List of top Mathematics Questions

The length of the latus-rectum of the ellipse, whose foci are $(2, 5)$ and $(2, -3)$ and eccentricity is $\frac{4}{5}$, is

The value of $\int_{-1}^{1} \frac{(1 + \sqrt{|x| - x})e^x + (\sqrt{|x| - x})e^{-x}}{e^x + e^{-x}} \, dx$ is equal to

Let the three sides of a triangle are on the lines \( 4x - 7y + 10 = 0,\quad x + y = 5,\quad 7x + 4y = 15 \).
Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines \( x = 0,\quad y = 0,\quad x + y = 1 \) is

Consider two vectors $\vec{u} = 3\hat{i} - \hat{j}$ and $\vec{v} = 2\hat{i} + \hat{j} - \lambda \hat{k}$, $\lambda>0$. The angle between them is given by $\cos^{-1} \left( \frac{\sqrt{5}}{2\sqrt{7}} \right)$. Let $\vec{v} = \vec{v}_1 + \vec{v}_2$, where $\vec{v}_1$ is parallel to $\vec{u}$ and $\vec{v}_2$ is perpendicular to $\vec{u}$. Then the value $|\vec{v}_1|^2 + |\vec{v}_2|^2$ is equal to

Considering the principal values of the inverse trigonometric functions, $\sin^{-1} \left( \frac{\sqrt{3}}{2} x + \frac{1}{2} \sqrt{1-x^2} \right)$, $-\frac{1}{2}<x<\frac{1}{\sqrt{2}}$, is equal to

In the expansion of \[ \left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n , \, n \in \mathbb{N}, \] if the ratio of the 15th term from the beginning to the 15th term from the end is \[ \frac{1}{6}, \] then the value of \[ {}^nC_3 \] is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $f(0) = 1$ and $f(2x) - f(x) = x$ for all $x \in \mathbb{R}$. If $\lim_{n \to \infty} \left\{ f(x) - f\left( \frac{x}{2^n} \right) \right\} = G(x)$, then $\sum_{r=1}^{10} G(r^2)$ is equal to

Let $A$ and $B$ be two distinct points on the line $L: \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2\sqrt{17}$ from the foot of perpendicular drawn from the point $(1, 2, 3)$ on the line $L$. If $O$ is the origin, then $\overrightarrow{OA} \cdot \overrightarrow{OB}$ is equal to:

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 ____.
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

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} $
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 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 the line $ L $ pass through $ (1, 1, 1) $ and intersect the lines $ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} $ and $ \frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z}{1}. $ Then, which of the following points lies on the line $ L $?