List of top Mathematics Questions

If \( 5 \sin^{-1} \alpha + 3 \cos^{-1} \alpha = \pi \), then \( \alpha = ? \)
Given that \( \vec{a} \parallel \vec{b} \), \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), and \( |\vec{a}| = 7 \), find \( |\vec{b}| \).
Find the focus of the parabola \( x^2 - 4x + 8y + 4 = 0 \).
If \( \cot^{-1} \left( \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \right) \), then find \( \sec^2 \theta \).
Evaluate the integral: \[ \int \frac{2x^2 + 4x + 3}{x^2 + x + 1} \, dx \]
The product of the first five terms of a GP is 32. What is the 3rd term?
Given: \[ \sum_{k=0}^{5} \binom{10}{2k} = \alpha \quad \text{and} \quad \sum_{k=0}^{4} \binom{10}{2k+1} = \beta \] \text{Find the value of} \( \alpha - \beta \).
Evaluate the integral: \[ \int \frac{\sin^1 x}{\sqrt{1 - x^2}} \, dx \]
Find \( \tan 15^\circ + \tan 45^\circ \).
An unbiased die is tossed until a sum \( S \) is obtained. If \( X \) denotes the number of times tossed, find the ratio \( \frac{P(X = 2)}{P(X = 5)} \).
Evaluate the limit: \[ \lim_{x \to \infty} \frac{\sqrt{\cos^2 x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \]
If \( -5<x \leq -1 \), then \( -21 \leq 5x + 4 \leq b \). Find \( b \).
Given the following information: \[ n(A \times B) = 160, \quad n(B \times C) = 80, \quad n(A \times C) = 240 \] Find \( n(A) \).
Find the standard deviation of the numbers: -3, 0, 3, 8.
If \( Z = \frac{2 - i}{\alpha + i} \) and \( 4 \, \text{Re}(Z) = 3 \, \text{Im}( \overline{Z} ) \), find \( \alpha \).
If \( f(x) = \cos x \), find the following expression: \[ \frac{1}{2} \left[ f(x + y) + f(y - x) - f(x) \cdot f(y) \right] \]
Evaluate the integral $ I = \int \frac{\sin 4x}{\sin 2x} \, dx $
If a box has 8 red balls, 12 white balls, 17 black balls. If two balls are taken one by one without replacement, then the probability of taking one red ball and one black ball is
Evaluate the integral: $ \int_{-1}^1 |x - 3| \, dx $
Solve the following differential equation and integrate: $ \frac{dy}{dx} + \frac{2x}{1 + x^2} \cdot y = x $
Evaluate the integral: $ \int_0^\pi \frac{\tan x}{\cos x} \, dx = \int_0^\pi \frac{\sin x}{\cos^2 x} \, dx $
Given $ Z_1 = \frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad Z_2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \quad \text{and} \quad w = Z_1 + Z_2, \quad \text{find} \, w. $
Find the value of $ \sin 60^\circ - \sin 80^\circ + \sin 100^\circ - \sin 120^\circ $
Solve for $ \alpha $ if: $ \cos^{-1}(2 \sin \alpha) = \frac{47}{12} $
Find the value of $ \cot^{-1}(1) + \cot^{-1}(2) + \cot^{-1}(3) $