List of top Questions asked in KEAM

The velocity is given as \( \mathbf{v} = 3\hat{i} + 3\hat{j} \). Find the acceleration \( \mathbf{a} \).
If \( \sin \alpha = \frac{12}{13} \), and \( \frac{\pi}{6} \leq \alpha \leq \frac{3\pi}{2} \), find \( \tan \alpha \).
Find \( \tan 15^\circ + \tan 45^\circ \).
Evaluate the integral: \[ \int_{\frac{\pi}{10}}^{\frac{2\pi}{5}} \frac{\cot^3 x}{1 + \cot^3 x} \, dx \]
Evaluate the integral: \[ \int \frac{\sin^1 x}{\sqrt{1 - x^2}} \, dx \]
If \( f(x) = \sqrt{x - 3} + 4 \sqrt{5 - x} \), find the domain of \( f(x) \).
If \( f(x) = \cos x \), find the following expression: \[ \frac{1}{2} \left[ f(x + y) + f(y - x) - f(x) \cdot f(y) \right] \]
Given: \[ \sum_{k=0}^{5} \binom{10}{2k} = \alpha \quad \text{and} \quad \sum_{k=0}^{4} \binom{10}{2k+1} = \beta \] \text{Find the value of} \( \alpha - \beta \).
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 \( C \).
Find the domain of the composite function \( f \circ g(x) \) where \( f(x) = \log(5x) \) and \( g(x) = \cos(x) \).
The eccentricity of the ellipse \[ p x^2 + 5y^2 = 80, \quad \text{where } p > 5, \] is \( \frac{\sqrt{3}}{2} \). Then the value of \( p \) is
How many grams of HCl will completely react with 17.4g of pure MnO\(_2\) (s) to liberate Cl\(_2\) (g)? (Atomic mass Mn = 55.0; H = 1; Cl = 35.5)
If \[ \lim_{x \to 9} f(x) = 6 \quad {and} \quad \lim_{x \to 9} g(x) = 3, \] then \[ \lim_{x \to 9} \frac{f(x) - 2g(x)}{g(x)} \] is equal to
If the acceleration due to gravity on the surface of a planet is 2.5 times that on Earth and radius, 10 times that of the Earth, then the ratio of the escape velocity on the surface of a planet to that on Earth is:
Two spheres A and B having respective charges 6 C and 12 C placed at a distance \( d \) repel each other by a force \( F \). The charge given to sphere A to reverse the force as \( -F \) is:
cos \( A \cos 2A \) is equal to:
If the maximum number of neighbours of a nucleon within the range of nuclear force is \( p \) and \( k \) is a constant, then the binding energy per nucleon is approximately:

14g of cyclopropane burnt completely in excess oxygen. The number of moles of water formed is:

For a paramagnetic material, the magnetic susceptibility \( \chi_m \) is
When light is reflected from an optically rarer medium:
Whenever light travels from rarer medium into denser medium its
In a single slit diffraction experiment, the width of the slit and the wavelength of the light are respectively 5 mm and 500 nm. If the focal length of the lens is 20 cm, then the size of the central bright fringe will be
When two coherent sources each of individual intensity \( I_0 \) interfere, the resultant intensity due to constructive and destructive interference are respectively
If a particle executing simple harmonic motion with period \( T \) and displacement \( x = A \cos(\omega t) \), then the acceleration and velocity of the particle at the time \( t = \frac{T}{2} \) are respectively:
Let a relation \( R \) on the set of natural numbers be defined by \( (x, y) \in R \) if and only if \( x^2 - 4xy + 3y^2 = 0 \) for all \( x, y \in \mathbb{N} \). Then the relation is: