List of top Mathematics Questions

If \( \overrightarrow{a} \) and \( \overrightarrow{b} \) are two unit vectors and if \( \frac{\pi}{4} \) is the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \), then \[ (\overrightarrow{a} + (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{b}) \cdot (\overrightarrow{a} - (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{b}) \] is:
Evaluate the integral \[ \int_{-500}^{500} \ln \left( \frac{1000 + x}{1000 - x} \right) dx \]
cos \( A \cos 2A \) is equal to:
Find the value of \[ \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right). \] The answer is:
Find the value of \[ \sin \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right). \] The answer is:
If \( z \) is a complex number of unit modulus, then \[ \left| \frac{1+z}{1+ \overline{z}} \right| \] equals:
The solution of \( \frac{dy}{\cos y} = dx \) is:
The value of \[ \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx \] is equal to:
Let \( \mathbb{N} \) be the set of all natural numbers. Let \( R \) be a relation defined on \( \mathbb{N} \) given by \( aRb \) if and only if \( a + 2b = 11 \). Then the relation \( R \) is:
If \[ \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2}, \] then \( x + y + z = \):
The integrating factor of

\[ (1 + 2e^{-x}) \frac{dy}{dx} - 2e^{-x} y = 1 + e^{-x} \] is:
The coefficient of \( x^{17} \) in

\[ (1 - x)^{13} (1 + x + x^2)^{12} \] is:

Let \( a, b, c \) be positive numbers such that \( abc = 1 \). Then the minimum value of \( a + b + c \) is:

Let \[ f(x) = |\sin x| + |\cos x|, \quad x \in \mathbb{R}. \] The period of \( f(x) \) is:
If \[ \theta \in \left( 0, \frac{\pi}{3} \right) \quad \text{and} \quad \begin{vmatrix} 0 & -\sin^2 \theta & -2 - 4 \cos 6\theta \\ 0 & \cos^2 \theta & -2 - 4 \cos 6\theta \\ 1 & \sin \theta & \cos 2\theta \end{vmatrix} = 0, \] then \( \theta \) is equal to:
The volume of the parallelepiped whose coterminous vectors are given by the vectors \[ \overrightarrow{a} = i - j + k, \quad \overrightarrow{b} = 3i + j - k, \quad \overrightarrow{c} = 5i + 2j - 7k \] is (in cubic units):
Let \( A \) be a symmetric matrix and \( B \) be a skew symmetric matrix. If

\[ A + B = \begin{pmatrix} 1 & 3 \\ -2 & 5 \end{pmatrix}, \] then \( A - B \) is equal to:
Variance of 6, 7, 8, 9 is
Let \( \left\lfloor x \right\rfloor \) be the greatest integer less than or equal to \( x \). Then \[ \lim_{x \to 0^-} \frac{x \left( \left\lfloor x \right\rfloor + |x| \right)}{|x|} \] is equal to:
If \( x = 5 \tan t \) and \( y = 5 \sec t \), then \( \frac{dy}{dx} \) at \( t = \frac{\pi}{3} \) is:
If \( \overrightarrow{a} \) and \( \overrightarrow{b} \) are two nonzero vectors and if \( |\overrightarrow{a} \times \overrightarrow{b}| = |\overrightarrow{a} \cdot \overrightarrow{b}| \), then the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \) is equal to
The value of \( \int_{-4}^{-2} \left[ (x+3)^3 + 2 + (x+3)\cos(x+3) \right] \, dx \) is equal to:

Let \( x \text{ and } y \) be two positive real numbers. Then
\[ \left( x + \frac{1}{x} \right) \left( y + \frac{1}{y} \right) \] is greater than or equal to:
 

If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]
The shortest distance between the parallel straight lines \[ \overrightarrow{r_1} = \hat{k} + s(\hat{i} + \hat{j}), \quad t, s \in \mathbb{R} \quad {and} \quad \overrightarrow{r_2} = \hat{j} + t(\hat{i} + \hat{j}), \] is: