List of top Mathematics Questions

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 $.

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.

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

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

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: $$ \text{If } I = \int_0^T f(x)\, dx,\quad \text{then } \int_0^{5T} f(2x)\, dx = ? $$

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) $.

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

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

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

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:

Evaluate: $$ \lim_{x \to \infty} \log_e(\cosh x) + x $$

If $ a, b, c $ are distinct real numbers and: $$ \lim_{x \to \infty} \frac{(b - c)x^2 + (c - a)x + (a - b)}{(a - b)x^2 + (b - c)x + (c - a)} = \frac{1}{2} $$ then what is the value of $ a + 2c $?

Evaluate: $$ \lim_{x \to \infty} \frac{3|x| - x}{|x| - 2x} - \lim_{x \to 0} \frac{\log(1 + x^3)}{\sin^3 x} $$

Assertion (A): $$ \frac{d}{dx} \left( \frac{x^2 \sin x}{\log x} \right) = \frac{x^2 \cos x \cdot \log x - x^2 \sin x \cdot \frac{1}{x}}{(\log x)^2} + \frac{2x \sin x}{\log x} \Rightarrow \text{(matches the structure of product + quotient rule)} $$ Reason (R): $$ \frac{d}{dx} \left( \frac{uv}{w} \right) = \frac{uv'}{w} + \frac{u'v}{w} - \frac{uvw'}{w^2} \Rightarrow \text{(correct general derivative formula)} $$

Find the radical centre of the three circles: $$ C_1: x^2 + y^2 - 1 = 0,\quad C_2: x^2 + y^2 - 8x + 15 = 0,\quad C_3: x^2 + y^2 + 10y + 24 = 0 $$

If the direction cosines of a line are: $$ \left( \frac{a}{\sqrt{83}},\ \frac{5}{\sqrt{83}},\ \frac{c}{\sqrt{83}} \right) $$ and it is given that $ c - a = 4 $, find $ ca $.

Which of the following equations represents a **parabola**?

If $ e_1 $ and $ e_2 $ are the eccentricities of the hyperbola $$ 16x^2 - 9y^2 = 1 $$ and its conjugate respectively, then find $ 3e_1 $ in terms of $ e_2 $.

Let the coordinates of points A and B satisfy:
- $ x^2 + 2x - a^2 = 0 $
- $ y^2 + 4y - b^2 = 0 $ Find the equation of the circle having AB as diameter.

A circle is such that it is tangent to every line of the form: $$ (x - 2)\cos\theta + (y - 2)\sin\theta = 1 $$ for all $ \theta $. Find the equation of the circle.