List of top Questions asked in KEAM

Evaluate the following integral: \[ \int e^x \left[\frac{1}{1+x(1+x)^2}\right] dx \]
Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]
Evaluate the following integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^4 + x^3 + x \right) \cos x \, dx \]
What is the value of n when \( t_n = \frac{n(n+6)}{n+4} \) and \( t_n = 5 \)?
Find the value of \( Z^2 \) if \( Z = \left( 1 + \frac{1}{i} \right) \).
A body of mass 0.8kg moves with velocity \( v = 2x^2 + 2 \, \text{m/s} \). What is the work done during its motion from \( x = 0 \) to \( x = 2 \, \text{m} \)?
If \( Z = \frac{2 - i}{\alpha + i} \) and \( 4 \, \text{Re}(Z) = 3 \, \text{Im}( \overline{Z} ) \), find \( \alpha \).
Given the system of equations: \[ Z + \bar{Z} = 4 \quad \text{and} \quad Z - \bar{Z} = 6 \] Find \( |Z| \).
If \( 5 \sin^{-1} \alpha + 3 \cos^{-1} \alpha = \pi \), then \( \alpha = ? \)
The fundamental frequency of a string of length \( l \) is \( n \). The string is cut into 3 parts \( l_1, l_2, \) and \( l_3 \), each having fundamental frequency \( n_1, n_2, n_3 \). Then, what is the relationship between \( n_1, n_2, n_3 \)?
Relation between the drift velocity and an electric field is ----?
A liquid of density \( d \) is moving down in a vessel of height \( h \) with an acceleration \( a<g \). Then the pressure at the bottom is?
Find the standard deviation of the numbers: -3, 0, 3, 8.
Given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \).
Evaluate the limit: \[ \lim_{x \to \infty} \frac{\sqrt{\cos^2 x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \]
A planet revolves around the sun with a time period 27 times that of planet B. Planet A is at \( x \) times the distance of planet B from the sun. Find the value of \( x \).
In a Carnot engine, efficiency is dependent on ________?
\( M^0 L T^{-1} \) is the dimension of ___________?
The product of the first five terms of a GP is 32. What is the 3rd term?
Find the domain of the composite function \( f \circ g(x) \) where \( f(x) = \log(5x) \) and \( g(x) = \cos(x) \).
Given the line equation \( Ax + By + C = 0 \), which passes through the point \( (-10, 7) \) and is perpendicular to the line \( 11x - 8y - 16 = 0 \), find \( C \).
Solve the system of equations: \[ \begin{bmatrix} 4 & 9 \\ 12 & -3 \\ 8 & -2 \end{bmatrix} \begin{bmatrix} 7 \\ 9 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \]
The velocity is given as \( \mathbf{v} = 3\hat{i} + 3\hat{j} \). Find the acceleration \( \mathbf{a} \).
Given the function \( h(x) = f(g(x)) \), where \( f(x) = f'(x) = 3 \), and \( g(x) = 9 \), find \( g'(3) \), \( f'(3) \), and \( h'(3) \).
Fehling's solution is a mixture of: