Q. A given object taken $n$ times more time to slide down a ${45}^o$ rough inclined plane as it takes to slide down a perfectly smooth ${45}^o$ incline. The coefficient of kinetic friction between the object and the incline is :

1. $\frac{1}{2-n^2}$
2. $\sqrt{1-\frac{1}{n^2}}$
3. $1-\frac{1}{n^2}$
4. $\sqrt{\frac{1}{1-n^2}}$

Q. An ideal capacitor of capacitance $0.2\mu F$ is charged to a potential difference of $10V$. The charging battery is then disconnected. The capacitor is then connected to an ideal inductor of self inductance $0.5mH$. The current at a time when the potential difference across the capacitor is $5V$, is:

1. $0.17A$
2. $0.15A$
3. $0.34A$
4. $0.25A$

Q. In a common emitter configuration with suitable bias, it is given than $R_L$ is the load resistance and $R_{BE}$ is small signal dynamic resistance (input side). Then, voltage gain, current gain and power gain are given, respectively, by:
$\beta$ is current gain, $I_B,I_C,I_E$ are respectively base, collector and emitter currents:

1. ${\beta }^2\frac{R_L}{R_{BE}},\frac{\Delta I_C}{\Delta I_B},\beta \frac{R_L}{R_{BE}}$
2. $\beta \frac{R_L}{R_{BE}},\frac{\Delta I_E}{\Delta I_B},{\beta }^2\frac{R_L}{R_{BE}}$
3. ${\beta }^2\frac{R_L}{R_{BE}},\frac{\Delta I_C}{\Delta I_E},{\beta }^2\frac{R_L}{R_{BE}}$
4. $\beta \frac{R_L}{R_{BE}},\frac{\Delta I_C}{\Delta I_B},{\beta }^2\frac{R_L}{R_{BE}}$

Q. The energy required to remove the electron from a singly ionized Helium atom is $2.2$ times the energy required to remove an electron from Helium atom. The total energy required to ionize the Helium atom completely is:

1. $109eV$
2. $20eV$
3. $79eV$
4. $34eV$

Q. Light wavelength $550nm$ falls normally on a slit of width $22.0\times {10}^{-5}cm$. The angular position of the second minima from the central maximum will be (in radians)

1. $\frac{\pi }{8}$
2. $\frac{\pi }{12}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

Q. The equivalent capacitance between $A$ and $B$ in the circuit given below is:

1. $2.4\mu F$
2. $4.9\mu F$
3. $3.6\mu F$
4. $5.4\mu F$

Q. A solution containing active cobalt ${}_{27}^{60}Co$ having activity of $0.8\mu Ci$ and decay constant $\lambda$ is injected in an animal's body. If $1{cm}^3$ of blood is drawn from the animal's body after $10$ hrs of injection, the activity found was $300$ decays per minute. What is the volume of blood that is flowing in the body? ($1Ci=3.7\times {10}^{10}$ decay per second and at $t=10hrs$ $e^{-\lambda t}=0.84$)

1. $6$ litres
2. $7$ litres
3. $4$ litres
4. $5$ litres

Q. A force of $40N$ acts on a point $B$ at the end of an L-shaped object, as shown in the figure. The angle $\theta$ that will produce maximum moment of the force about point $A$ is given by:

1. $\tan \theta =2$
2. $\tan \theta =\frac{1}{4}$
3. $\tan \theta =4$
4. $\tan \theta =\frac{1}{2}$

Q. A monochromatic beam of light has a frequency $v=\frac{3}{2\pi }\times {10}^{12}Hz$ and is propagating along the direction $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$. It is polarized along the $\hat{k}$ direction. The acceptable form for the magnetic field is:

1. $\hat{k}\frac{E_0}{C}\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos [{10}^4\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right).\overrightarrow{r}-(3\times {10}^{12})t]$
2. $\frac{E_0}{C}\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos [{10}^4\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right).\overrightarrow{r}-(3\times {10}^{12})t]$
3. $\frac{E_0}{C}\hat{k}\cos [{10}^4\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right).\overrightarrow{r}+(3\times {10}^{12})t]$
4. $\frac{E_0}{C}\frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\cos [{10}^4\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right).\overrightarrow{r}+(3\times {10}^{12})t]$

