List of top Mathematics Questions asked in JEE Main

The absolute difference between the squares of the radii of the two circles passing through the point $ (-9, 4) $ and touching the lines $ x + y = 3 $ and $ x - y = 3 $, is equal to:

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(\vec{SP} \cdot \vec{S'P}) + \max(\vec{SP} \cdot \vec{S'P}) $ 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:

If the system of equations: $$ \begin{aligned} 3x + y + \beta z &= 3 \\2x + \alpha y + z &= 2 \\x + 2y + z &= 4 \end{aligned} $$ has infinitely many solutions, then the value of $ 22\beta - 9\alpha $ is:

If $ \vec{a} $ is a non-zero vector such that its projections on the vectors $ 2\hat{i} - \hat{j} + 2\hat{k},\ \hat{i} + 2\hat{j} - 2\hat{k} $, and $ \hat{k} $ are equal, then a unit vector along $ \vec{a} $ is:

Let $ A $ be the set of all functions $ f: \mathbb{Z} \to \mathbb{Z} $ and $ R $ be a relation on $ A $ such that $$ R = \{ (f, g) : f(0) = g(1) \text{ and } f(1) = g(0) \} $$ Then $ R $ is:

For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:

If $ \theta \in [-2\pi,\ 2\pi] $, then the number of solutions of $$ 2\sqrt{2} \cos^2\theta + (2 - \sqrt{6}) \cos\theta - \sqrt{3} = 0 $$ is:

Let $ A = \begin{bmatrix} \alpha & -1 \\6 & \beta \end{bmatrix},\ \alpha > 0 $, such that $ \det(A) = 0 $ and $ \alpha + \beta = 1 $. If $ I $ denotes the $ 2 \times 2 $ identity matrix, then the matrix $ (1 + A)^5 $ is:

Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is:

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 tetrahedron ABCD is equal to:

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:

Let $ z $ be a complex number such that $ |z| = 1 $. If \[ \frac{2 + kz}{k + z} = kz,\ k \in \mathbb{R}, \] then the maximum distance of $ k + ik^2 $ from the circle $ |z - (1 + 2i)| = 1 $ is:

The sum of the series $ 2 \times 1 \times 20C_4 - 3 \times 2 \times 20C_5 + 4 \times 3 \times 20C_6 - 5 \times 4 \times 20C_7 + ... + 18 \times 17 \times 20C_{20}, \text{is equal to} $

The greatest value of $ n $, where $ n \in \mathbb{N} $, such that $ 3^n $ divides $ 50! $ is:

Let $ y = f(x) $ be the solution of the differential equation\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\] such that $ f(0) = 0 $. If \[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\] then $ \alpha^2 $ is equal to __________.

Let the system of equations $ x + 5y - z = 1 $ $ 4x + 3y - 3z = 7 $ $ 24x + y + \lambda z = \mu $ where $ \lambda, \mu \in \mathbb{R} $, have infinitely many solutions. Then the number of the solutions of this system, if $x, y, z$ are integers and satisfy $7 \leq x + y + z \leq 77$, is:

If the equation of the line passing through the point $ \left( 0, -\frac{1}{2}, 0 \right) $ and perpendicular to the lines $ \mathbf{r_1} = \lambda ( \hat{i} + a \hat{j} + b \hat{k}) \quad \text{and} \quad \mathbf{r_2} = ( \hat{i} - \hat{j} - 6 \hat{k} ) + \mu( -b \hat{i} + a \hat{j} + 5 \hat{k}), $ is $ \frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}, $ then $ a + b + c + d $ is equal to:

Consider the lines $ L_1: x - 1 = y - 2 = z $ and $ L_2: x - 2 = y = z - 1 $. Let the feet of the perpendiculars from the point $ P(5, 1, -3) $ on the lines $ L_1 $ and $ L_2 $ be $ Q $ and $ R $ respectively. If the area of the triangle $ PQR $ is $ A $, then $ 4A^2 $ is equal to:

Let $ e_1 $ and $ e_2 $ be the eccentricities of the ellipse $ \frac{x^2}{b^2} + \frac{y^2}{25} = 1 $ and the hyperbola $ \frac{x^2}{16} - \frac{y^2}{b^2} = 1, $ respectively. If $ b<5 $ and $ e_1 e_2 = 1 $, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

If the area of the region $ \{(x, y) : 1 + x^2 \leq y \leq \min(x + 7, 11 - 3x)\} $ is $ A $, then $ 3A $ is equal to:

If the range of the function $ f(x) = \frac{5 - x}{x^2 - 3x + 2}, \quad x \neq 1, 2 $ is $ (-\infty, \alpha] \cup [\beta, \infty) $, then $ \alpha^2 + \beta^2 $ is equal to: