List of top Mathematics Questions

If vectors \( 2\hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular to each other, then find the value of \( \lambda \).
If \( A = \{a, b, c\} \) and \( B = \{1, 2\} \), then find the number of relations from \( A \) to \( B \).
The order of the differential equation $2x^2 \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + y = 0$ will be:
The value of $\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})$ will be:
Differential coefficient of $\cos^{-1}(e^x)$ will be:
The value of $\int x e^x \, dx$ will be:
A relation $R = \{(a, b) : a = b - 1, b > 4\}$ is defined on set $\mathbb{N}$, then correct answer will be:

If \[ A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] \(\text{then prove that}\) \[ A \cdot \text{adj} \, A = |A| \cdot I. \]

If \[\begin{bmatrix} 2x - y & x + 2y \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\]
 then the value of $x$ and $y$ will be: 
 

Find the maximization of \( z = x + y \), under the following constraints: \[ x - y \leq -1, -x + y \leq 0, x \geq 0, y \geq 0. \]
For any positive constant \( a \), evaluate \[ \frac{dy}{dx}, \text{ where } y = a \frac{t+1}{t} \text{ and } x = (t + 1)^{\alpha}. \]
Let \[ A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}, B = \begin{pmatrix} 5 & 2 \\ 7 & 4 \end{pmatrix}, C = \begin{pmatrix} 2 & 5 \\ 3 & 8 \end{pmatrix}. \] Then find a matrix D such that \( CD - AB = 0 \).
Solve the system of linear equations by matrix method: \[ 3x + 2y + 3z = 5, \] \[ -2x + y + z = -4, \] \[ -x + 3y - 2z = 3. \]
Differentiate: \( y = x^{x} \).
An Apache helicopter of the enemy is flying along the curve \[ y = x^2 + 7. \] A soldier placed at the point \( (3, 7) \), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.
Coloured balls are distributed in three containers according to the following table: \[\begin{array}{|c|c|c|c|}\hline \text{Container} & \text{Black} & \text{White} & \text{Red} \\ \hline I & 3 & 4 & 5 \\ \hline II & 2 & 2 & 2 \\ \hline III & 1 & 2 & 3 \\ \hline \end{array}\] A ball is drawn out from a container randomly chosen. If the ball is black, then find the probability that the ball is drawn from Container-III.
There are 500 students in a school, of which 230 are girls. Also, 10% of 230 girls are studying in class XII. Find the probability that a randomly chosen student is of XII class and is a girl.
For any two vectors \( \vec{a} \) and \( \vec{b} \), prove that always \[ |\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}|. \]
By graphical method solve the LPP under the following constraints: \[ x + 2y \geq 10, \] \[ 3x + 4y \leq 24, \] \[ x \geq 0, \, y \geq 0, \] then find the minimum value of \( z = 200x + 500y \).
If the position vectors of the points A, B, C and D are \( \mathbf{A} = 3\hat{i} + 2\hat{j} - 3\hat{k}, \) \( \mathbf{B} = \hat{i} + \hat{j} + \hat{k}, \) \( \mathbf{C} = 2\hat{i} + 5\hat{j}, \) and \( \mathbf{D} = \hat{i} - 6\hat{j} - \hat{k}, \) respectively. Prove that the points are collinear.
Evaluate: \[ \int \left( \sqrt{\cot x} + \sqrt{\tan x} \right) dx \]
Find the shortest distance between the lines \[ \vec{r_1} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2t) \hat{k} \] \text{and} \[ \vec{r_2} = (s + 1) \hat{i} + (2s - 1) \hat{j} - (2s + 1) \hat{k}. \]
Find the equation of the curve which is passing through the point \( (1, 1) \), and whose differential equation is \[ x \, dy = (2x^2 + 1) \, dx, (x \neq 0). \]
Prove that \( f(x) = \tan x \) for all \( x \in \mathbb{R} \) is a continuous function.
Prove that \( f(x) = |x - 2| \) is not differentiable at \( x = 2 \).