List of practice Questions

The major product formed in the following reaction is 

The function $f(x)=\begin{cases}-x,\; -\pi<x<0 \\ x,\; 0<x<\pi\end{cases}$ is expanded as a Fourier series of the form $a_0 + \sum_{n=1}^{\infty}a_n\cos(nx)+\sum_{n=1}^{\infty}b_n\sin(nx)$. Which of the following is true? 
 

An infinitely long solenoid, with its axis along $\hat{k}$, carries a current $I$. In addition, there is a uniform line charge density $\lambda$ along the axis. If $\vec{S}$ is the energy flux, in cylindrical coordinates $(\hat{\rho}, \hat{\phi}, \hat{k})$, then
 

Given a spherically symmetric charge density $\rho(r)=\begin{cases}kr^2, & r<R \\ 0, & r>R\end{cases}$ (k being a constant), the electric field for $r<R$ is (take the total charge as $Q$) 
 

A long solenoid is carrying a time dependent current such that the magnetic field inside has the form $\vec{B}(t)=B_0 t^2 \hat{k}$, where $\hat{k}$ is along the axis of the solenoid. The displacement current at the point $P$ on a circle of radius $r$ in a plane perpendicular to the axis

Which one of the following graphs shows the correct variation of $v_o$ with $v_i$? Here, $v_d$ is the voltage drop across the diode and the Op-Amp is assumed to be ideal. 

Which one of the figures correctly represents the T–S diagram of a Carnot engine? 

Two vehicles A and B are approaching an observer O at rest with equal speed as shown in the figure. Both vehicles have identical sirens blowing at a frequency $f_s$. The observer hears these sirens at frequency $f_A$ and $f_B$, respectively. Which one of the following is correct? 

Which one of the following arrangements of optical components can be used to distinguish between unpolarised light and circularly polarised light? 

Three planets orbit a star at distances $a,\;4a,\;9a$. Their orbital periods are proportional to $r^{3/2}$. If the smallest planet has period $T$, after how long will all three be aligned again?
 

A curve is given by $\vec{r}(t)=t\hat{i}+t^{2}\hat{j}+t^{3}\hat{k}$. The unit vector of the tangent at $t=1$ is
 

Let $f(x,y)=x^{3}-2y^{3}$. The curve along which $\nabla^{2} f = 0$ is
 

Which one of the following curves correctly represents (schematically) the solution for the equation $\dfrac{df}{dx}+2f=3;\; f(0)=0$ ?

The mean momentum $\bar{p}$ of a nucleon in a nucleus of mass number $A$ and atomic number $Z$ depends on $A, Z$ as 
 

A particle of mass $m$ is in a one-dimensional potential $V(x)=\begin{cases}0,\; 0<x<L \\ \infty,\ \text{otherwise}\end{cases}$. 

At some instant its wave function is given by $\psi(x)=\dfrac{1}{\sqrt{3}}\psi_1(x) + i\sqrt{\dfrac{2}{3}}\,\psi_2(x)$, where $\psi_1,\psi_2$ are the ground and first excited states. Identify the correct statement. 
 

Consider a convex lens of focal length $f$. The lens is cut along a diameter into two parts. The two lens parts and an object are kept as shown in the figure. The images are formed at the following distances from the object: 

Two projectiles of identical mass are projected with same initial speed $u$ and angle $\alpha$ in the same plane. They collide at the highest point of their trajectories and stick to each other. Which statement is correct? 

A particle $P$ of mass $m$ is constrained to move on the surface of a cylinder under a force $-k\vec{r}$ as shown in the figure ($k$ is a positive constant). Neglect friction. Which of the following statements is correct?

Let matrix $M=\begin{pmatrix} 4 & x \\ 6 & 9\end{pmatrix}$. If $\det(M)=0$, then

Let the electric field in some region $R$ be given by $\vec{E} = e^{-y^2}\hat{i} + e^{-x^2}\hat{j}$. From this we conclude that
 

In presence of magnetic field $\vec{B}\hat{j}$ and electric field $(-E)\hat{k}$, a particle moves undeflected. Which statement is correct?
 

A particle of mass $m$ moves in the positive $x$ direction under the potential $V(x)=\frac{1}{2}kx^{2}+\frac{\lambda}{2x^{2}}$. If the particle is slightly displaced from equilibrium, the angular frequency $\omega$ is ............... (Give answer in units of $\sqrt{k/m}$ as an integer.)
 

A planet has the same average density as Earth but only $1/8$ the mass of Earth. If $g_p$ and $g_e$ are the surface gravities on the planet and Earth, then $\frac{g_p}{g_e} =$ ............ (Specify your answer up to one digit after decimal.)
 

In a grating with grating constant $d = a + b$, where $a$ is the slit width and $b$ is the separation between the slits, the diffraction pattern has the fourth order missing. The value of $\frac{b}{a}$ is ........... (Specify your answer as an integer.)
 

Consider an electromagnetic plane wave $\vec{E}=E_0(\hat{t}+b\hat{y})\cos\left[\frac{2\pi}{\lambda}(ct-(x-\sqrt{3}y))\right]$, where $\lambda$ is the wavelength and $c$ is the speed of light. The value of $b$ is ............. (Specify answer up to two digits after the decimal point.)