circle $(x-2\sqrt{3})^2 + y^2 = 12$ and outside the parabola $y^2 = 2\sqrt{3}x$ is:

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where gcd(m, n) = 1, then m + n is equal to:

Let $ A = \{1, 2, 3, \dots, 10\} $ and $ B = \left\{ \frac{m}{n} : m, n \in A, m <n \text{ and } \gcd(m, n) = 1 \right\} $. Then $ n(B) $ is equal to :

Let $ [\cdot] $ denote the greatest integer function. If \[ \int_0^3 \left\lfloor \frac{1}{e^x - 1} \right\rfloor \, dx = \alpha - \log_e 2, \] then $ \alpha^3 $ 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 for $ f(x) = 7\tan^8 x + 7\tan^6 x - 3\tan^4 x - 3\tan^2 x $, $ I_1 = \int_0^{\frac{\pi}{4}} f(x)dx $ and $ I_2 = \int_0^{\frac{\pi}{4}} x f(x)dx $. Then $ 7I_1 + 12I_2 $ is equal to:

Let the focal chord PQ of the parabola $ y^2 = 4x $ make an angle of $ 60^\circ $ with the positive x-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, $ S $ being the focus of the parabola, touches the y-axis at the point $ (0, \alpha) $, then $ 5\alpha^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 product of $ \omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta $ and $ \omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta $ be $ \alpha + i\beta $, where $ i = \sqrt{-1} $. Let $ p $ and $ q $ be the maximum and the minimum values of $ \alpha + \beta $ respectively.

Let $ a \in \mathbb{R} $ and $ A $ be a matrix of order $ 3 \times 3 $ such that $ \det(A) = -4 $ and \[ A + I = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix} \] where $ I $ is the identity matrix of order $ 3 \times 3 $.

If $ \det\left( (a + 1) \cdot \text{adj}\left( (a - 1) A \right) \right) $ is $ 2^m 3^n $, $ m, n \in \{ 0, 1, 2, \dots, 20 \} $, then $ m + n $ is equal to:

Let $ A = \{-3, -2, -1, 0, 1, 2, 3\} $ and $ R $ be a relation on $ A $ defined by $ xRy $ if and only if $ 2x - y \in \{0, 1\} $. Let $ l $ be the number of elements in $ R $. Let $ m $ and $ n $ be the minimum number of elements required to be added in $ R $ to make it reflexive and symmetric relations, respectively. Then $ l + m + n $ is equal to:

Let $ ABCD $ be a tetrahedron such that the edges $ AB $, $ AC $, and $ AD $ are mutually perpendicular. Let the areas of the triangles $ ABC $, $ ACD $, and $ ADB $ be 5, 6, and 7 square units respectively. Then the area (in square units) of the $ \triangle BCD $ 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:

Let the vertices Q and R of the triangle PQR lie on the line $ \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} $, $ QR = 5 $, and the coordinates of the point P be $ (0, 2, 3) $. If the area of the triangle PQR is $ \frac{m}{n} $, then:

If $ S $ and $ S' $ are the foci of the ellipse \[ \frac{x^2}{18} + \frac{y^2}{9} = 1 \] and $ P $ is a point on the ellipse, then \[ \min (SP \cdot S'P) + \max (SP \cdot S'P) \] is equal to:

Let $ f $ be a differentiable function on $ \mathbb{R} $ such that $ f(2) = 4 $. Let $ \lim_{x \to 0} \left( f(2+x) \right)^{3/x} = e^\alpha $. Then the number of times the curve $ y = 4x^3 - 4x^2 - 4(\alpha - 7)x - \alpha $ meets the x-axis is: