List of top Mathematics Questions

Two integers among 1 to 11 are selected at random. If their sum is even, then find the probability that both integers are odd.
If \( f : \mathbb{R} \to \mathbb{R} \), where \( f(x) = \sin x \) and \( g : \mathbb{R} \to \mathbb{R} \), where \( g(x) = x^2 \), then find the range of \( f(x) \) and \( g(x) \).
Find the area of the triangle whose two sides are represented by the vectors \[ \vec{a} = 3\hat{i} - \hat{j} + 5\hat{k}, \quad \vec{b} = \hat{i} + 2\hat{j} - \hat{k}. \]
If \( \vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k} \), \( \vec{b} = -\hat{i} + 2\hat{j} + \hat{k} \), and \( \vec{c} = 3\hat{i} + \hat{j} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then find the value of \( \lambda \).
If \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}, \] prove that \[ A^3 = \begin{bmatrix} \cos 3\theta & \sin 3\theta \\ -\sin 3\theta & \cos 3\theta \end{bmatrix}. \]
Prove: \[ \left| \begin{matrix} 1+\alpha & 1 & 1 \\ 1+\beta & 1 & 1 \\ 1 & 1 & 1+\gamma \\ \end{matrix} \right| = abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 \right) \]
Find the coordinates of the point which divides the line joining the points \( (2, -5, 1) \) and \( (1, 4, -6) \) internally in the ratio 2 : 3.
Solve the differential equation \( (x - y) \, dy - (x + y) \, dx = 0 \).
If the numbers of elements of two finite sets \( A \) and \( B \) are \( m \) and \( n \) respectively, then the total number of relations from \( A \) to \( B \) will be
If \( A = \{1, 2, 3\} \), \( B = \{2, 3, 4\} \), then the function from A to B will be
Prove that the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x - 1 \), when \( x>2 \), and \( f(1) = f(2) = 1 \), is onto but not one-to-one.
Solve the inequality \( 8x + 4<7x + 8 \).
The angle between the vectors \[ \mathbf{A} = 2\hat{i} + \hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{B} = 3\hat{i} - 2\hat{j} + \hat{k} \] will be
If \( P(A) = P(B) = \frac{5}{13} \) and \( P(A \cap B) = \frac{2}{5} \), then find \( P(A \cup B) \).
The value of \[ \int \cos^2 x \, dx \] will be
The degree of differential equation \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \] will be

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

If the mean deviation about median for the number $3,5,7,2 k , 12,16,21,24$ arranged in the ascending order, is 6 then the median is
Let $a_1=8, a_2, a_3, \ldots, a_n$ be an AP If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is______

Find the particular solution of the differential equation: \[ \frac{dy}{dx} + y \cot x = 4x \csc x \text{(} x \neq 0 \text{)}. \] Given that \( y = 0 \) \(\text{ when}\) \( x = \frac{\pi}{2} \).

Find the vector equation of a plane which passes through the point of intersection of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \] and the point $(1,1,1)$.

A relation \( R = \{(x, y) : \text{Number of pages in} \, x \text{ and } y \text{ are equal} \} \) is defined on the set \( A \) of all books in a college library. Prove that \( R \) is an equivalence relation.

Find the equation of the plane which passes through the intersecting point of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5, \] and the point \( (1, 1, 1) \).

Show that \( f(x) = |x| \, \textbf{is continuous at} \, x = 0.\)

If \[ P(A) = \frac{3}{13}, P(B) = \frac{5}{13}, \text{and} P(A \cap B) = \frac{2}{13}, \] \(\text{then find the value of }\) \( P(B|A) \).