List of top Mathematics Questions

Let $f: [0, \infty) \to \mathbb{R}$ be a differentiable function such that $f(x) = 1 - 2x + \int_0^x e^{x-t} f(t) \, dt$ for all $x \in [0, \infty)$. Then the area of the region bounded by $y = f(x)$ and the coordinate axes is

If $\lim_{x \to 1} \frac{(x-1)(6+\lambda \cos(x-1)) + \mu \sin(1-x)}{(x-1)^3} = -1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda + \mu$ is equal to

Let the shortest distance between the lines $\frac{x-3}{3} = \frac{y-\alpha}{-1} = \frac{z-3}{1}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-\beta}{4}$ be $3\sqrt{30}$. Then the positive value of $5\alpha + \beta$ is

The probability of forming a 12 persons committee from 4 engineers, 2 doctors, and 10 professors containing at least 3 engineers and at least 1 doctor is:
For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n-3}$ is 16, then the distance of the point $P(2n-1, n^2-4n)$ from the line $x + y = 8$ is:
Let $A = \{1, 6, 11, 16, \ldots\}$ and $B = \{9, 16, 23, 30, \ldots\}$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $n(A \cup B)$ is
Consider the sets $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$, $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$, $C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \leq 4\}$, and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:
Let $f, g: (1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x) = \frac{2x + 3}{5x + 2}$ and $g(x) = \frac{2 - 3x}{1 - x}$. If the range of the function $fog: [2, 4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta - \alpha}$ is equal to
The area of the region bounded by the curve $ y = \max\{|x|, |x-2|\} $, then x-axis and the lines x = -2 and x = 4 is equal to ____.
Let $ \vec{a} = \hat{i} + \hat{j} + \hat{k} $, $ \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k} $, $ \vec{c} = \lambda \hat{j} + \mu \hat{k} $ and $ \hat{d} $ be a unit vector such that $ \vec{a} \times \hat{d} = \vec{b} \times \hat{d} $ and $ \vec{c} \cdot \hat{d} = 1 $. If $ \vec{c} $ is perpendicular to $ \vec{a} $, then $ |3\lambda \hat{d} + \mu \vec{c}|^2 $ is equal to ____.
Line $ L_1 $ passes through the point (1, 2, 3) and is parallel to z-axis. Line $ L_2 $ passes through the point $ (\lambda, 5, 6) $ and is parallel to y-axis. Let for $ \lambda = \lambda_1, \lambda_2, \lambda_2<\lambda_1 $, the shortest distance between the two lines be 3. Then the square of the distance of the point $ (\lambda_1, \lambda_2, 7) $ from the line $ L_1 $ is

Let $ f(x) = \begin{cases} (1+ax)^{1/x} & , x<0 \\1+b & , x = 0 \\\frac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2} & , x>0 \end{cases} $ be continuous at x = 0. Then $ e^a bc $ is equal to

Let $ a_1, a_2, a_3, ... $ be a G.P. of increasing positive numbers. If $ a_3 a_5 = 729 $ and $ a_2 + a_4 = \frac{111}{4} $, then $ 24(a_1 + a_2 + a_3) $ is equal to
Let $ f(x) = \int x^3 \sqrt{3-x^2} dx $. If $ 5f(\sqrt{2}) = -4 $, then $ f(1) $ is equal to

If $ y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\27 & 28 & 27 \\1 & 1 & 1 \end{vmatrix} $, $ x \in \mathbb{R} $, then $ \frac{d^2y}{dx^2} + y $ is equal to

The number of solutions of the equation $ 2x + 3\tan x = \pi $, $ x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} $ is
The sum 1 + 3 + 11 + 25 + 45 + 71 + ... upto 20 terms, is equal to

Let A = $\{-3,-2,-1,0,1,2,3\}$. Let R be a relation on A defined by xRy if and only if $ 0 \le x^2 + 2y \le 4 $. Let $ l $ be the number of elements in R and m be the minimum number of elements required to be added in R to make it a reflexive relation. then $ l + m $ is equal to

Let $ \alpha $ and $ \beta $ be the roots of $ x^2 + \sqrt{3}x - 16 = 0 $, and $ \gamma $ and $ \delta $ be the roots of $ x^2 + 3x - 1 = 0 $. If $ P_n = \alpha^n + \beta^n $ and $ Q_n = \gamma^n + \delta^n $, then $ \frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}} \text{ is equal to} $

Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} $ at the point $ A(\alpha, \beta, \gamma) $ and the line $ L_2: x - 6 = y = -z + 4 $ at the point $ B(a, b, c) $. Then $ \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \text{ is equal to} $

For $ n \geq 2 $, let $ S_n $ denote the set of all subsets of $ \{1, 2, 3, \ldots, n\} $ with no two consecutive numbers. For example, $ \{1, 3, 5\} \in S_6 $, but $ \{1, 2, 4\} \notin S_6 $. Then, find $ n(S_5) $.
The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:
The number of relations on the set $ A = \{1, 2, 3\} $ containing at most 6 elements including $ (1, 2) $, which are reflexive and transitive but not symmetric, is:
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 the angle $ \theta, 0<\theta<\frac{\pi}{2} $ between two unit vectors $ \hat{a} $ and $ \hat{b} $ be $ \sin^{-1} \left( \frac{\sqrt{65}}{9} \right) $. If the vector $ \vec{c} = 3\hat{a} + 6\hat{b} + 9(\hat{a} \times \hat{b}) $, then the value of $ 9(\vec{c} \cdot \hat{a}) - 3(\vec{c} \cdot \hat{b}) $ is: