List of practice Questions

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?
\includegraphics[width=0.5\linewidth]{13.jpeg} \includegraphics[width=0.8\linewidth]{13.1.jpeg}

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

Let $ x_1 = -2, x_2 = 1 $ and $ x_3 = -1 $ be the observed values of a random sample of size three from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \frac{1}{2 \theta + 1}, & x \in \{-\theta, -\theta + 1, \dots, 0, \dots, \theta \}, \\ 0, & \text{otherwise}, \end{cases} \] where $ \theta \in \{ 1, 2, \dots \} $ is the unknown parameter. Then the method of moment estimate of $ \theta $ is

Let $ X $ be a Poisson random variable with mean $ \frac{1}{2} $. Then $ E((X + 1)!) $ equals

The value of \[ \lim_{n \to \infty} \left( 1 + \frac{2}{n} \right)^{n^2} e^{-2n} \text{ is } \]

Which one of the following matches is CORRECT between the inhibitors given in Group A with their modes of action in Group B?
\[\begin{array}{|c|c|} \hline Group A & Group B \\ \hline \text{(P) Antimycin A} & \text{(i) Inhibits cytochrome c oxidase} \\ \hline \text{(Q) Amytal} & \text{(ii) Blocks electron transfer from cyt b to cyt c1} \\ \hline \text{(R) Carbon monoxide} & \text{(iii) Inhibits adenine nucleotide translocase} \\ \hline \text{(S) Atractyloside} & \text{(iv) Prevents electron transfer from Fe-S centers of complex 1 to ubiquinone} \\ \hline \end{array}\]

Let y = $\sum_{n=1}^{3} nx^n$. The value of $\int_0^1$ y dx is .......... (decimal digits up to 2 places)

Rod shaped E. coli is 2 µm long and has diameter of 0.8 µm. The average density of E. coli is 1.1 x 10³ g/L. The mass of single E. coli cell in pg is ........... (decimal digits up to 1 place)

The length of each side of a regular octagon is 1 meter. The area of the octagon in m² is ............ (decimal digits up to 2 places)

Molecular weight of the genomic DNA of an organism is 3.96 x 10⁹ g/mol. The average molecular weight of nucleotide pair is 660 g/mol. If an average protein in this organism consists of a chain of 400 amino acids, the maximum number of proteins coded by DNA molecule will be ..............

The number of NADPH molecules required for the synthesis of palmitate (C16:0) from acetyl CoA is ..........

The number of peptide fragments generated from the sequence: Ala-Trp-Val-Ala-Phe-Thr-Gly-Lys-Glu-Tyr-Asp-Ser-Lys on treatment with chymotrypsin is ..........

The prevalence of a severe form of sickle-cell anemia (ss) in an African population is 16%. The percentage of the population resistant to malaria because of heterozygous (Ss) genotype for the sickle-cell gene is ............

The maximum velocity of an enzymatic reaction is 0.4 mole/sec. At 5 mM concentration of the substrate, the reaction velocity was found to be 0.2 mole/sec. If the enzyme shows standard Michaelis-Menten kinetics, the rate of the reaction at 10 mM substrate concentration in mole/sec is ............ (decimal digits up to 3 places)