List of practice Questions

A ball is dropped from some height and after first collision with the ground if it reaches $ \frac{3}{4} $th of its original height, then the % loss of its energy is: Options:
A solid sphere of mass 2 kg is rolling without slipping on a horizontal surface with a velocity 5 m/s. The rotational kinetic energy of the sphere is: Options:
A solid sphere and a solid cylinder roll down without slipping along an inclined plane. If they start from rest from the top of the inclined plane, the ratio of the velocities of the solid sphere and solid cylinder when they reach the bottom of the inclined plane is:
A pendulum of time period one second is losing its mechanical energy due to damping. Its mechanical energy at time $ t = 0 $ is 45 J. After completing 15 oscillations, its mechanical energy is 15 J. The ratio of the damping constant and the mass of the object making damped oscillations is:
Let $ a $ and $ b $ be arbitrary constants and $ C $ be a fixed constant. If $ y = ae^{2x} + bxe^{2x} + C $ is the general solution of a differential equation, then the order of that differential equation is:
The differential equation \[ y^2 \, dx + (3xy - 1) \, dy = 0 \] is:
The particular solution of the differential equation \[ \frac{dx}{dy} = \frac{\sin y(1 + y \cot y)}{x \log(x^2 e)}, \quad y(1) = 0 \] Options:
The potential difference across the ends of a conductor is $ (50 \pm 3) $ V and the current through it is $ (5 \pm 0.1) $ A. The percentage error in the measurement of the resistance of the conductor is: Options:
The velocity (v) of a particle starting from rest increases linearly with time (t) as $ v = 4t $, where v is in $ ms^{-1} $ and t is in second. The distance covered by the particle in the first 4 seconds is
The time taken by a boat to travel upstream to a certain distance and return back is 14 hours. If the velocity of boat in still water is 35 kmh$ ^{-1} $ and velocity of the stream is 5 kmh$ ^{-1} $, the distance travelled by the boat before it returns is
If the angle between two unit vectors $ \vec{A} $ and $ \vec{B} $ is $ \theta $, then \[ |\vec{A} + \vec{B}| = \]
$\int e^{-2x} \left( \frac{1 - \sin 2x}{1 + \cos 2x} \right) dx =$
If $ S_n = \int_0^{\frac{\pi}{2}} \frac{\sin((2n-1)x)}{\sin x} \, dx $, and $ n $ is an integer, then $ S_{n+1} - S_n = $:
If $ a = 2n $ and $ b = 2m+1 $ for all $ m, n \in \mathbb{N} $, then \[ \int_{-\pi}^{\pi} e^{\sin^2 x} \cot^{b}(2n+1) x \, dx = \]
If $ \int_{n}^{n+1} g(x) \, dx = n^2 $, $ \forall n \in Z $, then the value of $ \int_{-3}^{3} g(x) \, dx $ is
$ \int_{0}^{\frac{\pi}{2}} \frac{\sum_{n=0}^{4} \sin \left( \frac{n\pi}{4} + x \right)}{\cos x + \sin x} dx = $
The maximum value of $ a $ such that the second derivative of $ x^4 + ax^3 + \frac{3x^2}{2} + 1 $ is positive for all real $ x $ is
Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function. If $ px + my + n = 0 $ is a tangent drawn to the curve $ y = f(x) $ at $ x = \alpha $, then at $ x = 0 $, the value of \[ \frac{d}{dx} \left( f(\alpha e^{2x}) \right) \]
In the interval $ \left( \frac{1}{e}, e \right) $, a decreasing function among the following functions is:
If the height of a cone of greatest volume that can be inscribed in a sphere of radius $ R $ is $ kR $, then the ratio of the volume of the cone to the volume of the sphere is:
If $ f(x) = \frac{x}{(1 + nx^n)^{1/n}} $ for $ n \geq 2 $, then \[ \int x^{n-2} f(x) \, dx = \]
The integral \[ \int \frac{1 + x \cos x}{x \left[ 1 - x^2 \left( e^{\sin x} \right)^2 \right]} \, dx \] Options:
$\int \frac{dx}{(x-1)\sqrt{x+2}} =$
If a function $ f(x) $ defined on $ [a, b] $ is discontinuous at $ x = \alpha \in (a, b) $, then:
Assertion (A): If $ f(x) $ is not continuous at $ x = a $, then it is not differentiable at $ x = a $. Reason (R): If $ f(x) $ is differentiable at a point, then it is continuous at that point.