List of practice Questions

Two events occur in frame $S$ at $(t_1 = 0,\ r_1 = 0)$ and $(t_2 = 0,\ x_2 = 10^8\mathrm{m}, y_2 = 0, z_2 = 0)$. Another frame $S'$ moves with $v = 0.8c$ relative to $S$. The time difference $(t_2' - t_1')$ in $S'$ is ................... s. (Specify answer up to two digits after decimal.)
The lattice constant of NaCl crystal is $0.563$ nm. X-rays of wavelength $0.141$ nm are diffracted by this crystal. The angle at which the first order maximum occurs is .......... degrees. (Specify answer up to two digits after decimal.)
In a pn junction, dopant concentration on p-side is higher than n-side. Which statements are correct when the junction is unbiased?
Which of the combinations of crystal structure and coordination number is correct?
The coefficient of $x^3$ in the Taylor expansion of $\sin(\sin x)$ around $x = 0$ is .............. (Specify your answer up to two digits after the decimal point.)
Let $f(x)=3x^6 - 2x^2 - 8$. Which of the following statements is (are) true?
Two beams of visible light (400–700 nm) interfere at a point. The optical path difference is 5000 nm. Which wavelength interferes constructively?
Which of the following relations is (are) true for thermodynamic variables?
The Boolean expression $(AB)(\bar{A}+B)(A+\bar{B})$ can be simplified to
Consider the transformation to a new set of coordinates $(\xi,\eta)$ from rectangular coordinates $(x,y)$, where $\xi=2x+3y$ and $\eta=3x-2y$. In the $(\xi,\eta)$ coordinate system, the area element $dx\,dy$ is
A raindrop falls under gravity and captures water molecules from the atmosphere. Its mass changes at the rate $\lambda m(t)$, where $\lambda$ is a positive constant and $m(t)$ is the instantaneous mass. Assume gravity is constant and captured water is at rest before capture. Which of the following statements is correct?
Consider two waves $y_1=a\cos(\omega t-kz)$ and $y_2=a\cos[(\omega+\Delta \omega)t-(k+\Delta k)z]$. The group velocity of the superposed wave will be ($\Delta \omega \ll \omega$ and $\Delta k \ll k$)
Consider a convex lens of focal length $f$. A point object moves towards the lens along its axis between $2f$ and $f$. If the speed of the object is $V_o$, then its image would move with speed $V_i$. Which of the following is correct?
A disc of radius $R_1$ having uniform surface density has a concentric hole of radius $R_2<R_1$. If its mass is $M$, the principal moments of inertia are
The plane of polarisation of a plane-polarised light rotates by $60^\circ$ after passing through a wave plate. The pass-axis of the wave plate is at an angle $\alpha$ with respect to the plane of polarisation of the incident light. The wave plate and $\alpha$ are
A rectangular loop of dimensions $l$ and $w$ moves with a constant speed $v$ through a region containing a uniform magnetic field $B$ directed into the paper and extending a distance of $4w$. Which of the following figures correctly represents the variation of emf $(\varepsilon)$ with the position $(x)$ of the front end of the loop?
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The equation of state for one mole of a non-ideal gas is given by $PV = A\left(1 + \dfrac{B}{V}\right)$, where the coefficients $A$ and $B$ are temperature dependent. If the volume changes from $V_1$ to $V_2$ in an isothermal process, the work done by the gas is
An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that only one vibrational mode is excited. If $R$ denotes the gas constant, then the specific heat at constant volume of one mole of the gas at this temperature is
Consider an ensemble of thermodynamic systems, each of which is characterized by the same number of particles, pressure and temperature. The thermodynamic function describing the ensemble is
A current $I$ flows through the sides of an equilateral triangle of side $a$. The magnetic field at the centroid is
Three infinite plane sheets carrying uniform charge densities $-\sigma,\,2\sigma,\,3\sigma$ are placed parallel to the $xz$-plane at $y=a,\;3a,\;4a$, respectively. The electric field at the point $(0,2a,0)$ is
Two boxes A and B contain an equal number of molecules of the same gas. If the volumes are $V_A$ and $V_B$, and $\lambda_A$ and $\lambda_B$ denote respective mean free paths, then
Let $T_g$ and $T_e$ be the kinetic energies of the electron in the ground and the third excited states of a hydrogen atom. According to the Bohr model, the ratio $T_g/T_e$ is

Let \(a_n\) be a sequence of real numbers such that \( a_1 = 2 \), and for \( n \geq 1 \), \( a_{n+1} = \frac{2a_n + 1}{a_n + 1} \).
 

Let \( x_1 = 0, x_2 = 1, x_3 = 2, x_4 = 3 \) and \( x_5 = 0 \) be the observed values of a random sample of size 5 from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \frac{\theta}{3}, & x = 0, \\ \frac{2\theta}{3}, & x = 1, \\ \frac{1 - \theta}{2}, & x = 2, 3, \end{cases} \] where \( \theta \in [0,1] \) is the unknown parameter. Then the maximum likelihood estimate of \( \theta \) is