List of top Mathematics Questions

If the length of a rectangle is decreasing at the rate of 3 cm/min and width is increasing at the rate of 2 cm/min, then find the rate of change in perimeter of the rectangle when \( x = 10 \) cm and \( y = 6 \) cm, where \( x \) = length and \( y \) = width.
Prove that \( f(x) = \tan x \) for all \( x \in \mathbb{R} \) is a continuous function.
Evaluate: \[ \int \tan^4 x \sec^2 x \, dx. \]
Find the value of the expression \( i \cdot i + j \cdot j + 2k \cdot k \).
Find the degree of the differential equation \[ x y \left( \frac{d^2 y}{dx^2} \right)^2 + x \frac{dy}{dx} - y = 2. \]
If \( y = A \sin x + B \cos x \), then find the differential equation of it.
Find those points on the curve \[ \frac{x^2}{4} + \frac{y^2}{25} = 1, \] on which the normal is parallel to the x-axis.
The value of $\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)$ will be:
The value of \[ \int \frac{dx}{x^2 - a^2} \] will be:
If \( A = \begin{bmatrix} \cos a & -\sin a \\ \sin a & \cos a \end{bmatrix} \) and \( A + A' = I \), then find the value of \( a \).
Prove that a one-one function \( f : \{2, 3, 4\} \to \{2, 3, 4\} \) is onto.
The nature of the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y \] will be:
If $f: X \rightarrow Y$ is an onto function, if and only if the range of $f$ will be:
Prove that \[ \int_{0}^{\pi} \frac{x\sin x}{1+\cos^2 x}\, dx = \frac{\pi^2}{4}. \]
Minimize $Z = 6x + 3y$ under the constraints: \[ 4x + y \geq 80, x + 5y \geq 115, 3x + 2y \leq 150, x \geq 0, \; y \geq 0. \]
If the area bounded by the curve $x=y^{2}$ and the line $x=4$ is divided into two equal parts by the line $x=a$, find the value of $a$.
Prove that cone of the maximum volume inscribed in a sphere of radius $R$ is $\tfrac{8}{27}$ of the volume of the sphere.
From a well-shuffled pack of 52 cards, three cards were drawn one after another without any replacement. What is the probability that the first two cards be King and the third be Ace?
Find the shortest distance between the lines \[ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) \] and \[ \vec{r} = 4\hat{i} + 5\hat{j} + 6\hat{k} + \mu(2\hat{i} + 3\hat{j} + \hat{k}). \]
The probabilities of solving a particular problem by $A$ and $B$ independently are respectively $\tfrac{1}{2}$ and $\tfrac{1}{3}$. If both try to solve the problem independently, find the probability that (i) the problem is solved, (ii) only one of them solves the problem.
Define symmetric and skew-symmetric matrices. If $A$ and $B$ are symmetric matrices, prove that $(AB-BA)$ is a skew-symmetric matrix.
Solve by matrix method the system of equations: \[\begin{aligned} 2x - 3y + 5z &= 11 \\ 3x + 2y - 4z &= -5 \\ x + y - 2z &= -3 \end{aligned}\]
Prove that \[ \int \sin^{-1}\!\left(\frac{2x}{1+x^2}\right) dx = 2x \tan^{-1}x - \log(1+x^2) + C. \]
Find the intervals in which the function $f(x)=\sin x + \cos x,\; 0\leq x \leq 2\pi$ is increasing or decreasing.
Find the equation of the curve passing through $(-2,3)$ on which the gradient of the tangent at any point $(x,y)$ is \[ \frac{dy}{dx} = \frac{2x}{y^2}. \]