Q. When a rubber-band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=ax+b{x}^2$ where $a$ and $b$ are constants. The work done in stretching the unstretched rubber band by $L$ is

1. $\frac{a{L}^2}{2}+\frac{b{L}^3}{3}$
2. $\frac{1}{2}(\frac{a{L}^2}{2}+\frac{b{L}^3}{3})$
3. $a{L}^2+b{L}^3$
4. $\frac{1}{2}(a{L}^2+b{L}^3)$

Q. In the circuit shown here, the point 'C' is kept connected to point 'A' till the current flowing through the circuit becomes constant. Afterward, suddenly, point 'C' is disconnected from point 'A' and connected to point 'B' at time $t=0$. Ratio of the voltage across resistance and the inductor at $t$ $=L/R$ will be equal to:

1. $0$
2. $1$
3. $infinity$
4. $-1$

Q. A pipe of length 85 cm is closed from one end. Find the number of possible natural oscillations of the air column in the pipe whose frequencies lie below 1250 Hz. The velocity of sound in air is 340 m/s.

1. 8
2. 6
3. 4
4. 12

Q. A particle moves with simple harmonic motion in a straight line. In first $\tau$ s, after starting from rest, it travels a distance a, and in next $\tau$ s it travels 2a, in the same direction, then:

1. Amplitude of motion is 3a
2. Amplitude of motion is 4a
3. Time period of oscillations is 8$\tau$
4. Time period of oscillations is 6$\tau$

Q. A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

1. A screw gauge having 100 divisions in the circular scale and pitch as 1 mm.
2. A meter scale.
3. A screw gauge having 50 divisions in the circular scale and pitch as 1 mm.
4. A vernier calliper where the 10 divisions in vernier scale matches with 9 division in main scale and main scale has 10 divisions in 1 cm.

Q. A green light is incident from the water to the air - water interface at the critical angle $\left(\theta \right)$. Select the correct statement

1. the spectrum of visible light whose frequency is more than that of green light will come out to the air medium.
2. the spectrum of visible light whose frequency is less than that of green light will come out to the air medium.
3. the entire spectrum of visible light will come out of the water at various angle to the normal.
4. the entire spectrum of visible light will come out of the water at angle of 90${}^o$ to the normal.

Q. The forward biased diode connection is:

1. 
   
2. 
   
3. 
   
4. 
   

Q. The radiation corresponding to 3 $\to$ 2 transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field of $3\times {10}^{-4}$ T. If the radius of the largest circular path followed by these electrons is $10.0mm$, the work function of the metal is close to

1. $0.8eV$
2. $1.8eV$
3. $1.1eV$
4. $1.6eV$

Q. One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in figure. The process BC is adiabatic. The temperatures at A, B and C are $400K,800Kand600K$ respectively. Choose the correct statement:

1. The change in internal energy in the process AB is $-350R$.
2. The change in internal energy in the process BC is $-500R$.
3. The change in internal energy in the process CA is $700R$.
4. The change in internal energy in whole cyclin process is $250R$.

Q. A conductor lies along the z-axis at $-1.5m\le z<1.5m$ and carries a fixed current of $10.0A$ in $-{\hat{a}}_z$ direction (see figure). For a field $\overline{B}=3.0\times {10}^{-4}{e}^{-0.2x}{\hat{a}}_{y}T$ , the power required to move the conductor at constant speed to $x=2.0m$ and $y=0m$ in $5\times {10}^{-3}s$ is ( Assume parallel motion along the x-axis)

1. $14.85W$
2. $29.7W$
3. $1.57W$
4. $2.97W$

Q. A block of mass $m$ is placed on a surface with a vertical height given by $y=x^3/6.$ If the coefficient of friction is $0.5,$ the maximum height above the ground at which the block can be placed without slipping is

