List of practice Questions

The number of rotational degrees of freedom of a diatomic molecule is:

A hydraulic lift is shown in the figure. The movable pistons A, B and C are of radius 10 cm, 100 cm and 5 m respectively. If a body of mass 2 kg is placed on piston A, the maximum masses that can be lifted by pistons B and C are respectively:

An iron sphere of diameter $ D $ and mass $ M $ is immersed in hot water, causing its temperature to rise by $ \Delta T $. If $ \alpha $ is the coefficient of linear expansion, the change in surface area of the sphere is:

A small disc of mass $ m $ slides down with initial velocity zero from the top (A) of a smooth hill of height $ H $, having a horizontal portion (BC) as shown in the figure. If the height of the horizontal portion of the hill is $ h $, then the maximum horizontal distance covered by the disc from the point D is

A block of mass $ 50\, \text{kg} $ is pulled at a constant speed of $ 4\, \text{m/s} $ across a horizontal floor by an applied force of $ 500\, \text{N} $ directed $ 30^\circ $ above the horizontal. The rate at which the force does work on the block in watts is:
A particle is executing simple harmonic motion with instantaneous displacement: $$ x = A \sin^2\left( \omega t - \frac{\pi}{4} \right) $$ The time period of oscillation of the particle is:
Solve the differential equation: $$ \frac{dy}{dx} = \cos^2(3x + y) $$
The general solution of the differential equation: $$ \cos^2 x \cdot \frac{dy}{dx} + y = \tan x $$ is $$ y = \tan x - 1 + C e^{-\tan x} $$ If this solution satisfies $ y\left(\frac{\pi}{4}\right) = 1 $, then find $ C $.
Assertion (A): The order of the differential equation of a family of circles with constant radius is two.
Reason (R): An algebraic equation involving two arbitrary constants corresponds to the general solution of a second order differential equation.
The energy $ E $ of a system is a function of time $ t $ and is given by: $$ E(t) = \alpha t - \beta t^3 $$ where $ \alpha $ and $ \beta $ are constants. Find the dimensions of $ \alpha $ and $ \beta $.
Let $ T>0 $ be a fixed number. If $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function such that $ f(x + T) = f(x) $, then: $$ \text{If } I = \int_0^T f(x)\, dx,\quad \text{then } \int_0^{5T} f(2x)\, dx = ? $$
If $$ \int_1^3 x^n \sqrt{x^2 - 1} \, dx = 6 $$ Then find the value of $ n $.
If $ [\cdot] $ denotes the greatest integer function, then evaluate: $$ \int_{-1}^1 \left( x \cdot [1 + \sin \pi x] + 1 \right) dx $$
Given: $$ \lim_{n \to \infty} \left[ \frac{n}{(n+1)\sqrt{2n+1}} + \frac{n}{(n+2)\sqrt{2(n+2)}} + \frac{n}{(n+3)\sqrt{3(2n+3)}} + \cdots \text{(n terms)} \right] = \int_0^1 f(x)\, dx $$ Then find $ f(x) $.
If $ x = f(\theta) $, $ y = g(\theta) $, then find: $$ \frac{d^2 y}{dx^2} $$
If the normal drawn at a point $ P $ on the curve $ 3y = 6x - 5x^3 $ passes through the origin $(0, 0)$, then the positive integral value of the abscissa of point $ P $ is:
The line joining the points $ (0, 3) $ and $ (5, -2) $ is a tangent to the curve: $$ y = \frac{C}{x + 1} $$ Then find $ C $.
Given: $$ \int \frac{3x + 4}{x^3 - 2x + 4} \, dx = \log f(x) + C $$ Then: $$ f(3) = ? $$
Evaluate: $$ \lim_{x \to \infty} \log_e(\cosh x) + x $$
If $ a, b, c $ are distinct real numbers and: $$ \lim_{x \to \infty} \frac{(b - c)x^2 + (c - a)x + (a - b)}{(a - b)x^2 + (b - c)x + (c - a)} = \frac{1}{2} $$ then what is the value of $ a + 2c $?
Evaluate: $$ \lim_{x \to \infty} \frac{3|x| - x}{|x| - 2x} - \lim_{x \to 0} \frac{\log(1 + x^3)}{\sin^3 x} $$
Assertion (A): $$ \frac{d}{dx} \left( \frac{x^2 \sin x}{\log x} \right) = \frac{x^2 \cos x \cdot \log x - x^2 \sin x \cdot \frac{1}{x}}{(\log x)^2} + \frac{2x \sin x}{\log x} \Rightarrow \text{(matches the structure of product + quotient rule)} $$ Reason (R): $$ \frac{d}{dx} \left( \frac{uv}{w} \right) = \frac{uv'}{w} + \frac{u'v}{w} - \frac{uvw'}{w^2} \Rightarrow \text{(correct general derivative formula)} $$
Find the radical centre of the three circles: $$ C_1: x^2 + y^2 - 1 = 0,\quad C_2: x^2 + y^2 - 8x + 15 = 0,\quad C_3: x^2 + y^2 + 10y + 24 = 0 $$
Which of the following equations represents a **parabola**?
If $ e_1 $ and $ e_2 $ are the eccentricities of the hyperbola $$ 16x^2 - 9y^2 = 1 $$ and its conjugate respectively, then find $ 3e_1 $ in terms of $ e_2 $.