List of top Questions asked in KEAM- 2025

If \( \sin x + \sin y = a \), \( \cos x + \cos y = b \) and \( x + y = \frac{2\pi}{3} \), then the value of \( \frac{a}{b} \) is equal to

If \( x + z = 2y \) and \( y = \frac{\pi}{4} \), then \( \tan x \tan y \tan z = \)

\( \tan 15^\circ + \tan 75^\circ = \)

The following system of equations \[ x + y + z = 1 \] \[ 2x + 3y - mz = 2 \] \[ 3x + 5y + 3z = 3 \] has no unique solution. Then the value of \( m \) is equal to

If \( \begin{pmatrix} -1 & 2 3 & -4 -5 & 6 \end{pmatrix} \begin{pmatrix} 7 8 \end{pmatrix} = \begin{pmatrix} \alpha \beta 13 \end{pmatrix} \), then the value of \( \alpha + \beta \) is equal to

If the matrix \( \begin{pmatrix} 8-k & 2 -2 & 4-k \end{pmatrix} \) is singular, then the value of \( k \) is equal to

The set of all \( x \) satisfying the inequalities \( -4 \le 2 - 3x<7 \) is

\( -5<x \le -1 \) implies \( -21<5x + 4 \le b \), the least value of \( b \) is

If \( ^{11}P_r = 7920 \), then the value of \( r \) is equal to

Let \( A = \{0,2,4,6,8\} \). The number of 5-digit numbers that can be formed using the digits in \( A \) without replacement, is

Let \( \alpha = \sum_{k=0}^{5} {^{10}C_{2k}} \) and \( \beta = \sum_{k=0}^{4} {^{10}C_{2k+1}} \). Then \( \alpha - \beta \) is equal to

If \( \alpha = {^nC_r} \) and \( \beta = {^nC_{r-1}} \), then \( 1 + \frac{\alpha}{\beta} \) is equal to

In the binomial expansion of \( (2x + \alpha)^8 \), the co-efficients of \( x^2 \) and \( x^3 \) are equal. Then the value of \( \alpha \) is equal to

Let \( A \) be a \(3 \times 3\) matrix and let \( B = 3A \). If \( |A| = 5 \), then the value of \( \frac{|\text{adj } B|}{|3A|} \) is equal to

The general term of a sequence is \( t_n = \frac{n(n+6)}{n+4}, \, n = 1,2,3,\ldots \). If \( t_n = 5 \), then the value of \( n \) is

In a G.P., the first and third terms are 4 and 8 respectively. Then the \(21^{\text{st}}\) term is

Let \( a_1, a_2, a_3, \ldots \) be in G.P. If \( a_1 \cdot a_2 \cdot a_3 = 64 \) and \( a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 = 32 \), then common ratio is

If \( z + \bar{z} = 6 \) and \( z - \bar{z} = 4i \), then \( |z|^2 = \)

The product of first 5 terms of a G.P., whose terms are increasing, is 32. The third term of the G.P. is

Let \( z = \frac{2 - i}{\alpha + i} \), where \( \alpha \) is a real number. If \( 4\text{Re}(z) = 3\text{Im}(\bar{z}) \), then the value of \( \alpha \) is

Let \( f(x) = x^2 - 10x - 19, \, x \in \mathbb{R} \). Then the inverse image of 5, \( f^{-1}(5) = \)

Let \( f(x)=\log_5 x \, (x > 0) \) and \( g(x)=\cos^{-1}(x) \, (-1\le x \le 1) \). Then the domain of \( g \circ f \) is

Let \( z = 1 - i \). Then the value of \( z^4 \) is equal to

The modulus of the complex number \( (2\sqrt{2} + i2\sqrt{2})^2 \) is equal to

Let \( f(x) = \cos x \). Then the value of \( \frac{1}{2}[f(x+y) + f(y-x)] - f(x)f(y) \) is equal to