List of top Mathematics Questions

If \( \log_2 3 = p \), express \( \log_8 9 \) in terms of \( p \):
If \(a = 1 + \sum_{r=1}^{6} (-3)^{r-1} \binom{12}{2r-1}\), then the distance of the point \((12, \sqrt{3})\) from the line \(\alpha x - \sqrt{3}y + 1 = 0\) is:
If \( A = \begin{bmatrix} 2 & 1
3 & 4 \end{bmatrix} \), then the determinant of matrix \( A \) is:

If vector \( \mathbf{a} = 3 \hat{i} + 2 \hat{j} - \hat{k} \) \text{ and } \( \mathbf{b} = \hat{i} - \hat{j} + \hat{k} \),  then which of the following is correct?

Find the values of \( a \) for which \( f(x) = \sqrt{3} \sin x - \cos x - 2ax + b \) is decreasing on \( \mathbb{R} \).
If \( \int_0^1 \frac{e^x}{1+x} \, dx = \alpha \), then \( \int_0^1 \frac{e^x}{(1+x)^2} \, dx \) is equal to:
A system of linear equations is represented as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix. Then the system of equations is:
Find: \[ \int \frac{\sin^{-1} \left( \frac{x}{\sqrt{a + x}} \right)}{ \sqrt{a + x}} \, dx \]
If: \[ \begin{bmatrix} x + y & 3y \\ 3x + 3 & x + 3 \end{bmatrix} \begin{bmatrix} 9 & x + y \\ 4x + y & y \end{bmatrix} \] then \( (x - y) = ? \)
A spherical medicine ball when dropped in water dissolves in such a way that the rate of decrease of volume at any instant is proportional to its surface area. Calculate the rate of decrease of its radius.
Solve for \( x \), \[ 2 \tan^{-1} x + \sin^{-1} \left( \frac{2x}{1 + x^2} \right) = 4\sqrt{3} \]
The order and degree of the differential function \[ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^5 = \frac{d^2y}{dx^2} \]
Find: \[ \int \frac{\cos 2x}{(\sin x + \cos x)^2} \, dx \]
Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx \]
The unit vector perpendicular to the vectors \( \hat{i} - \hat{j} \) and \( \hat{i} + \hat{j} \) is:
Find the values of \( a \) for which \( f(x) = x^2 - 2ax + b \) is an increasing function for \( x>0 \).
Evaluate: \( \sin^{-1} \left( \sin \frac{3\pi}{5} \right) \).
Find: \[ \frac{dy}{dx}, \quad \text{if} \quad y = x \tan x + \frac{\sqrt{x^2 + 1}}{2} \]
The area of an expanding rectangle is increasing at the rate of \( 4 \, \text{cm}^2/\text{s} \). The length of the rectangle is always square of its breadth. At what rate is the length of the rectangle increasing at an instant, when breadth \( = 4.5 \, \text{cm} \)?
Differentiate: \[ \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \quad \text{w.r.t.} \quad \cos^{-1}(2x\sqrt{1 - x^2}), \quad x \in \left(\frac{1}{\sqrt{2}}, 1\right) \]
Solve the following linear programming problem graphically: Maximise \( Z = 20x + 30y \) Subject to the constraints: \[ x + y \leq 0, \quad 2x + 3y \geq 100, \quad x \geq 14, \quad y \geq 14. \]
A school wants to allocate students into three clubs: Sports, Music, and Drama, under the following conditions:
- The number of students in the Sports club should be equal to the sum of the number of students in the Music and Drama clubs.
- The number of students in the Music club should be 20 more than half the number of students in the Sports club.
- The total number of students to be allocated in all three clubs is 10.
Find the number of students allocated to different clubs, using the matrix method.}
Let \( A = [a_{ij}] \) be a square matrix of order 3 such that \( a_{ij} = \hat{i} - 2 \hat{j} \). Then, which of the following is true?
Let \( A \) be a matrix of order \( m \times n \) and \( B \) be a matrix such that \( A^T B \) and \( B A^T \) are defined. Then, the order of \( B \) is:
If \( \vec{\alpha} = \hat{i} - 4 \hat{j} + 9 \hat{k} \) and \( \vec{\beta} = 2\hat{i} - \hat{j} + \lambda \hat{k} \) are two mutually parallel vectors, then \( \lambda \) is equal to: