List of top Mathematics Questions

A six faced die is biased such that
3 × P (a prime number) = 6 × P (a composite number) = 2 × P (1).
Let X be a random variable that counts the number of times one gets a perfect square on some
throws of this die. If the die is thrown twice, then the mean of X is :

Let S = {1, 2, 3, …, 2022}. Then the probability that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is
Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :
If the transformation $z = \log \tan \frac{x}{2}$ reduces the differential equation $\frac{d^2y}{dx^2} + \cot x \frac{dy}{dx} + 4y \csc^2 x = 0$ into the form $\frac{d^2y}{dz^2} + ky = 0$, then $k$ is equal to
If $\mathbf{a} = \hat{i} + \hat{j} - \hat{k}$, $\mathbf{b} = \hat{i} - \hat{j} + \hat{k}$, and $\mathbf{c}$ is a unit vector perpendicular to $\mathbf{a}$ and coplanar with $\mathbf{a}$ and $\mathbf{b}$, then the unit vector $\mathbf{d}$ perpendicular to both $\mathbf{a}$ and $\mathbf{c}$ is
Let $S$, $T$, $U$ be three non-void sets, where $f: S \to T$, $g: T \to U$, and the composed mapping $g \circ f: S \to U$ is defined. If $g \circ f$ is an injective mapping, then
If $a$, $b$ are odd integers, then the roots of the equation $2ax^2 + (2a + b)x + b = 0$, where $a \neq 0$, are
The value of $a$ for which the sum of the squares of the roots of the equation $x^2 - (a - 2)x - (a - 1) = 0$ assumes the least value is
If $a$, $b$, and $c$ are in GP, and $\log a - \log 2b$, $\log 2b - \log 3c$, $\log 3c - \log a$ are in A.P., then $a$, $b$, and $c$ are the lengths of the sides of a triangle which is
Let $f(n) = 2n + 1$, $g(n) = 1 + (n + 1)^{2n}$ for all $n \in \mathbb{N}$. Then
Consider the equation $y - y_1 = m(x - x_1)$. If $m$ and $x_1$ are fixed, and different lines are drawn for different values of $y_1$, then
$f: X \to \mathbb{R}, X = \{x | 0<x<1\}$ is defined as $f(x) = \frac{2x - 1}{1 - |2x - 1|}$. Then
The maximum value of $f(x) = e^{\sin x} + e^{\cos x}$, where $x \in \mathbb{R}$, is
A curve passes through the point $(3, 2)$ for which the segment of the tangent line contained between the coordinate axes is bisected at the point of contact. The equation of the curve is
The line $x - 2y + 4z + 4 = 0$, $x + y + z - 8 = 0$ intersects the plane $x - y + 2z + 1 = 0$ at the point
The value of $\int_0^{\pi/2} \frac{(\cos x)\sin x}{(\cos x)\sin x + (\sin x)\cos x} \, dx$ is
For the mapping $f: \mathbb{R} \setminus \{1\} \to \mathbb{R} \setminus \{2\}$ given by $f(x) = \frac{2x}{x - 1}$, which of the following is correct?
Let $f$ be a non-negative function defined in $[0, \pi/2]$, $f'$ exists and is continuous for all $x$, and $\int_0^x \sqrt{1 - (f'(t))^2} dt = \int_0^x f(t) dt$ and $f(0) = 0$. Then
If $I$ is the greatest of $I_1 = \int_0^1 e^{-x} \cos^2 x \, dx$, $I_2 = \int_0^1 e^{-x^2} \cos^2 x \, dx$, $I_3 = \int_0^1 e^{-x^2} \, dx$, $I_4 = \int_0^1 e^{-\frac{x^2}{2}} \, dx$, then
If $x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}$, then $|f(xy)|$ is equal to
The area of the figure bounded by the parabola $y^2 + 8x = 16$ and $y^2 - 24x = 48$ is
$AB$ is a chord of a parabola $y^2 = 4ax$, $(a > 0)$ with vertex $A$, $BC$ is drawn perpendicular to $AB$ meeting the axis at $C$. The projection of $BC$ on the axis of the parabola is
The point of contact of the tangent to the parabola $y^2 = 9x$ which passes through the point $(4, 10)$ and makes an angle $\theta$ with the positive side of the axis of the parabola, where $\tan \theta>2$, is
If $P_1P_2$ and $P_3P_4$ are two focal chords of the parabola $y^2 = 4ax$, then the chords $P_1P_3$ and $P_2P_4$ intersect on the
Let the tangent and normal at any point $P(at^2, 2at), (a>0)$, on the parabola $y^2 = 4ax$ meet the axis of the parabola at $T$ and $G$ respectively. Then the radius of the circle through $P$, $T$, and $G$ is