1. $\frac{1}{3}m$
2. $\frac{1}{2}m$
3. $\frac{1}{6}m$
4. $\frac{2}{3}m$

Q. The corecivity of a small magnet where the ferromagnet gets demagnetized is $3\times {10}^{3}A{m}^{-1}$. The current required to be passed in a solenoid of length $10cm$ and number of turns $100$, so that the magnet gets demagnetized when inside the solenoid is:

1. $30mA$
2. $3A$
3. $6A$
4. $60mA$

Q. A thin convex lens made from crown glass $(\mu =\frac{3}{2})$ has focal length f. When it is measured in two different liquids having refractive indices $\frac{4}{3}$ and $\frac{5}{3}$, it has the focal lengths $f_1$ and $f_2$ respectively. The correct relation between the focal lengths is:

1. $f_1$ and $f_2$ both become negative
2. $f_1>f$ and $f_2$ becomes negative
3. $f_1=f_2<f$
4. $f_2>f$ and $f_1$ becomes negative

Q. Three rods of Copper, Brass and Steel are welded together to from a Y-shaped structure. Area of cross section of each rod $=4c{m}^2$. End of copper rod is maintained at $100$${}^o$C where as ends of brass and steel are kept at $0^o$C. Lengths of the copper, brass and steel rods are $46,13$ and $12$ cms respectively. The rods are thermally insulated from surroundings except at ends. Thermal conductivities of copper, brass and steel are $0.92,0.26$ and $0.12$ CGS units respectively. Rate of heat flow through copper rod is:

1. $4.8cal/s$
2. $6.0cal/s$
3. $1.2cal/s$
4. $2.4cal/s$

Q. The current voltage relation of diode is given by $I=(e^{1000V/T}-1)$ mA, where the applied voltage V is in volts and the temperature T is in degree Kelvin. If a student makes an error measuring $\pm$ 0.01V while measuring the current of 5mA at 300 K, what will be the error in the value of current in mA?

1. 0.05 mA
2. 0.5 mA
3. 0.2 mA
4. 0.02 mA

Q. A parallel plat capacitor is made of two circular plates separated by a distance of $5mm$ and with a dielectric of dielectric constant $2.2$ between them. When the electric field in the dielectric is $3\times {10}^4$V/m, the charge density of the positive plate will be close to:

1. $3\times {10}^{4}C/m^2$
2. $6\times {10}^{4}C/m^2$
3. $6\times {10}^{-7}C/m^2$
4. $3\times {10}^{-7}C/m^2$

Q. During the propagation of electromagnetic waves in a medium:

1. Electric energy density is equal to the magnetic energy density
2. Electric energy density is double of the magnetic energy density
3. Both electric and magnetic energy densities are zero
4. Electric energy density is half of the magnetic energy density

Q. There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends ${90}^o$ angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical . The ratio $d_1/d_2$ is

1. $\frac{1+\tan \alpha }{1-\tan \alpha }$
2. $\frac{1+\sin \alpha }{1-\sin \alpha }$
3. $\frac{1+\sin \alpha }{1-\cos \alpha }$
4. $\frac{1+\cos \alpha }{1-\cos \alpha }$

Q. A mass '$m$' is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

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

Q. Assume that an electric field $\overline{E}=30x^2\hat{i}$ exists in space. Then the potential difference $V_A-V_O$, where $V_O$ is the potential at the origin and $V_A$ the potential at $x=2$m is :

1. $-80V$
2. $80V$
3. $120V$
4. $-120V$

Q. A bob of mass, $m$ attached to an inextensible string of length, $l$, is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed, $\omega rad/s$ about the vertical. About the point of suspension:

1. angular momentum changes both in direction and magnitude
2. angular momentum changes in direction but not in magnitude
3. angular momentum is conserved
4. angular momentum changes in magnitude but not in direction

Q. Hydrogen ${(}_1{H}^1)$, Deuterium ${(}_1{H}^2)$, singly ionised Helium ${(}_{2}H{e}^4{)}^{+}$ and double ionised lithium ${(}_{3}L{i}^6{)}^{++}$ all have one electron around the nucleus. Consider an electron transition from $n=2$ to $n=1$. If the wave lengths of emitted radiation are ${\lambda }_1,{\lambda }_2,{\lambda }_3$ and ${\lambda }_4$ respectively then approximately which one of the following is correct?

