Q. In Young's double slit experiment, the distance between slits and the screen is $1.0m$ and monochromatic light of $600nm$ is being used. A person standing near the slits is looking at the fringe pattern. When the separation between the slits is varied, the interference pattern disappears for a particular distance $d_0$ between the slits. If the angular resolution of the eye is $\frac{1}{60}$, the value of $d_0$ is close to:

1. $2mm$
2. $3mm$
3. $1mm$
4. $4mm$

Q. The ratio of work done by an ideal monoatomic gas to the heat supplied to it in an isobaric process is:

1. $\frac{2}{3}$
2. $\frac{3}{2}$
3. $\frac{3}{5}$
4. $\frac{2}{5}$

Q. The potential (in volts) of a charge distribution is given by
$V\left(z\right)=30-5z^2$ for $\left|z\right|\le 1m$
$V\left(z\right)=35-10\left|z\right|$ for $\left|z\right|\ge 1m$
$V\left(z\right)$ does not depend on $x$ and $y$. If this potential is generated by a constant charge per unit volume ${\rho }_0$ (in units of ${\epsilon }_0$) which is spread over a certain region, then choose the correct statement.

1. ${\rho }_0=20{\epsilon }_0$ for $\left|z\right|\le 1m$ and ${\rho }_0=0$ elsewhere
2. ${\rho }_0=40{\epsilon }_0$ in the entire region
3. ${\rho }_0=20{\epsilon }_0$ in the entire region
4. ${\rho }_0=10{\epsilon }_0$ for $\left|z\right|\le 1m$ and ${\rho }_0=0$ elsewhere

Q. An experiment is performed to determine the $I-V$ characteristics of a Zener diode, which has a protective resistance of $R=100\Omega$, and a maximum power of dissipation rating of $1W$. The minimum voltage range of the DC source in the circuit is

1. $0-24V$
2. $0-5V$
3. $0-12V$
4. $0-8V$

Q. A cubical block of side $30cm$ is moving with velocity $2{ms}^{-1}$ on a smooth horizontal surface. The surface has a bump at a point $O$ as shown in figure. The angular velocity (in $rad/s$) of the block immediately after it hits the bump, is:

1. $13.3$
2. $5.0$
3. $9.4$
4. $96.7$

Q. A rocket is fired vertically from the earth with an acceleration of $2g$, where $g$ is the gravitational acceleration. On an inclined plane inside the rocket, making an angle $\theta$ with the horizontal, a point object of mass $m$ is kept. The minimum coefficient of friction ${\mu }_{m}in$ between the mass and the inclined surface such that the mass does not move is:

1. $3\tan \theta$
2. $\tan 2\theta$
3. $\tan \theta$
4. $2\tan \theta$

Q. Figure shows elliptical path $abcd$ of a planet around the sun $S$ such that the area of triangle csa is $\frac{1}{4}$ the area of the ellipse. (See figure) With $db$ as the major axis, and $ca$ as the minor axis. If $t_1$ is the time taken for planet to go over path $abc$ and $t_2$ for path taken over $cda$ then

1. $t_1=t_2$
2. $t_1=2t_2$
3. $t_1=4t_2$
4. $t_1=3t_2$

Q. An audio signal consists of two distinct sounds : one a human speech signal in the frequency band of $200Hz$ to $2700Hz$, while the other is a high frequency music signal in the frequency band of $10200Hz$ to $15200Hz$. The ratio of the AM signal bandwidth required to send both the signals together to the AM signal bandwidth required to send just the human speed is:

1. $5$
2. $2$
3. $3$
4. $6$

Q. In the following $I$ refers to current and other symbols have their usual meaning. Choose the option that corresponds to the dimensions of electrical conductivity:

1. $M{L}^{-3}{T}^{-3}{I}^2$
2. $M^{-1}{L}^{-3}{T}^3{I}^2$
3. $M^{-1}{L}^3{T}^{3}I$
4. $M^{-1}{L}^{-3}{T}^{3}I$

Q. The truth table given in figure represents:

1. AND - Gate
2. NOR - Gate
3. NAND - Gate
4. OR - Gate

Q. Which of the following option correctly describes the variation of the speed $v$ and acceleration ${}^{\prime }{a}^{\prime }$ of a point mass falling vertically in a viscous medium that applies a force $F=-kv$, where ${}^{\prime }{k}^{\prime }$ is a constant, on the body? (Graphs are schematic and not drawn to scale)

1. 
   
2. 
   
3. 
   
4. 
   

Q. A $50\Omega$ resistance is connected to a battery of $5V$. A galvanometer of resistance $100\Omega$ is to be used as an ammeter to measure current through the resistance, for this a resistance $r_s$ is connected to the galvanometer. Which of the following connections should be employed if the measured current is within $1$% of the current without the ammeter in the circuit?

1. $r_s=1\Omega$ in parallel with galvanometer
2. $r_s=0.5\Omega$ in parallel with the galvanometer
3. $r_s=1\Omega$ in series with galvanometer
4. $r_s=0.5\Omega$ in series with the galvanometer

Q. Microwave oven acts on the principle of:

1. giving rotational energy to water molecules
2. transferring electrons from lower to higher energy levels in water molecule
3. giving vibrational energy to water molecules
4. giving translational energy to water molecules

Q. In the circuit shown, the resistance $r$ is a variable resistance. If for $r=fR$, the heat generation in $r$ is maximum then the value of $f$ is:

1. $1$
2. $\frac{1}{4}$
3. $\frac{1}{2}$
4. $\frac{3}{4}$

Q. Consider a water jar of radius $R$ that has water filled up to height $H$ and is kept on a stand of height $h$ (see figure). Through a hole of radius $r$ $(r\ll R)$ at its bottom, the water leaks out and the stream of water coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of water stream when it hits the ground is $x$. Then:

1. $x=r{\left(\frac{H}{H+h}\right)}^{\frac{1}{4}}$
2. $x=r{\left(\frac{H}{H+h}\right)}^2$
3. $x=r{\left(\frac{H}{H+h}\right)}^{\frac{1}{2}}$
4. $x=r\left(\frac{H}{H+h}\right)$

Q. A series LR circuit is connected to a voltage source with $V\left(t\right)=V_0\sin \Omega t$. After very large time, current $I\left(t\right)$ behaves as $(t_0\gg \frac{L}{R})$

1. 
   
2. 
   
3. 
   
4. 
   

Q. A convex lens, of focal length $30cm$, a concave lens of focal length $120cm$, and a plane mirror are arranged as shown. For an object kept at a distance of $60cm$ from the convex lens, the final image, formed by the combination, is a real image, at a distance of :

1. $60cm$ from the convex lens
2. $60cm$ from the concave lens
3. $70cm$ from the convex lens
4. $70cm$ from the concave lens

Q. A magnetic dipole is acted upon by two magnetic fields which are inclined to each other at an angle of ${75}^o$. One of the fields has a magnitude of $15mT$. The dipole attains stable equilibrium at an angle of ${30}^o$ with this field. The magnitude of the other field (in $mT$) is close to:

1. $36$
2. $11$
3. $1060$
4. $1$

Q. A uniformly tapering conical wire is made from a material of Young's modulus $Y$ and has a normal, unextended length $L$. The radii, at the upper and lower ends of this conical wire, have values $R$ and $3R$, respectively. The upper end of the wire is fixed to a rigid support and a mass $M$ is suspended from its lower end. The equilibrium extended length, of this wire, would equal to:

1. $L(1+\frac{2}{9}\frac{Mg}{\pi {\rm Y}{R}^2})$
2. $L(1+\frac{1}{9}\frac{Mg}{\pi {\rm Y}{R}^2})$
3. $L(1+\frac{1}{3}\frac{Mg}{\pi {\rm Y}{R}^2})$
4. $L(1+\frac{2}{3}\frac{Mg}{\pi {\rm Y}{R}^2})$

Q. A car of weight $W$ is on an inclined road that rises by $100m$ over a distance of $1km$ and applies a constant frictional force $\frac{W}{20}$ on the car. While moving uphill on the road at a speed of $10{ms}^{-1}$, the car needs power $P$. If it needs power $\frac{P}{2}$ while moving downhill at speed $v$ then value of $v$ is:

1. $15{ms}^{-1}$
2. $5{ms}^{-1}$
3. $10{ms}^{-1}$
4. $20{ms}^{-1}$

Q. Three capacitors each of $4\mu F$ are to be connected in such a way that the effective capacitance is $6\mu F$. This can be done by connecting them:

1. all in parallel
2. two in parallel and one in series
3. all in series
4. two in series and one in parallel

Q. A hydrogen atom makes a transition from $n=2$ to $n=1$ and emits a photon. This photon strikes a doubly ionized lithium atom $(z=3)$ in excited state and completely removes the orbiting electron. The least quantum number for the excited state of the ion for the process is

1. $3$
2. $2$
3. $5$
4. $4$

Q. Two engines pass each other moving in opposite directions with uniform speed of $30m/s$. One of them is blowing a whistle of frequency $540Hz$. Calculate the frequency heard by driver of second engine before they pass each other. Speed of sound is $330m/sec$.

1. $648Hz$
2. $540Hz$
3. $270Hz$
4. $450Hz$

Q. When photons of wavelength ${\lambda }_1$ are incident on an isolated sphere, the corresponding stopping potential is found to be $V$. When photons of wavelength ${\lambda }_2$ are used, the corresponding stopping potential was thrice that of the above value. If light of wavelength ${\lambda }_3$ is used then find the stopping potential for this case:

1. $\frac{hc}{e}[\frac{1}{{\lambda }_3}+\frac{1}{{\lambda }_2}-\frac{1}{{\lambda }_1}]$
2. $\frac{hc}{e}[\frac{1}{{\lambda }_3}+\frac{1}{{2\lambda }_2}-\frac{3}{{2\lambda }_1}]$
3. $\frac{hc}{e}[\frac{1}{{\lambda }_3}-\frac{1}{{\lambda }_2}-\frac{1}{{\lambda }_1}]$
4. $\frac{hc}{e}[\frac{1}{{\lambda }_3}+\frac{1}{2{\lambda }_2}-\frac{1}{{\lambda }_1}]$

Q. To know the resistance $G$ of a galvanometer by half deflection method, a battery of emf $V_E$ and resistance $R$ is used to deflect the galvanometer by angle $\theta$. If a shunt of resistance $S$ is needed to get half deflection then $G,R$ and $S$ are related by the equation

1. $2S=G$
2. $S(R+G)=RG$
3. $2S(R+G)=RG$
4. $2G=S$

Q. To find the focal length of a convex mirror, a student records the following data:
The focal length of the convex lens is $f_1$ and that of mirror is $f_2$. Then taking index correction to be negligibly small, $f_1$ and $f_2$ are close to :

1. $f_1=7.8cm{f}_2=12.7cm$
2. $f_1=7.8cm{f}_2=25.4cm$
3. $f_1=15.6cm{f}_2=25.4cm$
4. $f_1=12.7cm{f}_2=7.8cm$

Q. Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are same and equal to $A$ and $T$, respectively. At time $t=0$ one particle has displacement $A$ while the other one has displacement $\frac{-A}{2}$ and they are moving towards each other. If they cross each other at time $t$, then $t$ is:

1. $\frac{5T}{6}$
2. $\frac{T}{4}$
3. $\frac{T}{6}$
4. $\frac{T}{3}$

Q. A simple pendulum made of a bob of mass $m$ and a metallic wire of negligible mass has time period $2s$ at $T=0$. If the temperature of the wire is increased and the corresponding change in its time period is plotted against its temperature, the resulting graph is a line of slope $S$. If the coefficient of linear expansion of metal is $\alpha$ then the value of $S$ is

1. $2\alpha$
2. $\frac{1}{\alpha }$
3. $\alpha$
4. $\frac{\alpha }{2}$

Q. An unknown transistor needs to be identified as a $npn$ or $pnp$ type. A multimeter, with $+ve$ and $-ve$ terminals, is used to measure resistance between different terminals of transistor. If terminal $2$ is the base of the transistor then which of the following is correct for a $pnp$ transistor?

1. $+ve$ terminal $1$, $-ve$ terminal $2$, resistance high
2. $+ve$ terminal $3$, $-ve$ terminal $2$, resistance high
3. $+ve$ terminal $2$, $-ve$ terminal $3$, resistance low
4. $+ve$ terminal $2$, -$ve$ terminal $1$, resistance high

Q. $200g$ water is heated from ${40}^{o}C$ to ${60}^{o}C$. Ignoring the slight expansion of water, the change in its internal energy is close to:

1. $16.7kJ$
2. $167.4kJ$
3. $8.4kJ$
4. $4.2kJ$