Q. A uniform rod $AB$ is suspended from a point $X$, at a variable distance $x$ from $A$, as shown. To make the rod horizontal, a mass $m$ is suspended from its end $A$. A set of $(m,x)$ values is recorded. The appropriate variables that give a straight line, when plotted, are:

1. $m,\frac{1}{x}$
2. $m,\frac{1}{x^2}$
3. $m,x$
4. $m,x^2$

Q. Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de Broglie wavelengths are ${\lambda }_1$ and ${\lambda }_2$, their de Broglie wavelength in the frame of reference attached to their centre of mass is:

1. $\frac{1}{{\lambda }_{CM}}=\frac{1}{{\lambda }_1}+\frac{1}{{\lambda }_2}$
2. ${\lambda }_{CM}={\lambda }_1={\lambda }_2$
3. ${\lambda }_{CM}=\frac{2{\lambda }_1{\lambda }_2}{\sqrt{{{\lambda }_1}^2+{{\lambda }_2}^2}}$
4. ${\lambda }_{CM}=\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)$

Q. A planoconvex lens becomes an optical system of $28cm$ focal length when its plane surface is silvered and illuminated from left to right as shown in Fig-A.
If the same lens is instead silvered on the curved surface and illuminated from other side as in Fig. B, it acts like an optical system of focal length $10cm$. The refractive index of the material of lens if:

1. $1.50$
2. $1.55$
3. $1.75$
4. $1.51$

Q. A body of mass $M$ and charge $q$ is connected to a spring of spring constant $k$. It is oscillating along x-direction about its equilibrium position, taken to be at $x=0$, with an amplitude $A$. An electric field $E$ is applied along the x-direction. Which of the following statements is correct?

1. The total energy of the system is $\frac{1}{2}m{\omega }^2{A}^2+\frac{1}{2}\frac{q^2{E}^2}{k}$
2. The new equilibrium position is at a distance: $\frac{2qE}{k}$ from $x=0$
3. The new equilibrium position is at a distance: $\frac{qE}{2k}$ from $x=0$
4. The total energy of the system is $\frac{1}{2}m{\omega }^2{A}^2-\frac{1}{2}\frac{q^2{E}^2}{k}$

Q. A thin uniform tube is bent into a circle of radius $r$ in the vertical plane. Equal volumes of two immiscible liquids, whose densities are ${\rho }_1$ and ${\rho }_2$ $({\rho }_1>{\rho }_2)$ fill half the circle. The angle $\theta$ between the radius vector passing through the common interface and the vertical is:

1. $\theta ={\tan }^{-1}\left[\frac{\pi }{2}\left(\frac{{\rho }_1-{\rho }_2}{{\rho }_1+{\rho }_2}\right)\right]$
2. None of above
3. $\theta ={\tan }^{-1}\frac{\pi }{2}\left(\frac{{\rho }_1-{\rho }_2}{{\rho }_1+{\rho }_2}\right)$
4. $\theta ={\tan }^{-1}\pi \left(\frac{{\rho }_1}{{\rho }_2}\right)$

Q. The velocity-time graphs of a car and a scooter are shown in the figure. (i) the difference between the distance travelled by the car and the scooter in $15s$ and (ii) the time at which the car will catch up with the scooter are, respectively

1. $337.5m$ and $25s$
2. $225.5m$ and $10s$
3. $112.5m$ and $22.5s$
4. $\mathrm{11.2.5}m$ and $15s$

Q. An automobile, travelling at $40km/h$, can be stopped at a distance of $40m$ by applying brakes. If the same automobile is travelling at $80km/h$, the minimum stopping distance, in metres, is (assume no skidding)

1. $75m$
2. $160m$
3. $100m$
4. $150m$

Q. In the given circuit all resistances are of value $R$ ohm each. The equivalent resistance between $A$ and $B$ is:

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

Q. A particle is oscillating on the X-axis with an amplitude $2cm$ about the point $x_0=10cm$ with a frequency $\omega$. A concave mirror of focal length $5cm$ is placed at the origin (see figure)
Identify the correct statements:
(A) The image executes periodic motion
(B) The image executes no n-periodic motion
(C) The turning points of the image are asymmetric w.r.t the image of the point at $x=10cm$
(D) The distance between the turning points of the oscillation of the image is $\frac{100}{21}$

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

Q. A charge $Q$ is placed at a distance $a/2$ above the centre of the square surface of edge $a$ as shown in the figure.
The electric flux through the square surface is:

1. $\frac{Q}{6{ϵ}_0}$
2. $\frac{Q}{3{ϵ}_0}$
3. $\frac{Q}{2{ϵ}_0}$
4. $\frac{Q}{{ϵ}_0}$

Q. A body of mass $m$ is moving in a circular orbit of radius $R$ about a planet of mass $M$. At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius $\frac{R}{2}$, and the other mass, in a circular orbit of radius $\frac{3R}{2}$. The difference between the final initial total energies is:

1. $-\frac{GMm}{6R}$
2. $-\frac{GMm}{2R}$
3. $+\frac{GMm}{6R}$
4. $\frac{GMm}{2R}$

Q. A Carnot's engine works as a refrigerator between $250K$ and $300K$. It receives $500cal$ heat from the reservoir at the lower temperature.The amount of work done in each cycle to operate the refrigerator is:

1. $2100J$
2. $420J$
3. $772J$
4. $2520J$

Q. The relative error in the determination of the surface area of a sphere is $\alpha$. Then the relative error in the determination of its volume is :

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

Q. In a meter bridge, as shown in the figure, it is given that resistance $Y=12.5\Omega$ and that the balance is obtained at a distance $39.5cm$ from end $A$ (by jockey J). After interchanging the resistances $X$ and $Y$, a new balance point is found at a distance $l_2$ from end $A$. What are the values of $X$ and $l_2$?

1. $8.16\Omega$ and $39.5cm$
2. $19.15\Omega$ and $39.5cm$
3. $19.15\Omega$ and $60.5cm$
4. $8.16\Omega$ and $60.5cm$

Q. Take the mean distance of the moon and the sun from the earth to be $0.4\times {10}^{6}km$ and $150\times {10}^{6}km$ respectively. Their masses are $8\times {10}^{22}kg$ and $2\times {10}^{30}kg$ respectively. The radius of the earth is $6400km$. Let $\Delta F_1$ be the difference in the forces exerted by the moon at the nearest and farthest points on the earth and $\Delta F_2$ be the difference in the force exerted by the sun at the nearest and farthest points on the earth. Then, the number closest to $\frac{\Delta F_1}{\Delta F_2}$ is:

1. $2$
2. $6$
3. ${10}^{-2}$
4. $0.6$

Q. The B-H curve for a ferromagnet is shown in the figure. The ferromagnet is placed inside a ling solenoid with $1000$ turns/cm.. The current that should be passed in the solenoid to demagnetise the ferromagnet completely is:

1. $2mA$
2. $1mA$
3. $3mA$
4. None of these

Q. The number of amplitude modulated broadcast stations that can be accommodated in a $300kHz$ band width for the highest modulating frequency $15kHz$ will be:

1. $20$
2. $10$
3. $8$
4. $15$

Q. A Helmholtz coil has pair of loops, each with $N$ turns and radius $R$. They are placed coaxially at distance $R$ and the same current $I$ flows through the loops in the same direction. The magnitude of magnetic field at $P$, midway between the centres $A$ and $C$, is given by (Refer to figure):

1. $\frac{4N{\mu }_{0}I}{5^{3/2}R}$
2. $\frac{8N{\mu }_{0}I}{5^{3/2}R}$
3. $\frac{4N{\mu }_{0}I}{5^{1/2}R}$
4. $\frac{8N{\mu }_{0}I}{5^{1/2}R}$

Q. In a screw gauge, $5$ complete rotations of the screw cause it to move a linear distance of $0.25cm$. There are $100$ circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of $4$ main scale divisions and $30$ circular scale divisions. Assuming negligible zero error, the thickness of the wire is:

1. $0.0430cm$
2. $0.4300cm$
3. $0.3150cm$
4. $0.2150cm$

Q. A tuning fork vibrates with frequency $256Hz$ and gives one best per second with the third normal mode of vibration of an open pipe. What is the length of the pipe? (Speed of sound of air is $340m{s}^{-1}$)

1. $200cm$
2. $180cm$
3. $190cm$
4. $220cm$

Q. One mode of an ideal monoatomic gas is compressed isothermally in a rigid vessel to double its pressure at room temperature, ${27}^{o}C$. The work done on the gas will be:

1. $300R\ln 6$
2. $300R$
3. $300R\ln 7$
4. $300R\ln 2$