Q. A man in a car at location $Q$ on a straight highway is moving with speed $v$. He decides to reach a point $P$ in a field at a distance $d$ from highway (point $M$) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance $RM$, so that the time taken to reach $P$ is minimum?

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

Q. A solid ball of radius $R$ has a charge density $\rho$ given by $\rho ={\rho }_0(1-r/R)$ for $0\le r\le R$. The electric field outside the ball is:

1. $\frac{4rh{o}_0{R}^3}{3{\in }_0{r}^2}$
2. $\frac{3{\rho }_0{R}^3}{4{\in }_0{r}^2}$
3. $\frac{{\rho }_0{R}^3}{12{\in }_0{r}^2}$
4. $\frac{{\rho }_0{R}^3}{{\in }_0{r}^2}$

Q. $5$ beats/ second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either $0.95m$ or $1m$. The frequency of the fork will be:

1. $195Hz$
2. $300Hz$
3. $251Hz$
4. $150Hz$

Q. The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants $G$, $h$ and $c$. Which of the following correctly gives the Planck length?

1. ${\left(\frac{G\hslash }{c^3}\right)}^{\frac{1}{2}}$
2. $G^2\hslash c$
3. $G^{\frac{1}{2}}{\hslash }^{2}c$
4. $G{\hslash }^2{c}^3$

Q. Truth table for the given circuit will be

1. 
   
2. 
   
3. 
   
4. 
   

Q. When an air bubble of radius $r$ rises from the bottom to the surface of a lake, its radius becomes $\frac{5r}{4}$. Taking the atmospheric pressure to be equal to $10m$ height of water column, the depth of the lake would approximately be (ignore the surface tension and the effect of temperature):

1. $10.5m$
2. $8.7m$
3. $11.2m$
4. $9.5m$

Q. Two Carnot engines $A$ and $B$ are operated in series. Engine $A$ receives heat from a reservoir at $600K$ and rejects heat to a reservoir heat to reservoir at temperature $T$. Engine $B$ receives heat rejected by engine $A$ and in turn rejects it to a reservoir at $100K$. If the efficiencies of the two engines $A$ and $B$ are represented by ${\eta }_A$ and ${\eta }_B$ respectively, then what is the value of $\frac{{\eta }_B}{{\eta }_A}$

1. $\frac{12}{7}$
2. $\frac{12}{5}$
3. $\frac{5}{12}$
4. $\frac{7}{12}$

Q. A plane polarized light is incident on a polariser with its pass axis making angle $\theta$ with x-axis, as shown in the figure. At four different values of $\theta ,\theta =8^o,{38}^o,{188}^o$ and ${218}^o$, the observed intensities are same. What is the angle between the direction of polarization and x-axis

1. ${203}^o$
2. ${45}^o$
3. ${98}^o$
4. ${128}^o$

Q. A capacitor $C_1=1.0\mu F$ is charged up to a voltage $V=60V$ by connecting it to battery $B$ through switch (1), Now $C_1$ is disconnected from battery and connected to a circuit consisting of two uncharged capacitors $C_2=3.0\mu F$ and $C_3=6.0\mu F$ through a switch (2) as shown in the figure. The sum of final charges on $C_2$ and $C_3$ is:

1. $40\mu C$
2. $36\mu C$
3. $20\mu C$
4. $54\mu C$

Q. A disc rotates about its axis of symmetry in a horizontal plane at a steady rate of $3.5$ revolutions per second. A coin placed at a distance of $1.25cm$ from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is $(g=10m/s^2)$

1. $0.5$
2. $0.7$
3. $0.3$
4. $0.6$

Q. A proton of mass $m$ collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of ${90}^o$ with respect to each other. The mass of unknown particle is:

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

Q. As shown in the figure, forces of ${10}^{5}N$ each are applied in opposite directions, on the upper and lower faces of a cube of side $10cm$, shifting the upper face parallel to itself by $0.5cm$. If the side of another cube of the same material is $20cm$, then under similar conditions as above, the displacement will be:

1. $1.00cm$
2. $0.25cm$
3. $0.37cm$
4. $0.75cm$

Q. A thin rod $MN$, free to rotate in the vertical plane about the fixed end $N$, is held horizontal. When the end $M$ is released the speed of this end, when the rod makes an angle $\alpha$ with the horizontal, will be proportional to: (see figure)

1. $\sqrt{\cos \alpha }$
2. $\cos \alpha$
3. $\sin \alpha$
4. $\sqrt{\sin \alpha }$

Q. A plane polarized monochromatic EM wave is travelling a vacuum along $z$ direction such that at $t=t_1$ it is found that the electric field is zero at a spatial point $z_1$. The next zero that occurs in its neighborhood. is at $z_2$. The frequency of the electromagnetic wave is:

1. $\frac{3\times {10}^8}{|z_2-z_1|}$
2. $\frac{6\times {10}^8}{|z_2-z_1|}$
3. $\frac{1.5\times {10}^8}{|z_2-z_1|}$
4. $\frac{1}{t_1+\frac{|z_2-z_1|}{3\times {10}^8}}$

Q. An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of $8:27$. The ratio of the radii of the nuclei (assumed to be spherical ) is:

1. $2:3$
2. $3:2$
3. $4:9$
4. $8:27$

Q. The value closest to the thermal velocity of a Helium atom at room temperature ($300K$) in ${ms}^{-1}$ is:
$[k_B=1.4\times {10}^{-23}J/K:m_{He}=7\times {10}^{-27}kg]$

1. $1.3\times {10}^5$
2. $1.3\times {10}^4$
3. $1.3\times {10}^3$
4. $1.3\times {10}^2$

Q. The carrier frequency of a transmitter is provided by a tank circuit of a coil of inductance $49\mu H$ and a capactiance of $2.5nF$. It is modulated by an audio signal of $12kHz$. The frequency range occupied by the side bands is:

1. $63kHz-75kHz$
2. $18kHz-30kHz$
3. $13482kHz-13494kHz$
4. $442kHz-466kHz$

Q. A parallel plate capacitor with area $200{cm}^2$ and separation between the plates $1.5cm$, is connected across a battery of emf $V$. If the force of attraction between the plates is $25\times {10}^{-6}N$, the value of $V$ is approximately:
$({ϵ}_0=8.85\times {10}^{--12}\frac{C^2}{N.m^2})$

1. $150V$
2. $100V$
3. $300V$
4. $250V$

Q. At the centre of a fixed large circular coil of radius $R$, a much smaller circular coil of radius $r$ is placed. The two coils are concentric and are in the same plane. The larger coil carries a current $I$. The smaller coil is set to rotate with a constant angular velocity $\omega$ about an axis along their common diameter. Calculate the emf induced in the smaller coil after a time $t$ of its start of rotation.

1. $\frac{{\mu }_{0}I}{2R}\omega r^2\sin \omega t$
2. $\frac{{\mu }_{0}I}{4R}\omega \pi r^2\sin \omega t$
3. $\frac{{\mu }_{0}I}{2R}\omega \pi r^2\sin \omega t$
4. $\frac{{\mu }_{0}I}{4R}\omega r^2\sin \omega t$

Q. A thin uniform bar of length $L$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ moving in the same horizontal plane from opposite sides of the bar with speeds $2v$ and $v$ respectively. The masses stick to the bar after collision at a distance $\frac{L}{3}$ and $\frac{L}{6}$ respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be:

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

Q. Two simple harmonic motions, as shown, are at right angles. They are combined to form Lissajous figures.
$x\left(t\right)=A\sin (at+\delta )$
$y\left(t\right)=B\sin \left(bt\right)$
Identify the correct match below

1. Parameters: $A=B,a=2b;\delta =\frac{\pi }{2}$; Curve: Circle
2. Parameters: $A=B,a=b;\delta =\frac{\pi }{2}$; Curve: Line
3. Parameters: $A\ne B,a=b;\delta =0$; Curve: Parabola
4. Parameters: $A\ne B,a=b;\delta =\frac{\pi }{2}$; Curve: Ellipse

Q. A current of $1A$ is flowing on the sides of an equilateral triangle of side $4.5\times {10}^{-2}m$. The magnetic field at the centre of the triangle will be:

1. Zero
2. $2\times {10}^{-5}Wb/m^2$
3. $4\times {10}^{-5}Wb/m^2$
4. $8\times {10}^{-5}Wb/m^2$

Q. A constant voltage is applied between two ends of a metallic wire. If the length is halved and the radius of the wire is doubled, the rate of heat developed in the wire will be:

1. Increased 8 times
2. Unchanged
3. Halved
4. Doubled

Q. A body of mass $2kg$ slides down with an acceleration of $3m/s^2$ on a rough inclined plane having a slope of ${30}^o$. The external force required to take the same body up the plane with the same acceleration will be: $(g=10m/s^2)$

1. $14N$
2. $4N$
3. $6N$
4. $20N$

Q. A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of $10cm$. The separation between the two lenses is $2cm$. The focal lengths of the component lenses

1. $18cm,20cm$
2. $10cm,12cm$
3. $12cm,14cm$
4. $16cm,18cm$

Q. A body takes $10$ minutes to cool from ${60}^{o}C$ to ${50}^{o}C$. The temperature of surroundings is constant at ${25}^{o}C$. Then, the temperature of the body after next $10$ minutes will be approximately

1. ${43}^{o}C$
2. ${41}^{o}C$
3. ${47}^{o}C$
4. ${45}^{o}C$

Q. A copper rod of mass $m$ slides under gravity on two smooth parallel rails, with separation $1$ and set at an angle of $\theta$ with the horizontal. At the bottom, rails are joined by a resistance $R$.There is a uniform magnetic field $B$ normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is:

1. $\frac{mgR\sin \theta }{B^2{l}^2}$
2. $\frac{mgR\tan \theta }{B^2{l}^2}$
3. $\frac{mgR\cos \theta }{B^2{l}^2}$
4. $\frac{mgR\cot \theta }{B^2{l}^2}$

Q. Muon $\left({\mu }^{-1}\right)$ is negatively charged $(\left|q\right|=\left|e\right|)$ with a mass $m_{\mu }=200m_e$, where $m_e$ is the mass of the electron and $e$ is the electronic charge. If ${\mu }^{-1}$ is bound to a proton to form a hydrogen like atom, identify the correct statements
(A) Radius of the muonic orbit is $200$ times smaller than that of the electron
(B) the speed of the ${\mu }^{-1}$ in the nth orbit is $\frac{1}{200}$ times that of the electron in the nth orbit
(C) The ionization energy of muonic atom is $200$ times more than that of an hydrogen atom
(D) The momentum of the muon in the nth orbit is $200$ times more than that of the electron

1. (A), (B), (D)
2. (A), (C), (D)
3. (B), (D)
4. (C), (D)

Q. If the de Broglie wavelengths associated with a proton and an $\alpha -$ particle are equal, then the ratio of velocities of the proton and the $\alpha -$ particle will be:

1. $1:4$
2. $4:1$
3. $1:2$
4. $2:1$

Q. A copper rod of cross-sectional area $A$ carries a uniform current $I$ through if. At temperature $T$, if the volume charge density of the rod is $\rho$, how long will the charges take to travel a distance $d$?

1. $\frac{2\rho dA}{I}$
2. $\frac{\rho dA}{I}$
3. $\frac{\rho dA}{IT}$
4. $\frac{2\rho dA}{IT}$