List of top Questions asked in KEAM

If \( f(x)=\begin{vmatrix} \frac{1}{2x} & \frac{1}{x-1} & \frac{1}{x} \\ 3x(x-1) & (x-1)(x-2) & x(x-1) \end{vmatrix} \), then \( f(50) \) is

Coin tossed and die rolled: probability of head and 3

If determinant of matrix is zero, then system is

If \( A=\begin{bmatrix}2x & 0 x & x \end{bmatrix} \) and \( A^{-1}=\begin{bmatrix}1 & 0 -1 & 2 \end{bmatrix} \), find \(x\)

If \( A = \begin{bmatrix} 1 & 2 & 3 0 & 1 & 4 5 & 6 & 0 \end{bmatrix} \), then the sum of the diagonal elements of \( A^{-1} \) is

If \( A = \begin{bmatrix} 2 & 1 3 & 2 \end{bmatrix} \), then \( A^{-1} \) is

If set \( S \) has 10 elements and \( A=\{(x,y):x,y\in S, x\ne y\} \), number of elements in \( A \)

Variance of first 20 natural numbers

If \( x = A\cos 4t + B\sin 4t \), then \( \frac{d^2x}{dt^2} = \)

The arithmetic mean of \( ^nC_0, ^nC_1, \ldots, ^nC_n \) is

Find plane at distance 5 from origin perpendicular to \(2\hat{i}+\hat{j}+2\hat{k}\)

The real part of \( (i - \sqrt{3})^{13} \) is

\( \int \frac{(\sin x + \cos x)(2 - \sin 2x)}{\sin^2 2x} \, dx \)

\( \frac{\sin A - \sin B}{\cos A + \cos B} \) is equal to

\( \lim_{x\to0} \frac{1+x-e^x}{x^2} \)

The differential equation whose general solution is \( y = e^x(A\cos x + B\sin x) \) is

\( \lim_{x\to0} \frac{\int_0^{x^2} \sin(\sqrt{t}) \, dt}{x^2} \)

\( \int_{0}^{\pi/2} \frac{2\sin x}{2\sin x + 2\cos x} dx \)

Area bounded by \( y=\sin^2 x \), \( x=\frac{\pi}{2} \), \( x=\pi \)

\( \int_{\pi/4}^{3\pi/4} \frac{x}{1+\sin x} \, dx \) is equal to

If \( f \) is differentiable and \( \lim_{h\to0} \frac{f(1+h)-f(1)}{h}=5 \), find \( f'(1) \)

\( \lim_{x\to\infty} \left(\sqrt{x^2+1} - \sqrt{x^2-1}\right) \)

If \( \int f(x)\cos x \, dx = \frac{1}{2}\{f(x)\}^2 + c \), then \( f\left(\frac{\pi}{2}\right) \) is

Maximum value of \( 2x^3 -15x^2 +36x +4 \)

Let \( f:(-1,1)\to(-1,1) \) be continuous, \( f(x)=f(x^2) \), and \( f(0)=\frac{1}{2} \). Find \( 4f\left(\frac{1}{4}\right) \)