List of top Mathematics Questions

If \[ A = \begin{pmatrix} 2 & 3 \\ 3 & 5 \end{pmatrix} \] then the value of \[ |A^{2025} - 3A^{2024} + A^{2023}| \] is:

If the image of the point \( P(3, 2, a) \) reflected about the line \[ \frac{x-3}{2} = \frac{y-5}{5} = \frac{z-2}{-2} \] is \( (5, b, c) \), then the value of \( \sigma^2 + b^2 + c^2 \) is:

The value of \[ \int_{\frac{\pi}{2}}^{\pi} \frac{1}{\left\lfloor x \right\rfloor + 4} \, dx \] where \( \left\lfloor x \right\rfloor \) denotes the greatest integer function, is:

Number of solutions of the equation \[ \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0 \] in \( \theta \in \left[ -3\pi, 2\pi \right] \):

If \( A \) is a square matrix of order 3 and \( |A| = 6 \), it is given that \[ \left| \text{adj} \left( 3 \, \text{adj} \left( A^2 \, \text{adj} (2A) \right) \right) \right| = 2^m \cdot 3^n \] where \( m \) and \( n \) are natural numbers. Then find \( m + n \). Here, \( \text{adj}(X) \) denotes the adjoint of matrix \( X \), and \( |X| \) denotes the determinant of matrix \( X \).

If \[ \begin{vmatrix} 0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} = \begin{vmatrix} 1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} \] then the value of \( \cos^2 \alpha + \cos^3 \beta + \cos^2 \gamma \) is:

If \( 3 \leq |2z + 3(1 + i)| \leq 7 \) and if the maximum and minimum values of \( \left| z + \frac{1}{2}(5 + 3i) \right| \) are \( \alpha \) and \( \beta \) respectively, then \( (\alpha + 2\beta) \) is:

Find the area bounded by \( y = \max \{ \sin x, \cos x \} \) when \( x \in \left[ 0, \frac{3\pi}{2} \right] \) with the x-axis:

Variates are given as \( -10, -7, -1, x, y, 2, 9, 16 \). If the mean \( (\mu) = \frac{7}{2} \) and variance = \( \frac{293}{4} \), find the mean of \( (1 + x + y), x, y, |y - x| \):

The value of \[ \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{1}{1 + \sqrt{\tan 2x}} \, dx \] is:

If the area of the region \[ \{(x,y): x^2+1 \le y \le 3-x\} \] is divided by the line \(x=-1\) in the ratio \(m:n\) (where \(m\) and \(n\) are coprime natural numbers), then find the value of \(m+n\).

If \[ \int (\cos x)^{-5/2}(\sin x)^{-11/2}\,dx = \frac{p_1}{q_1}(\cot x)^{9/2} + \frac{p_2}{q_2}(\cot x)^{5/2} + \frac{p_3}{q_3}(\cot x)^{1/2} - \frac{p_4}{q_4}(\cot x)^{-3/2} + C, \] where \(C\) is the constant of integration, then find the value of \[ \frac{15\,p_1p_2p_3p_4}{q_1q_2q_3q_4}. \]

The largest value of \( n \in \mathbb{N} \) such that \( 7^{n} \) divides \( (101)! \) is _____.

If \(a_1 = 1\) and for all \(n \ge 1\), \[ a_{n+1} = \frac{1}{2}a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2}, \] then the value of \[ \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \] is equal to:

In the binomial expansion of \[ (ax^2 + bx + c)(1 - 2x)^{26}, \] the coefficients of \(x, x^2, x^3\) are \(-56, 0\) and \(0\) respectively. Then the value of \( (a + b + c) \) is:

If the domain of the function \[ \cos^{-1}\!\left(\frac{2x-5}{11x-7}\right) + \sin^{-1}\!\left(2x^2 - 3x + 1\right) \] is \[ [0,a] \cup \left[\frac{12}{13},\, b\right], \] then the value of \( \dfrac{1}{ab} \) is:

If \( f(3)=18,\; f'(3)=0 \) and \( f''(3)=4 \), then the value of \[ \lim_{x\to 1}\ln\left(\frac{f(x+2)}{f(3)}\right)^{\frac{18}{(x-1)^2}} is: \]

The value of \[ \int_{\frac{\pi}{6}}^{\pi} \frac{\pi + 4x^{11}}{1 - \sin\left(|x| + \frac{\pi}{6}\right)}\,dx is: \]

If the mean and variance of observations \( x, y, 12, 14, 4, 10, 2 \) is \(8\) and \(16\) respectively, where \( x>y \), then the value of \( 3x - y \) is:

If \( x^2 + x + 1 = 0 \), then evaluate \[ \left(x + \frac{1}{x}\right)^4 + \left(x^2 + \frac{1}{x^2}\right)^4 + \left(x^3 + \frac{1}{x^3}\right)^4 + \cdots + \left(x^{25} + \frac{1}{x^{25}}\right)^4. \]

If \( O \) is the vertex of the parabola \( x^2 = 4y \), \( Q \) is a point on the parabola. If \( C \) is the locus of a point which divides \( OQ \) in the ratio \( 2:3 \), then the equation of the chord of \( C \) which is bisected at the point \( (1,2) \) is:

If \( a_1, a_2, a_3, \ldots \) are in increasing geometric progression such that \[ a_1 + a_3 + a_5 = 21, a_1 a_3 a_5 = 64, \] then \( a_1 + a_2 + a_3 \) is:

The locus of the point of intersection of tangents drawn to the circle \[ (x - 2)^2 + (y - 3)^2 = 16, \] which subtends an angle of \(120^\circ\), is:

Consider a set \( S = \{a, b, c, d\} \). Then the number of reflexive as well as symmetric relations from \( S \to S \) is: