Q. A progressive wave travelling along the positive x-direction is represented by $y(x,t)=A\sin (kx-\omega t+\varphi )$. Its snapshot $t=0$ is given in the figure:

1. $0$
2. $=\frac{\pi }{2}$
3. $\frac{\pi }{2}$
4. $\pi$

Q.

A person of mass $M$ is sitting on a swing of length $L$ and swinging with an angular amplitude ${\theta }_0$. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance $\ell (\ell <<L)$, is close to :

1. $Mg\ell$
2. $Mg\ell (1+{\theta }_0^2)$
   
3. $Mg\ell (1-{\theta }_0^2)$
4. $Mg\ell (1+\frac{{\theta }_0^2}{2})$

Q. A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as $\left(\frac{{\sigma }_0}{r}\right)$, then the radius of gyration of the disc about its axis passing through the centre is :

1. $\frac{a+b}{2}$
2. $\frac{a+b}{3}$
3. $\sqrt{\frac{a^2+b^2+ab}{3}}$
4. $\sqrt{\frac{a^2+b^2+ab}{2}}$

Q. Shown in the figure is a shell made of a conductor. It has inner radius $a$ and outer radius $b$, and carries charge $Q$. At its centre is a dipole $\overrightarrow{p}$ as shown. In this case :

1. Electric field outside the shell is the same as that of a point charge at the centre of the shell.
2. Surface charge density on the inner surface of the shell is zero everywhere.
3. Surface charge density on the inner surface (Q/ 2) is uniform and equal to $\frac{\left(Q/2\right)}{4\pi a^2}$
4. Surface charge density on the outer surface depends on $|\overrightarrow{p}|$

Q. A concave mirror has radius of curvature of $40cm$. It is at the bottom of a glass that has water filled up to $5cm$ (see figure). If a small particle is floating on the surface of water, its image as seen, from directly above the glass, is at a distance $d$ from the surface of water. The value of $d$ is close to : (Refractive index of water $=1.33$)

1. $8.8cm$
2. $11.7cm$
3. $6.7cm$
4. $13.4cm$

Q. The value of numerical aperature of the objective lens of a microscope is $1.25$. If light of wavelength $5000\mathring{A}$ is used, the minimum separation between two points, to be seen as distinct, will be :

1. $0.24\mu m$
2. $0.48\mu m$
3. $0.12\mu m$
4. $0.38\mu m$

Q. Two identical parallel plate capacitors, of capacitance $C$ each, have plates of area $A$, separated by a distance $d$. The space between the plates of the two capacitors, is filled with three dielectrics, of equal thickness and dielectric constants $K_1,K_2$ and $K_3$. The first capacitor is filled as shown in fig.-I, and the second one is filled as shown in fig.-II.
If these two modified capacitors are charged by the same potential $V$, the ratio of the energy stored in the two, would be ($E_1$ refers to capacitor (I) and $E_2$ to capacitor (II)) :

Q. A point dipole $\overrightarrow{p}=p_0\hat{x}$ is kept at the origin. The potential and electrical field due to this dipole on the y-axis at a distance $d$ are, respectively:
(Take $V=0$ at infinity):

Q. When $M_1$ gram of ice at $-{10}^{0}C$ (specific heat $=0.5cal{g}^{10}{C}^{-1}$) is added to $M_2$ gram of water at ${50}^{o}C$, finally no ice is left and the water is at $0^{o}C$. The value of latent heat of ice, in cal $g^{-1}$ is:

Q. The trajectory of a projectile near the surface of the earth is given as $y=2x-9x^2$. If it were launched at an angle ${\theta }_0$ with speed $v_0$ then $(g=10m{s}^{-2})$:

Q. The truth table for the circuit given in the fig. is:

Q. A shell is fired from a fixed artillery gun with an initial speed $u$ such that it hits the target on the ground at a distance $R$ from it. If $t_1$ and $t_2$, are the values of the time taken by it to hit the target in two possible ways, the product $t_1{t}_2$ is:

Q. A submarine (A) travelling at $18km/hr$ is being chased along the line of its velocity by another submarine (B) travelling at $27km/hr$. $B$ sends a sonar signal of $500Hz$ to detect $A$ and receives a reflected sound of frequency $v$. The value of v is close to :
(Speed of sound in water $=1500m{s}^{-1}$)

Q. Two moles of helium gas is mixed with three moles of hydrogen molecules (taken to be rigid). What is the molar specific heat of mixture at constant volume ? $(R=8.3J/molK)$

