List of practice Questions

If a force \( \vec{F} = (3\hat{i} + 2\hat{j} + 5\hat{k})~\text{N} \) acting on a body displaces it through \( \vec{d} = (2\hat{i} + 2\hat{j} + 1\hat{k})~\text{m} \), then the work done by the force on the body is
If two bodies A and B are projected with same velocity but with different angles \( \theta_1 \) and \( \theta_2 \) respectively with the horizontal such that both will have same range, then the ratio of times of flight of the bodies A and B is
A body falls freely on to a hard horizontal surface. If the coefficient of restitution between the surface and the body is \( 0.8 \), then the ratio of the maximum height to which the body rises after second impact and the initial height of the body is
Two bodies of masses \( M \) and \( 4M \) initially at rest, start moving towards each other due to their mutual attraction. The velocity of their centre of mass when the first body attains a velocity \( v_0 \) is
Evaluate the integral: \[ \int_0^1 x^{5/2} (1 - x)^{3/2} \, dx = \]
Evaluate the limit: \[ \lim_{n \to \infty} \left[ \frac{1}{n^2} \sec^2 \frac{1}{n^2} + \frac{2}{n^2} \sec^2 \frac{4}{n^2} + \frac{3}{n^2} \sec^2 \frac{9}{n^2} + .s + \frac{1}{n^2} \sec^2 1 \right] = \]
The general solution of the differential equation \( \left(x \sin \frac{y}{x} \right) dy = \left( y \sin \frac{y}{x} - x \right) dx \) is
The general solution of the differential equation \( \cos(x + y) \, dy = dx \) is
If \( Ax^3 + Bxy = 4 \) (A and B are arbitrary constants) is the general solution of the differential equation \[ F(x)\frac{d^2y}{dx^2} + G(x)\frac{dy}{dx} - 2y = 0, \] then \( F(1) + G(1) = \)
The physical quantity having the dimensions of the square root of the ratio of the kinetic energy and surface tension is
If the displacement \( s \) (in metre) of a moving particle in terms of time \( t \) (in second) is \( s = t^3 - 6t^2 + 18t + 9 \), then the minimum velocity attained by the particle is
\( \int x \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \, dx \, (x>0) = \)
\( \int \frac{dx}{(1+\sqrt{x}) \sqrt{x-x^2}} = \)
\( \int \sin^{-1}\left(\frac{x}{\sqrt{a+x}}\right) \, dx = \)
If \( \int \frac{x}{x \tan x + 1} \, dx = \log f(x) + k \), then \( f\left(\frac{\pi}{4}\right) = \)
Evaluate the definite integral: \[ \int_0^1 \frac{2x + 5}{x^2 + 3x + 2} \, dx = \]
The area (in sq. units) of the region given by \( R = \left\{ (x, y) : \dfrac{y^2}{2} \leq x \leq y + 4 \right\} \) is
If \( x = \sqrt{2 \cosec^{-1} t} \) and \( y = \sqrt{2 \sec^{-1} t} \), \( |t| \geq 1 \), then \( \dfrac{dy}{dx} = \)
If \( (a + \sqrt{2}b \cos x)(a - \sqrt{2}b \cos y) = a^2 - b^2 \), where \( a>b>0 \), then at \( \left( \dfrac{\pi}{4}, \dfrac{\pi}{4} \right) \), \( \dfrac{dy}{dx} = \)
Consider the quadratic equation \( ax^2 + bx + c = 0 \), where \( 2a + 3b + 6c = 0 \) and let \[ g(x) = \frac{a x^3}{3} + \frac{b x^2}{2} + c x \] Statement-I: The given quadratic equation \( ax^2 + bx + c = 0 \) has at least one root in \( (0, 1) \). Statement-II: Rolle's theorem is applicable to \( g(x) \) on \( [0, 1] \). Then:
The difference between the absolute maximum and absolute minimum values of the function \( f(x) = 2x^3 - 15x^2 + 36x - 30 \) on \( [-1, 4] \) is:
If \( f(x) = x e^{x(1-x)}, \, x \in \mathbb{R} \), then \( f(x) \) is:
The angle between the curves \( y^2 = x \) and \( x^2 = y \) at the point \( (1,1) \) is:
If \( \int \frac{5 \tan x}{\tan x - 2} \, dx = a x + b \log |\sin x - 2 \cos x| + c \), then \( a + b = \)
The circumradius of the triangle formed by the points \( (2, -1, 1) \), \( (1, -3, -5) \), and \( (3, -4, -4) \) is