List of top Mathematics Questions

Evaluate the following integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^4 + x^3 + x \right) \cos x \, dx \]
Find the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \sin^2 x)}{x^2} \]
Evaluate the derivative of the function: \[ f(x) = x |x|, \quad \text{find} \quad f'(-10) \]
Find the integral: \[ \int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx \]
Evaluate the following sum: \[ \sum_{n=1}^{2025} i^n (1+i) \]
If \[ (a - 2)^2 + (b - 2)^2 + (c - 2)^2 = 0, \text{ then } a + b + c = \]
Find the range of \( f(x) = \log_e(4x^2 - 4x + 1) \)
Given that \( a, \frac{3}{4} a r^2, a r^3 \) are in G.P. Product of first four terms = \( \frac{3^6}{4^3} \), then find \( a \):

Find the mean deviation of the following data: 

Find the center of the ellipse given by the equation \[ 4x^2 + 24x + 9y^2 - 18y + 9 = 0 \]
Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]
Coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \)
Evaluate the integral: \[ \int \frac{\sin(2x)}{\sin(x)} \, dx \]
Find the probability of getting at most 2 heads when 4 coins are tossed.
Given that \( \vec{a} \parallel \vec{b} \), \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), and \( |\vec{a}| = 7 \), find \( |\vec{b}| \).
Given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \).
Find the standard deviation of the numbers: -3, 0, 3, 8.
Solve the system of equations: \[ \begin{bmatrix} 4 & 9 \\ 12 & -3 \\ 8 & -2 \end{bmatrix} \begin{bmatrix} 7 \\ 9 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \]
Given the system of equations: \[ Z + \bar{Z} = 4 \quad \text{and} \quad Z - \bar{Z} = 6 \] Find \( |Z| \).
If \( f(x) = \cos x \), find the following expression: \[ \frac{1}{2} \left[ f(x + y) + f(y - x) - f(x) \cdot f(y) \right] \]
If \( Z = \frac{2 - i}{\alpha + i} \) and \( 4 \, \text{Re}(Z) = 3 \, \text{Im}( \overline{Z} ) \), find \( \alpha \).
Evaluate the integral: \[ \int_{\frac{\pi}{10}}^{\frac{2\pi}{5}} \frac{\cot^3 x}{1 + \cot^3 x} \, dx \]
The velocity is given as \( \mathbf{v} = 3\hat{i} + 3\hat{j} \). Find the acceleration \( \mathbf{a} \).
An unbiased die is tossed until a sum \( S \) is obtained. If \( X \) denotes the number of times tossed, find the ratio \( \frac{P(X = 2)}{P(X = 5)} \).
If \(x + z = 2y\) and \(y = \frac{\pi}{4}\), what is \( \tan x \cdot \tan y \cdot \tan z \)?