List of top Mathematics Questions

Let A and B be two non zero square matrices and AB and BA both are defined. It means

If $ AX = D $ represents the system of simultaneous linear equations: $$ \begin{aligned} x + y + z &= 6 \\ 5x - y + 2z &= 3 \\ 2x + y - z &= -5 \end{aligned} $$ Then compute: $ (\text{Adj } A) D $

If $$ A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} $$ then find: $$ \det(A^6 + B^6) $$

If $$ A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix} $$ then $ AA^T $ is:

If $$ f(x) = \log\left(\left(\frac{2x^2 - 3}{x}\right) + \sqrt{\frac{4x^4 - 11x^2 + 9}{|x|}}\right) $$ then $ f(x) $ is:

Let $ f : \mathbb{R} - \left\{-\frac{1}{2}\right\} \rightarrow \mathbb{R} $ be defined by $$ f(x) = \frac{x - 2}{2x + 1} $$ If $ \alpha, \beta $ satisfy the equation $$ f(f(x)) = -x $$ then evaluate: $$ 4(\alpha^2 + \beta^2) $$

The trigonometric function y = tan x in the II quadrant

The general solution of the differential equation: $$ \cos^2 x \cdot \frac{dy}{dx} + y = \tan x $$ is $$ y = \tan x - 1 + C e^{-\tan x} $$ If this solution satisfies $ y\left(\frac{\pi}{4}\right) = 1 $, then find $ C $.

Solve the differential equation: $$ \frac{dy}{dx} = \cos^2(3x + y) $$

Assertion (A): The order of the differential equation of a family of circles with constant radius is two.
Reason (R): An algebraic equation involving two arbitrary constants corresponds to the general solution of a second order differential equation.

Evaluate: $$ \int \frac{e^{\tan^{-1}x}}{1+x^2} \left[\left(\sec^{-1}(\sqrt{1+x^2})\right)^2 + \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)\right] dx $$

If $ [\cdot] $ denotes the greatest integer function, then evaluate: $$ \int_{-1}^1 \left( x \cdot [1 + \sin \pi x] + 1 \right) dx $$

Given: $$ \lim_{n \to \infty} \left[ \frac{n}{(n+1)\sqrt{2n+1}} + \frac{n}{(n+2)\sqrt{2(n+2)}} + \frac{n}{(n+3)\sqrt{3(2n+3)}} + \cdots \text{(n terms)} \right] = \int_0^1 f(x)\, dx $$ Then find $ f(x) $.

If $$ \int_1^3 x^n \sqrt{x^2 - 1} \, dx = 6 $$ Then find the value of $ n $.

Let $$ I_n = \int (\cos^n x + \sin^n x) \, dx,\quad \text{and} \quad \frac{n - 1}{n} I_{n - 2} = \frac{\sin x \cos x}{n} f(x) $$ Then, what is $ f(x) $?

Evaluate: $$ \int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} = ? $$

Let $ T>0 $ be a fixed number. If $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function such that $ f(x + T) = f(x) $, then: $$ \text{If } I = \int_0^T f(x)\, dx,\quad \text{then } \int_0^{5T} f(2x)\, dx = ? $$

The point on the curve $ y = x^2 + 4x + 3 $ that is closest to the line $ y = 3x + 2 $ is:

Given: $$ \int \frac{3x + 4}{x^3 - 2x + 4} \, dx = \log f(x) + C $$ Then: $$ f(3) = ? $$

The function $ y = x^3 - ax^2 + 48x + 7 $ is increasing for all real values of $ x $. Find the interval in which $ a $ lies:

If $ a, b>0 $, then the minimum value of: $$ y = \frac{b^2}{a - x} + \frac{a^2}{x},\quad \text{for } 0<x<a $$

If $ x = f(\theta) $, $ y = g(\theta) $, then find: $$ \frac{d^2 y}{dx^2} $$

If the normal drawn at a point $ P $ on the curve $ 3y = 6x - 5x^3 $ passes through the origin $(0, 0)$, then the positive integral value of the abscissa of point $ P $ is:

The line joining the points $ (0, 3) $ and $ (5, -2) $ is a tangent to the curve: $$ y = \frac{C}{x + 1} $$ Then find $ C $.

Given: $$ 3f(\cos x) + 2f(\sin x) = 5x $$ Find: $$ f'(\cos x) + f'(\sin x) $$