List of practice Questions

A bag has a total of 108 toys. These toys are of 3 different colours: black, green and white. If a toy is taken out randomly, the probabilities of getting a black and a green are $ \frac{1}{3} $ and $ \frac{4}{9} $ respectively. What is the number of white toys?

A spherical dielectric shell with inner radius $a$ and outer radius $b$, has polarization $\vec{P} = \dfrac{k}{r^2}\hat{r}$, where $k$ is a constant and $\hat{r}$ is the unit vector along the radial direction. Which of the following statements is/are correct?

For electric and magnetic fields, $\vec{E}$ and $\vec{B}$, due to a charge density $\rho(\vec{r},t)$ and a current density $\vec{J}(\vec{r},t)$, which of the following relations is/are always correct?

An object executes simple harmonic motion along the x-direction with angular frequency $\omega$ and amplitude $a$. The speed of the object is 4 cm/s and 2 cm/s when it is at distances 2 cm and 6 cm, respectively, from the equilibrium position. Which of the following is/are correct?

A thin rod of uniform density and length $2\sqrt{3}\, \text{m}$ is undergoing small oscillations about a pivot point. The time period of oscillation ($T_m$) is minimum when the distance of the pivot point from the center-of-mass of the rod is $x_m$. Which of the following is/are correct? (Assume acceleration due to gravity $g = 10\, \text{m/s}^2$)

Consider a vector function $\vec{u}(\vec{r})$ and two scalar functions $\psi(\vec{r})$ and $\phi(\vec{r})$. The unit vector $\hat{n}$ is normal to the elementary surface $dS$, $dV$ is an infinitesimal volume, $d\vec{l}$ is an infinitesimal line element, and $\partial / \partial n$ denotes the partial derivative along $\hat{n}$. Which of the following identities is/are correct?

In the circuit shown in the figure, both OPAMPs are ideal. The output for the circuit $V_{out}$ is:

For an unbiased Silicon $n\text{-}p\text{-}n$ transistor in thermal equilibrium, which one of the following electronic energy band diagrams is correct? ($E_c$ = conduction band minimum, $E_v$ = valence band maximum, $E_F$ = Fermi level.)

Consider a one-dimensional infinite potential well of width $a$. This system contains five non-interacting electrons, each of mass $m$, at temperature $T = 0 \, K$. The energy of the highest occupied state is:

A linear operator $\hat{O}$ acts on two orthonormal states of a system $\psi_1$ and $\psi_2$ as per the following: \[ \hat{O}\psi_1 = \psi_2, \hat{O}\psi_2 = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2) \] The system is in a superposed state defined by \[ \psi = \frac{1}{\sqrt{2}}\psi_1 + \frac{i}{\sqrt{2}}\psi_2 \] The expectation value of $\hat{O}$ in the state $\psi$ is:

A thin conducting square loop of side $L$ is placed in the first quadrant of the $xy$-plane with one of the vertices at the origin. If a changing magnetic field $\vec{B}(t) = B_0(52yt\,\hat{x} + xzt\,\hat{y} + 3y^2t^2\,\hat{z})$ is applied, where $B_0$ is a constant, then the magnitude of the induced electromotive force in the loop is:

A sum of INR 15000 is put on compound interest. This sum becomes 2 times in 5 years at this rate. What would the original sum become after 20 years?

Which one of the following statements is correct?
Given, $\displaystyle \binom{n}{m} = \frac{n!}{m!(n-m)!}$ is the binomial coefficient.

Consider the motion of a quantum particle of mass $m$ and energy $E$ under the influence of a step potential of height $V_0$. If $R$ denotes the reflection coefficient, which one of the following statements is true?

The figure below shows a cubic unit cell with lattice constant $a$. The shaded crystallographic plane intersects the x-axis at 0.5a. The Miller indices of the shaded plane are:

Two stationary point particles with equal and opposite charges are at some fixed distance from each other. The points having zero electric potential lie on:

Let $X$ be a random variable having the Poisson(4) distribution and let $E$ be an event such that $P(E|X = i) = 1 - 2^{-i}$, $i = 0, 1, 2, \ldots$. Then $P(E)$ equals ........... (round off to two decimal places).

Let $X_1, X_2,$ and $X_3$ be independent random variables such that $X_1 \sim N(47, 10)$, $X_2 \sim N(55, 15)$, and $X_3 \sim N(60, 14)$. Then $P(X_1 + X_2 \ge 2X_3)$ equals ............ (round off to two decimal places).

Let $U \sim F_{5,8}$ and $V \sim F_{8,5}$. If $P[U > 3.69] = 0.05$, then the value of $c$ such that $P[V > c] = 0.95$ equals ................ (round off to two decimal places).

Let the sample mean based on a random sample from Exp($\lambda$) distribution be 3.7. Then the maximum likelihood estimate of $1 - e^{-\lambda}$ equals ........... (round off to two decimal places).

Let $X$ be a single observation drawn from $U(0, \theta)$ distribution, where $\theta \in \{1, 2\}$. To test $H_0: \theta = 1$ against $H_1: \theta = 2$, consider the test procedure that rejects $H_0$ if and only if $X > 0.75$. If the probabilities of Type-I and Type-II errors are $\alpha$ and $\beta$, respectively, then $\alpha + \beta$ equals ......... (round off to two decimal places).

Let $\alpha$ be the real number such that the coefficient of $x^{125}$ in Maclaurin's series of $(x + \alpha^3)^3 e^x$ is $\dfrac{28}{124!}$. Then $\alpha$ equals ..............

Consider the matrix $M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & 1 & 4 \end{bmatrix}$. Let $P$ be a nonsingular matrix such that $P^{-1}MP$ is a diagonal matrix. Then the trace of the matrix $P^{-1}M^3P$ equals ...........

Let $P$ be a $3 \times 3$ matrix having characteristic roots $\lambda_1 = -\dfrac{2}{3}$, $\lambda_2 = 0$ and $\lambda_3 = 1$. Define $Q = 3P^3 - P^2 - P + I_3$ and $R = 3P^3 - 2P$. If $\alpha = \det(Q)$ and $\beta = \text{trace}(R)$, then $\alpha + \beta$ equals .......... (round off to two decimal places).

Let $X$ and $Y$ be independent random variables with respective moment generating functions $M_X(t) = \dfrac{(8 + e^t)^2}{81}$ and $M_Y(t) = \dfrac{(1 + 3e^t)^3}{64}$, $-\infty < t < \infty$. Then $P(X + Y = 1)$ equals .............. (round off to two decimal places).