1. $\lambda ={\lambda }_2=4{\lambda }_3=9{\lambda }_4$
2. ${\lambda }_1=2{\lambda }_2=3{\lambda }_3=4{\lambda }_4$
3. $4{\lambda }_1=2{\lambda }_2=2{\lambda }_3={\lambda }_4$
4. ${\lambda }_1=2{\lambda }_2=2{\lambda }_3={\lambda }_4$

Q. In a large building, there are 15 bulbs of 40 W, 5 bulbs of 100 W, 5 fans of 80 W and 1 heater of 1kW. The voltage of the electric mains is 220 V. The minimum capacity of the main fuse of the building will be:

1. 12 A
2. 14 A
3. 8 A
4. 10 A

Q. The pressure that has to be applied to the ends of a steel wire of length $10cm$ to keep its length constant when its temperature is raised by 100${}^o$C is:
(For steel Young's modulus is $2\times {10}^{11}N{m}^{-2}$ and coefficient of thermal expansion is $1.1\times {10}^{-5}{K}^{-1}$)

1. $2.2\times {10}^7$Pa
2. $2.2\times {10}^6$Pa
3. $2.2\times {10}^8$Pa
4. $2.2\times {10}^9$Pa

Q. An open glass tube is immersed in mercury in such a way that a length of 8 cm extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically up by additional 46 cm. What will be length of the air column above mercury in the tube now? (Atmospheric pressure $=$ 76 cm of Hg)

1. 38 cm
2. 16 cm
3. 6 cm
4. 22 cm

Q. On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius $R$ and making a circular contact of radius r with the bottom of the vessel. If $r<<R$, and the surface tension of water is $T$, value of $r$ just before bubbles detach is: (density of water is ${\rho }_w$)

1. $R^2\sqrt{\frac{{\rho }_{w}g}{T}}$
2. $R^2\sqrt{\frac{3{\rho }_{w}g}{T}}$
3. $R^2\sqrt{\frac{{\rho }_{w}g}{6T}}$
4. $R^2\sqrt{\frac{2{\rho }_{w}g}{3T}}$

Q. Four particles, each of mass $M$ and equidistant from each other, move along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle is:

1. $\sqrt{\frac{GM}{R}(1+2\sqrt{2})}$
2. $\sqrt{\frac{GM}{R}}$
3. $\sqrt{2\sqrt{2}\frac{GM}{R}}$
4. $\frac{1}{2}\sqrt{\frac{GM}{R}(1+2\sqrt{2})}$

Q. From a tower of height $H$, a particle is thrown vertically upwards with a speed $u$. The time taken by the particle, to hit the ground, is $n$ times that taken by it to reach the highest point of its path. The relation between $H$, $u$ and $n$ is:

1. $2gH=n{u}^2(n-2)$
2. $gH=(n-2)u^2$
3. $2gH=n^2{u}^2$
4. $gH=(n-2{)}^2{u}^2$

Q. Two beams, A and B, of plane polarized light with mutually perpendicular planes of polarization are seen through a polaroid. From the position when the beam A has maximum intensity (and beam B has zero intensity), a rotation of Polaroid through 30${}^o$ makes the two beams appear equally bright. If the initial intensities of the two beams are $I_A$ and $I_B$ respectively, then $I_A/I_B$ equals:

1. $I/3$
2. $I$
3. $3$
4. $3/2$

Q. Match List-I (Electromagnetic wave type) with List-II (Its association/application) and select the correct option from the choices given below the lists:

1. a - iii, b - ii, c - i, d - iv
2. a - iv, b - iii, c - ii, d - i
3. a - i, b - ii, c - iii, d - iv
4. a - i, b - ii, c - iv, d - iii