List of practice Questions

A sphere of surface area \( 4 \, \text{m}^2 \), temperature 400 K, emissivity 0.5 is in an environment of 200 K. The net rate of energy exchange is (take \( \sigma = 5.67 \times 10^{-8} \, \text{Wm}^{-2} \text{K}^{-4} \))
Two rectangular blocks of masses 40 kg and 60 kg are connected by a string on a frictionless table. A force of 1000 N is applied on the 60 kg block away from the 40 kg block. Find the tension in the string.
A ball of mass 0.5 kg moving horizontally at \( 10\, \text{ms}^{-1} \) strikes a vertical wall and rebounds with speed \( V \). The change in linear momentum is 8.0 kg·m/s. Find the magnitude of \( V \).
A mass of 1 kg falls from a height of 1 m and lands on a massless platform supported by a spring with spring constant \( 15 \, \text{Nm}^{-1} \). The maximum compression of the spring is (take \( g = 10 \, \text{ms}^{-2} \))
A mass of 1 kg falls from a height of 1 m
A bead of mass 400 g is moving along a straight line under a force delivering constant power 1.2 W. If the bead is initially at rest, the speed attained after 6 seconds is (in \( \text{ms}^{-1} \))
Masses \( m \left(\frac{1}{3}\right)^N \) are placed at \( x = N \), for \( N = 2, 3, 4, \dots \). If the total mass is \( M \), then the center of mass of the system is:
Consider a disc of radius \( R \) and mass \( M \). A hole of radius \( \frac{R}{3} \) is created in the disk such that the center of the hole is \( \frac{R}{3} \) away from the center of the disc. The moment of inertia of the remaining system about the axis perpendicular to the disc and passing through the original center is:
\( \lim_{n \to \infty} \left( \frac{1}{1 + n^5} + \frac{2^4}{2^5 + n^5} + \frac{3^4}{3^5 + n^5} + \ldots + \frac{n^4}{n^5 + n^5} \right) = \)
If the solution of \( \frac{dy}{dx} = y \log_e 0.5 = 0 \), \( y(0) = 1 \), and \( y(x) \to k \) as \( x \to \infty \), then \( k = \)
At any point \( (x, y) \) on a curve, if the length of the subnormal is \( (x - 1) \) and the curve passes through \( (1, 2) \), then the curve is a conic. A vertex of the curve is:
The function \( y = Ae^x + Be^{-2x} \) satisfies which of the following differential equations?
If \( N_A, N_B, N_C \) are the number of significant figures in \( A = 0.001204 \) m, \( B = 43120000 \) m and \( C = 1.200 \) m respectively, then which of the following is correct?
A car covers a distance at a speed of 60 km/h. It returns to the starting point at a speed of \( V \). If the average speed for the entire trip is 48 km/h, then what is the magnitude of \( V \)?
A car travels with a speed of 40 km/h. Rain drops fall vertically but the traces on the side window make an angle of \( 30^\circ \) with the vertical. Find the magnitude of rain’s velocity with respect to the car.
Assertion (A): \( I_n = \int \cot^n x \, dx \), then \( I_6 + I_4 = -\frac{\cot^5 x}{5} \)
Reason (R): \( \int \cot^n x \, dx = \frac{-\cot^{n-1} x}{n - 1} - \int \cot^{n - 2} x \, dx \)
If \( I_n = \int \tan^n x \, dx \), and \( I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \), then \( n = \)
\( \int \frac{e^{2x}}{\sin^2 x} \left( 2 \log \csc x + \sin 2x \right) dx = \)
The parametric form of a curve is \( x = \frac{t^3}{t^2 - 1},\ y = \frac{t}{t^2 - 1} \), then \( \int \frac{dx}{x - 3y} = \)
\( \int_0^1 \alpha^k x^k dx = \)
Let \( \alpha \) and \( \beta \) (\( \alpha<\beta \)) be roots of \( 18x^2 - 9\pi x + \pi^2 = 0 \), \( f(x) = x^2, g(x) = \cos x \). Then
\[ \int_{\alpha}^{\beta} x (g \circ f(x)) \, dx = ? \]
\( \int_0^{\pi} x \left( \sin^2(\sin x) + \cos^2(\cos x) \right) dx = \)
If \( x^3 - 2x^2y + 5x + y - 5 = 0 \), then at (1,1), \( y'(1) = \)
If the curves \( y = x^3 - 3x^2 - 8x - 4 \) and \( y = 3x^2 + 7x + 4 \) touch each other at a point \( P \), then the equation of the common tangent at \( P \) is
If \( ax + by = 1 \) is a normal to the parabola \( y^2 = 4px \), then the condition is
The condition that \( f(x) = ax^3 + bx^2 + cx + d \) has no extreme value is