List of top Mathematics Questions

Evaluate
\[ \lim_{x \to 0} \sqrt{\frac{x - \sin x}{x + \sin^2 x}} \]

Number of solutions of the equation
\[ \tan^{-1}(1+x) + \tan^{-1}(1-x) = \frac{\pi}{2} \]
are:

If
\[ A = \frac{1}{3} \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix} \] is an orthogonal matrix, then:

The points represented by the complex numbers \[ 1 + i, \quad -2 + 3i, \quad \frac{5}{3} i \] on the Argand plane are:

If matrix
\[ A = \begin{bmatrix} 3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 \end{bmatrix}, \quad \text{and} \quad A^{-1} = \frac{1}{k} \, \text{adj}(A), \]
then \(k\) is:

If \(x, y, z\) are complex numbers, and
\[ \Delta = \begin{vmatrix} 0 & -y & -z \\ \overline{y} & 0 & -x \\ \overline{z} & \overline{x} & 0 \end{vmatrix}, \]
then \(\Delta\) is:

If \[ y = \frac{x}{x+1} + \frac{x+1}{x}, \] then \[ \frac{d^2 y}{dx^2} \text{ at } x = 1 \text{ is equal to:} \]

Let \(y = e^{2x}\). Then \(\frac{d^2 y}{dx^2} \cdot \frac{d^2 x}{dy^2}\) is:

The value of the integral
\[ \int_a^b \frac{\sqrt{x} \, dx}{\sqrt{x} + \sqrt{a + b - x}} \]
is:

Evaluate
\[ \int e^{x^2} (2x + x^3) (3 + x^2)^2 \, dx \]

If
\[ \int_0^a f(2a - x) \, dx = m \quad \text{and} \quad \int_0^a f(x) \, dx = n, \]
then
\[ \int_0^{2a} f(x) \, dx \]
is equal to:

The integrating factor of the differential equation
\[ \sin x \frac{dy}{dx} + 2y \cos x = 1 \]
is:

The expression satisfying the differential equation \[ (x^2 - 1) \frac{dy}{dx} + 2xy = 1 \] is:

Let \(\mathbf{a} = \mathbf{i} - \mathbf{k}, \quad \mathbf{b} = x\mathbf{i} + \mathbf{j} + (1-x)\mathbf{k}, \quad \mathbf{c} = y\mathbf{i} + x\mathbf{j} + (1+x-y)\mathbf{k}\). Then \([\mathbf{a}, \mathbf{b}, \mathbf{c}]\) depends on:

The sum \[ 1 + \frac{1+a}{2!} + \frac{1+a+a^2}{3!} + \cdots \] is equal to:

If
\( f(x) = \begin{cases} 1, & 0 < x \le \dfrac{3\pi}{4} \\ 2\sin\left(\dfrac{2x}{9}\right), & \dfrac{3\pi}{4} < x < \pi \end{cases} \)
then:

If f(x)= begincases sin x, & when x is rational
cos x, & when x is irrational endcases then the function is: