List of practice Questions

A long solenoid of initial radius $R_0$ is put in a region of uniform magnetic field $\mathbf{B}$ with the axis of the solenoid aligned along the magnetic field. The solenoid is a part of a closed circuit that has no initial current running through it. If the radius of the solenoid starts increasing at a uniform rate, how do the magnetic field strength $B_{in}$ and the associated magnetic energy $U_{in}$ inside the solenoid change?

The position of a particle of mass 1 kg at time $t$ is given by $\mathbf{r} = t\hat{i} + \hat{j} + 2t^2\hat{k}$, where $t$ is in seconds and the coefficients have the proper units for $\mathbf{r}$ to be in metres. What is the component of the angular momentum (with respect to the origin) in kg m$^2$ s$^{-1}$ along the vector $(\hat{i} + \hat{j})$?

A planet is revolving in a circular orbit with a time period $T$ around the center of a star solely under the gravity of the star. Suppose the distance between the star and the planet is halved. The individual radii of the star and the planet are also halved, keeping their uniform mass densities unchanged. What will be the time period of the new orbit of the planet?

An exotic spherical jellyfish has a bulk modulus $B$. Close to the surface of the sea (depth $d=0$), its radius is $R$. When it dives to a depth $d$ ($d \gg R$), its radius is reduced by $\Delta R > 0$. Given the density of the incompressible sea water $\rho$, and the uniform acceleration due to gravity $g$ such that $\rho g d \ll B$, what is $\frac{\Delta R}{R}$?

A particle of mass $m_1$ and electric charge $q$ starts from rest under the influence of a uniform external electric field $\mathbf{E}$ to travel a distance $d$ in time $t_1$. If the particle had mass $m_2$, it would take time $t_2$ to travel the same distance. What is the ratio $\frac{t_1}{t_2}$?

A simple pendulum of length $L$, mass $m$ and electric charge $q$ on its bob is oscillating with a time period $T$ under uniform gravity which is in the $-\hat{z}$ direction. Upon applying a uniform electric field $|E|\hat{n}$ (where $\hat{n}$ is a unit vector in the plane of oscillation), the time period of the pendulum decreases. Which of the following statements is NOT correct?

Let $f : \mathbb{R} \to \mathbb{R}$ be the function given by \[ f(x) = |x - 2| + 3|x - 1| + ||x - 2| - 1| . \] What is the number of points where $f$ is NOT differentiable?

Suppose there are two boxes $B_1$ and $B_2$, each having 3 red and 4 black balls. One ball is drawn at random from $B_1$. If it is red, 4 red balls are put into $B_2$, otherwise 3 black balls are put into $B_2$. Then one ball is randomly drawn from $B_2$. If this ball is red, what is the conditional probability that the ball drawn from $B_1$ was also red?

Let $n = 20^{26}$. What is the remainder when $49^n + 41^n + 10n$ is divided by 100?

Let $a_1, a_2, a_3, \dots$ be a geometric progression of positive integers such that $a_1 = 3$ and $a_{n+2} - 2a_n = a_{n+1}$ for all positive integers $n$. What is the value of $a_1 + a_2 + a_3 + a_4 + a_5$?

For a $2 \times 2$ matrix $A$, whose elements are real numbers, denote by $A^m$ the product $AA\dots A$ ($m$ times), where $m$ is a positive integer. Define $x_0 = 0$, $x_1 = 1$, $x_n = x_{n-1} + x_{n-2}$, for all $n \ge 2$ and \[ A_n = \begin{bmatrix} x_{n+1} & x_n\\ x_n & x_{n-1} \end{bmatrix}, \text{ for all } n \ge 1. \] Which of the following statements is TRUE for all $m \ge 3$?

For real numbers $a$ and $b$, consider the function $f : \mathbb{R} \to \mathbb{R}$ given by \[ f(x) = \begin{cases} -ax - b ;& \text{if } x \\ 5x + 1 ;& \text{if } -1 \le x \le 1, \\ a^2x + 3b ;& \text{if } x > 1 . \end{cases} \] How many pairs $(a, b)$ are there for which $f$ is continuous at every point of $\mathbb{R}$?

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sin^2(7x) - \sin^2(5x)$. Which of the following statements is NOT TRUE?

Consider the data of scores obtained by students in an examination. If the score of every student is increased by 2 marks, then which of the following statements is TRUE?

What is the value of $\int_{-1}^{2} \min\{1 - x, 1 - x^3\} \, dx$?

Let $\mathcal{C}$ be the set of all the circles in a plane. If \[ \mathcal{R} = \{(C_1, C_2) \in \mathcal{C} \times \mathcal{C} \mid C_1 \text{ and } C_2 \text{ intersect}\} , \] then which of the following statements is TRUE?

Let $r, l$ be two integers such that $r \ge l \ge 3$. What is the total number of functions \[ f : \{1, 2, \dots, r\} \to \{1, 2, \dots, r\} \] such that $f(1), f(2), \dots, f(l)$ are all distinct?

Let $l_1$ be the line joining $(1, 1, 1)$ and $(3, 1, 3)$ and let $l_2$ be the line joining $(0, 2, -1)$ and $(2, 0, 3)$. What is the angle between $l_1$ and $l_2$?

Consider the points $A(4\hat{i} + \hat{j} + 3\hat{k})$, $B(2\hat{j})$ and $C(-4\hat{i} + 3\hat{j} - 3\hat{k})$. Which of the following statements is TRUE?

Consider the following sets of points in the complex plane \[ A = \left\{ \cos \left( \frac{2n\pi}{5} \right) + i \sin \left( \frac{2n\pi}{5} \right) : n \in \mathbb{Z} \right\} \text{ and} \] \[ B = \left\{ \cos \left( \frac{2n}{5} \right) + i \sin \left( \frac{2n}{5} \right) : n \in \mathbb{Z} \right\} . \] Which of the following statements is TRUE?

Let $p(x)$ be a quadratic polynomial such that $p(1) = p(-1) = 0$. What is the coefficient of $x$ in $p(x)$?

The rate constant of a reaction at 600 K with an activation energy of 191.47 kJ mol$^{-1}$ is 5.0 $\times$ 10$^{-5}$ s$^{-1}$. What is the temperature at which the half-life of the reaction becomes 152 s? [Consider pre-exponential factor and activation energy to be independent of temperature. R = 8.314 J K$^{-1}$mol$^{-1}$]}

For two pure volatile liquids X and Y, attractive intermolecular interactions of both X-X and Y-Y are weaker than those of X-Y. The total vapour pressure of an equimolar solution of X and Y is p$_{\text{total}$. The vapour pressure of pure X and pure Y are p$^0_{\text{X}}$ and p$^0_{\text{Y}}$, respectively. Which one of the following relations is correct?

What is the ratio of the velocity of an electron in the fourth orbit of Be$^{3+}$ to the velocity of the electron in the second orbit of He$^{+}$?

An ideal gas goes through a reversible isothermal expansion (solid line) followed by a reversible adiabatic expansion (dashed line). Which of the following diagram(s) closely depict(s) the entire process?