List of practice Questions

A body of mass 2 kg is moving towards north with a velocity of 20 m/s and another body of mass 3 kg is moving towards east with a velocity of 10 m/s. The magnitude of the velocity of the centre of mass of the system of the two bodies is

The number of significant figures in 0.03240 is

Three vectors each of magnitude $3\sqrt{1.5}$ units are acting at a point. If the angle between any two vectors is $\frac{\pi}{3}$, then the magnitude of the resultant vector of the three vectors is

Evaluate: $$ \int_{-\pi/2}^{\pi/2} \sin \left(x - [x]\right) dx $$ where $[x]$ is the greatest integer function.

Evaluate the integral: $$ \int_0^2 x^2 (2 - x)^5 \, dx $$

Find the general solution of: $$ (2x - y)^2 dy - 2(2x - y)^2 dx - 2 dx = 0 $$

Find the general solution of the differential equation: $$ x \log x \, dy = (x \log x - y) dx $$

If the function $$ f(x) = \sin x - \cos^2 x $$ is defined on the interval $ [-\pi, \pi] $, then $ f $ is strictly increasing in the interval:

If Lagrange's mean value theorem is applied to the function $$ f(x) = e^x $$ defined on the interval $ [1, 2] $ and the value of $ c \in (1, 2) $ is $ k $, then find $ e^{k-1} $.

If $$ \int \frac{x^4 + 1}{x^2 + 1} dx = Ax^3 + Bx^2 + Cx + D \tan^{-1} x + E, $$ then find $ A + B + C + D $.

If $$ \int \frac{x^2 - x + 2}{x^2 + x + 2} dx = x - \log(f(x)) + \frac{2}{\sqrt{7}} \tan^{-1}(g(x)) + c, $$ then find $$ f(-1) + \sqrt{7} g(-1). $$

Evaluate the integral: $$ \int \sec \left(x - \frac{\pi}{3}\right) \sec \left(x + \frac{\pi}{6}\right) dx $$

If a real valued function $$ f(x) = \begin{cases} \frac{\sin a(x - [x])}{e^{x - [x]}}, & x<1 \\ b + 1, & x = 1 \\ \frac{|x^2 + x - 2|}{x - 1}, & x>1 \end{cases} $$ is continuous at $ x=1 $, then find $ b $. Here, $ [x] $ denotes the greatest integer function.

If $$ \sin x \sqrt{\cos y} - \cos y \sqrt{\sin x} = 0, $$ then find $$ \frac{dy}{dx}. $$

If $$ f(x) = 2 + |\sin^{-1} x|, $$ and $$ A = \{ x \in \mathbb{R} \mid f'(x) \text{ exists} \}, $$ then find $ A $.

If $$ y = (\log_x \sin x)^x, $$ then find $$ \frac{dy}{dx}. $$

If the area of a square is 575 square units, then the approximate value of its side is:

If $ \theta $ is the angle subtended by a latus rectum at the center of the hyperbola having eccentricity $$ \frac{2}{\sqrt{7} - \sqrt{3}}, $$ then find $ \sin \theta $.

The points $ A(-1, 2, 3), B(2, -3, 1), C(3, 1, -2) $ are:

If the equation of the plane passing through point $ (3, 2, 5) $ and perpendicular to the planes $$ 2x - 3y + 5z = 7, \quad 5x + 2y - 3z = 11 $$ is $$ x + by + cz + d = 0, $$ then find $ 2b + 3c + d $.