List of top Mathematics Questions asked in KEAM

Given the function \( h(x) = f(g(x)) \), where \( f(x) = f'(x) = 3 \), and \( g(x) = 9 \), find \( g'(3) \), \( f'(3) \), and \( h'(3) \).
The product of the first five terms of a GP is 32. What is the 3rd term?
If \( A \) is a \( 3 \times 3 \) matrix and \( |B| = 3|A| \) and \( |A| = 5 \), then find \( \left| \frac{\text{adj} B}{|A|} \right| \).
Given the function \( F(x) = |\sin(3x)| - \cos(3x) \), for \( \frac{\pi}{6} \leq x \leq \frac{\pi}{3} \), find \( f' \left( \frac{\pi}{4} \right) \).
If \( -5<x \leq -1 \), then \( -21 \leq 5x + 4 \leq b \). Find \( b \).
An unbiased die is tossed until a sum \( S \) is obtained. If \( X \) denotes the number of times tossed, find the ratio \( \frac{P(X = 2)}{P(X = 5)} \).
Solve the system of equations: \[ \begin{bmatrix} 4 & 9 \\ 12 & -3 \\ 8 & -2 \end{bmatrix} \begin{bmatrix} 7 \\ 9 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \]
Evaluate the integral: \[ \int e^x \sec(x) \left( \tan(x) + 1 \right) \, dx \]
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 \(x + z = 2y\) and \(y = \frac{\pi}{4}\), what is \( \tan x \cdot \tan y \cdot \tan z \)?
What is the value of n when \( t_n = \frac{n(n+6)}{n+4} \) and \( t_n = 5 \)?
If \( 5 \sin^{-1} \alpha + 3 \cos^{-1} \alpha = \pi \), then \( \alpha = ? \)
Find the value of \( Z^2 \) if \( Z = \left( 1 + \frac{1}{i} \right) \).
The cubic polynomial \( 2x^3 - 3x^2 - 36x + 28 \) is increasing in the range of \( x \). Find the interval where the function is increasing.
If \( \cot^{-1} \left( \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \right) \), then find \( \sec^2 \theta \).
If \( Z = \frac{2 - i}{\alpha + i} \) and \( 4 \, \text{Re}(Z) = 3 \, \text{Im}( \overline{Z} ) \), find \( \alpha \).
Find the vertex of the parabola \( 4y = x^2 - 6x + 17 \).
Find the focus of the parabola \( x^2 - 4x + 8y + 4 = 0 \).
Given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \).
Find the equation of the circle touching the x-axis at \( (9, 0) \) and the line \( y = 14 \).
If \( \sin \alpha = \frac{12}{13} \), and \( \frac{\pi}{6} \leq \alpha \leq \frac{3\pi}{2} \), find \( \tan \alpha \).
Evaluate the limit: \[ \lim_{x \to \infty} \frac{\sqrt{\cos^2 x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \]
Find \( \tan 15^\circ + \tan 45^\circ \).
The velocity is given as \( \mathbf{v} = 3\hat{i} + 3\hat{j} \). Find the acceleration \( \mathbf{a} \).