List of practice Questions

A particle performs S.H.M. Its potential energies are \( U_1 \) and \( U_2 \) at displacements \( x_1 \) and \( x_2 \) respectively. At displacement \( x_1 + x_2 \), its potential energy \( U \) is

The acceleration due to gravity on moon is \( \frac{1}{6} \)th times the acceleration due to gravity on earth. If the ratio of the density of earth \( \rho_e \) to the density of moon \( \rho_m \) is \( \frac{5}{3} \), then the radius of moon \( R_m \) in terms of the radius of earth \( R_e \) is

A resonance tube completely filled with water has a small hole at the bottom. Length of the tube is 80 cm. A vibrating tuning fork of frequency 500 Hz is held near the open end of the tube. Water is slowly removed from the bottom. The maximum number of resonances heard will be (neglect end correction). Speed of sound in air is 340 m/s

If \( \Delta Q \) is the amount of heat supplied to \( n \) moles of a diatomic gas at constant pressure, \( \Delta U \) is the change in internal energy and \( \Delta W \) is the work done, then \( \Delta W : \Delta U : \Delta Q \) is

The width of the depletion layer of a p-n junction diode, when it is (i) forward biased and (ii) reverse biased respectively

The relative angular speed of hour hand and second hand of a clock is

Water rises to height 2.2 cm in glass capillary tube. The height to which same water rises in another capillary having \( \frac{1}{4} \)th area of cross-section is

A thin, uniform metal rod of mass \( M \) and length \( L \) is swinging about a horizontal axis passing through its end. Its maximum angular velocity is \( \omega \). Its centre of mass rises to a maximum height of \( \frac{L^2 \omega^2}{6g} \). (where \( g \) is the acceleration due to gravity)

A series LCR circuit has \( R = 200 \, \Omega \), \( L = 663 \, \text{mH} \), and \( C = 265 \, \mu\text{F} \). The applied alternating voltage has an amplitude of 50 V and a frequency of 60 Hz so that \( X_L = 250 \, \Omega \) and \( X_C = 100 \, \Omega \). The peak current is

Two masses of 1 gram and 4 gram are moving with equal kinetic energy. The ratio of the magnitudes of their momenta is

If the frequency of incident radiation is kept constant and the experiment is repeated by using incident light of different intensities, then stopping potential (\( V_s \))

If \( A(0, 4, 0) \), \( B(0, 0, 3) \), and \( C(0, 4, 3) \) are the vertices of \( \Delta ABC \), then its incenter is:

The probability distribution of a discrete r.v. \( X \) is: \[ \begin{array}{c|ccccc} X = x & 0 & 1 & 2 & 3 & 4 \\ P(X = x) & k & 2k & 4k & 2k & k \\ \end{array} \] Then, the value of \( P(X \leq 2) \) is:

The equation of the line passing through the points \( (3, 4, -7) \) and \( (6, -1, 1) \) is:

If \( \frac{x}{x-y} = \log \left( \frac{a}{x-y} \right) \), then \( \frac{dy}{dx} = \)

Evaluate the integral: \[ \int_{-1}^{1} \left[ \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right] dx \]

The co-ordinates of the foot of the perpendicular from the point \( (0, 2, 3) \) on the line
\[ \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \]

The approximate value of \( \log_{10}99 \) is (Given \( \log_{10}e = 0.4343 \))

The maximum value of \(Z = 10x + 25y\) subject to \[ 0 \le x \le 3,\; 0 \le y \le 3,\; x + y \le 5,\; x \ge 0,\; y \ge 0 \] is

Evaluate the integral: \[ \int \frac{\sin x}{\sin\left(x-\frac{\pi}{4}\right)}\,dx \]

If \((\sim p \wedge q)\rightarrow r\) is false, then the truth values of \(p,q,r\) respectively are

The value of \(x\) such that the matrix \[ \begin{bmatrix} x & 2 & 3 \\ 4 & 5 & 6 \\ 2 & 3 & 5 \end{bmatrix} \] is not invertible is

What is the approximate ratio of roasted ore, coke, and lime stone respectively in the charge to be added into the blast furnace for extraction of iron?

Relation between \( \Delta H \) and \( \Delta U \) for the reaction \[ \text{2SO}_3 (g) \rightarrow \text{2SO}_2 (g) + \text{O}_2 (g) \] is

Evaluate the integral: \[ \int \left[ \log(1+\cos x) - x \tan\left(\frac{x}{2}\right) \right] dx \]