List of practice Questions

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).

A computer lab has two printers handling certain types of printing jobs. Printer-I and Printer-II handle 40% and 60% of the jobs, respectively. For a typical printing job, printing time (in minutes) of Printer-I follows $N(10, 4)$ and that of Printer-II follows $U(1, 21)$. If a randomly selected printing job is found to have been completed in less than 10 minutes, then the conditional probability that it was handled by the Printer-II equals .......... (round off to two decimal places).

Let $x_1 = 1, x_2 = 0, x_3 = 0, x_4 = 1, x_5 = 0, x_6 = 1$ be the data on a random sample of size 6 from Bin(1, $\theta$) distribution, where $\theta \in (0, 1)$. Then the uniformly minimum variance unbiased estimate of $\theta(1 + \theta)$ equals .............

Let $x_1 = 1, x_2 = 4$ be the data on a random sample of size 2 from a Poisson($\theta$) distribution, where $\theta \in (0, \infty)$. Let $\hat{\psi}$ be the uniformly minimum variance unbiased estimate of $\psi(\theta) = \sum_{k=4}^{\infty} e^{-\theta} \dfrac{\theta^k}{k!}$ based on the given data. Then $\hat{\psi}$ equals ............ (round off to two decimal places).

Let $X$ be a random variable having $N(\theta,1)$ distribution, where $\theta \in \mathbb{R}$. Consider testing $H_0: \theta = 0$ against $H_1: \theta \neq 0$ at $\alpha = 0.617$ level of significance. The power of the likelihood ratio test at $\theta = 1$ equals ............ (round off to two decimal places).

Let $(X, Y)$ have the joint pdf \[ f(x, y) = \begin{cases} \dfrac{3}{4}(y - x), & 0 < x < y < 2, \\ 0, & \text{otherwise.} \end{cases} \] Then the conditional expectation $E(X|Y = 1)$ equals ........... (round off to two decimal places).

The value of the integral \[ \int_0^1 \int_{y^2}^1 \frac{e^x}{\sqrt{x}} \, dx \, dy \] equals ............ (round off to two decimal places):

Evaluate: \[ \lim_{n \to \infty} \left(\frac{\sqrt{n^2 + 1} + n}{\sqrt[3]{n^6 + 1}}\right)^2 \]

Consider the two probability density functions (pdfs): \[ f_0(x) = \begin{cases} 2x, & 0 \le x \le 1, \\ 0, & \text{otherwise}, \end{cases} f_1(x) = \begin{cases} 1, & 0 \le x \le 1, \\ 0, & \text{otherwise.} \end{cases} \]

Let $X$ be a random variable with pdf $f \in \{f_0, f_1\}$. Consider testing $H_0: f = f_0(x)$ against $H_1: f = f_1(x)$ at $\alpha = 0.05$ level of significance. For which observed value of $X$, the most powerful test would reject $H_0$?