Q. The stopping potential $V_0$ (in volt) as a function of frequency $\left(v\right)$ for a sodium emitter, is shown in the figure. The work function of sodium, from the data plotted in the figure, will be :
(Given : Planck's constant $\left(h\right)=6.63\times {10}^{-34}Js$, electron charge $e=1.6\times {10}^{-19}C$)

Q. At ${40}^{o}C$, a brass wire of $1mm$ radius is hung from the ceiling. A small mass, $M$ is hung from the free end of the wire. When the wire is cooled down from ${40}^{o}C$ to ${20}^{o}C$ it regains its original length of $0.2m$. The value of $M$ is close to :
(Coefficient of linear expansion and Young's modulus of brass are ${10}^{-5}{/}^{o}C$ and ${10}^{11}N/m^2$, respectively; $g=10m{s}^{-2}$)

Q. A uniform rod of length $\ell$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is $T\left(x\right)$ at a distance $x$ from the axis, then which of the following graphs depicts it most closely?

Q. Which of the following combinations has the dimension of electrical resistance (${\epsilon }_0$ is the permittivity of vacuum and ${\mu }_0$ is the permeability of vacuum) ?

Q. In a double slit experiment, when a thin film of thickness $t$ having refractive index $\mu$ is introduced in front of one of the slits, the maximum at the centre of the fringe pattern shifts by one fringe width. The value of $t$ is ($\lambda$ is the wavelength of the light used) :

Q. An electromagnetic wave is represented by the electric field $\overrightarrow{E}=E_0\hat{n}\sin [\omega t+(6y-8z\left)\right]$. Taking unit vectors in $x,y$ and $z$ directions to be $\hat{i},\hat{j},\hat{k}$, the direction of propogation $\hat{s}$, is:

Q. The figure shows a square loop $L$ of side $5cm$ which is connected to a network of resistances. The whole setup is moving towards right with a constant speed of $1cm{s}^{-1}$. At some instant, a part of L is in a uniform magnetic field of $1T$, perpendicular to the plane of the loop. If the resistance of $L$ is $1.7\Omega$, the current in the loop at that instant will be close to :

Q. The transfer characteristic curve of a transistor, having input and output resistance $100\Omega$ and $100k\Omega$ respectively, is shown in the figure. The Voltage and Power gain, are respectively :

Q. A galvanometer of resistance $100\Omega$ has $50$ divisions on its scale and has sensitivity of $20\mu A$/division. It is to be converted to a voltmeter with three ranges, of $0-2V,0-10V$ and $0-20V$. The appropriate circuit to do so is :

Q. To verify Ohm's law, a student connects the voltmeter across the battery as, shown in the figure. The measured voltage is plotted as a function of the current, and the following graph is obtained:
If $V_0$ is almost zero, identify the correct statement:

Q. An excited $H{e}^{+}$ ion emits two photons in succession, with wavelengths $108.5nm$ and $30.4nm$, in making a transition to ground state. The quantum number $n$, corresponding to its initial excited state is (for photon of wavelength $\lambda$, energy $E=\frac{1240eV}{\lambda \left(innm\right)}$)

Q. A sample of an ideal gas is taken through the cyclic process $abca$ as shown in the figure. The change in the internal energy of the gas along the path $ca$ is - $180J$. The gas absorbs $250J$ of heat along the path ab and $60J$ along the path $bc$. The work done by the gas along the path $abc$ is :

Q. A thin ring of $10cm$ radius carries a uniformly distributed charge. The ring rotates at a constant angular speed of
$40\pi rad{s}^{-1}$ about its axis, perpendicular to its plane. If the magnetic field at its center is $3.8\times {10}^{-9}T$, then the
charge carried by the ring is close to $({\mu }_0=4\pi \times {10}^{-7}N/A^2)$

Q. A magnetic compass needle oscillates $30$ times per minute at a place where the dip is ${45}^o$, and $40$ times per minute where the dip is ${30}^o$. If $B_1$ and $B_2$ are respectively the total magnetic field due to the earth at the two places, then the ratio $B_1/B_2$ is best given by :

Q. A man (mass = 50 kg) and his son (mass = 20 kg) are standing on a frictionless surface facing each other. The man pushes his son so that he starts moving at a speed of $0.70m{s}^{-1}$ with respect to the man. The speed of the man with respect to the surface is :

Q. The resistive network shown below is connected to a D.C. source of $16V$. The power consumed by the network is $4$ Watt. The value of $R$ is: