List of top Mathematics Questions

If the matrix \[ \begin{pmatrix} 0 & a & a\\ 2b & b & -b\\ c & -c & c \end{pmatrix} \] is orthogonal, then the values of \( a, b, c \) are:
If \( a, b, c \) are positive real numbers each distinct from unity, then the value of the determinant \[ \left| \begin{matrix} 1 & \log_a b & \log_a c \\ \log_b a & 1 & \log_b c \\ \log_c a & \log_c b & 1 \end{matrix} \right| \] is:
Given the set $ S = \{ a, b, c, d, e, f \} $, find the total number of subsets with an odd number of elements.
Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b} \), \( |\mathbf{a} + \mathbf{b}| = \sqrt{29} \), \( \mathbf{a} \cdot \mathbf{b} = ? \)
Let \( A = \begin{pmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{pmatrix} \). If \( |A|^2 = 25 \), then \( |\alpha| \) equals to
Coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \)
Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \neq 0 \)) and they are in A.P. Then the rank of the matrix \( \begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix} \) is:
Axis of parabola is \( x = 0 \). If the vertex is at a distance of 3 from the origin above the x-axis, find the vertex.
Find the integral: \[ \int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx \]
Evaluate: \[ \sec \left( \cos^{-1} \left( \frac{2024}{2025} \right) \right) \]
Evaluate the expression: \[ \frac{\sin \frac{\pi}{7} + \sin \frac{3\pi}{7}}{1 + \cos \frac{\pi}{7} + \cos \frac{2\pi}{7}} \]
Find the values of: \[ \cos 75^\circ, \cos 15^\circ, \cos 45^\circ \]
Find the probability of getting at most 2 heads when 4 coins are tossed.
Find the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \sin^2 x)}{x^2} \]
Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]
Integrating factor of \[ \frac{dy}{dx} - 2y = 2x - 3 \]
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 \( a \):
If \[ (a - 2)^2 + (b - 2)^2 + (c - 2)^2 = 0, \text{ then } a + b + c = \]
Equation of line through (0, 0, 1) and (1, 1, 0)
Equation of line through (0, 0, 1) and (1, 1, 0)
If \[ (a - 2)^2 + (b - 2)^2 + (c - 2)^2 = 0, \text{ then } a + b + c = \]
Find the range of \( f(x) = \log_e(4x^2 - 4x + 1) \)
Evaluate the integral: \[ \int \frac{\log(1+x)}{1+x} \, dx \]
Length of latus rectum of ellipse \[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]
Evaluate the integral: \[ \int_{-3}^{3} \left\lfloor x \right\rfloor \, dx \]