Q. Let $N_{\beta }$ be the number of $\beta$ particles emitted by $1$ gram of ${}^{24}Na$ radioactive nuclei (half life $=15hrs)$ in $7.5$ hours, $N_{\beta }$ is close to (Avogadro number $=6.023\times {10}^{23}/g$ mole)

1. $1.75\times {10}^{22}$
2. $6.2\times {10}^{21}$
3. $7.5\times {10}^{21}$
4. $1.25\times {10}^{22}$

Q. A wire carrying current I is tied between points P and Q and is in the shape of a circular arc of radius R due to a uniform magnetic field B (perpendicular to the plane of the paper, shown by xxx) in the vicinity of the wire. If the wire subtends an angle $2{\theta }_0$ at the centre of the circle (of which it forms an arch) then the tension in the wire is :

1. $\frac{IBR}{2\sin {\theta }_0}$
2. $IBR$
3. $\frac{IBR{\theta }_0}{\sin {\theta }_0}$
4. $\frac{IBR}{\sin {\theta }_0}$

Q. Using equipartition of energy, the specific heat (in $Jk{g}^{-1}{K}^{-1})$ of aluminium at high temperature can be estimated to be (atomic weight of aluminium $=27$)

1. 25
2. 1850
3. 410
4. 923

Q. A thin convex lens of focal length '$f$' is put on a plane mirror as shown in the figure. When an object is kept at a distance 'a' from the lens - mirror combination, its image is formed at a distance $\frac{a}{3}$ in front of the combination. The value of 'a' is

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

Q. A source of sound emits sound waves at frequency $f_0$. It is moving towards an observer with fixed speed $v_s(v_s<v$, where v is the speed of sound in air). If the observer were to move towards the source with speed $v_0$, one of the following two graphs (A and B) will give the correct variation of the frequency f heard by the observer as $v_0$ is changed.
The variation of f with $v_0$ is given correctly by :

1. graph A with slope $=\frac{f_0}{(v-v_s)}$
2. graph B with slope $=\frac{f_0}{(v+v_s)}$
3. graph A with slope $=\frac{f_0}{(v+v_s)}$
4. graph B with slope $=\frac{f_0}{(v-v_s)}$

Q. A large number (n) of identical beads, each of mass m and radius r are strung on a thin smooth rigid horizontal rod of length L (L >> r) and are at rest at random positions. The rod is mounted between two rigid supports (see figure). If one of the beads is now given a speed v, the average force experienced by each support after a long time is (assume all collisions are elastic)

1. $\frac{m{v}^2}{L-nr}$
2. $\frac{m{v}^2}{L-2nr}$
3. $\frac{m{v}^2}{2(L-nr)}$
4. $zero$

Q. Which of the following most closely depicts the correct variation of the gravitation potential $V\left(r\right)$ due to a large planet of radius $R$ and uniform mass density? (figures are not drawn to scale)

1. 
   
2. 
   
3. 
   
4. 
   

Q. A particle is moving in a circle of radius r under the action of a force $F=\alpha r^2$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy $=0$ for $r=0)$

1. $\frac{1}{2}\alpha r^3$
2. $\frac{5}{6}\alpha r^3$
3. $\frac{4}{3}\alpha r^3$
4. $\alpha r^3$

Q. A uniform thin rod AB of length L has linear mass density $\mu \left(x\right)=a+\frac{bx}{L}$, where x is measured from A. If the CM of the rod lies at a distance of $\left(\frac{7}{12}L\right)$ from A, then a and b are related as

1. $2a=b$
2. $a=2b$
3. $a=b$
4. $3a=2b$

Q. The de - Broglie wavelength associated with the electron in the $n=4$ level is :

1. half of the de-Broglie wavelength of the electron in the ground state
2. four times the de-Broglie wavelength of the electron in the ground state
3. $1/4^{th}$ of the de-Broglie wavelength of the electron in the ground state
4. two times the de-Broglie wavelength of the electron in the ground state

Q. A cylindrical block of wood $(density=650kg{m}^{-3})$, of base area $30c{m}^2$ and height $54cm$, floats in a liquid of density $900kg$ $m^{-3}$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly)

1. $52cm$
2. $26cm$
3. $39cm$
4. $65cm$

Q. An experiment takes $10$ minutes to raise the temperature of water in a container from $0^{o}C$ to ${100}^{o}C$ and another $55$ minutes to convert it totally into steam by a heater supplying heat at a uniform rate. Neglecting the specific heat of the container and taking specific heat of water to be $1cal/g^{o}C$, the heat of vapourization according to this experiment will come out to be

1. 530 cal/g
2. 540 cal/g
3. 550 cal/g
4. 560 cal/g

Q. A short bar magnet is placed in the magnetic meridian of the earth with north pole pointing north. Neutral points are found at a distance of 30 cm from the magnet on the East - West line, drawn through the middle point of the magnet. The magnetic moment of the magnet in $A{m}^2$ is close to (Given $\frac{{\mu }_0}{4\pi }={10}^{-7}$ in SI units and $B_H=$ Horizontal component of earth's magnetic field $=3.6\times {10}^{-5}Tesla.)$

1. 4.9
2. 14.6
3. 19.4
4. 9.7

Q. A vector $\overrightarrow{A}$ is rotated by a small angle $\Delta \theta$ radians $(\Delta \theta <<1)$ to get a new vector $\overrightarrow{B}$. In that case $|\overrightarrow{B}-\overrightarrow{A}|$ is

1. 0
2. $|\overrightarrow{A}|\Delta \theta$
3. $|\overrightarrow{A}|(1-\frac{\Delta {\theta }^2}{2})$
4. $|\overrightarrow{B}|\Delta \theta -|\overrightarrow{A}|$

Q. A beaker contains a fluid of density $\rho kg/m^3$, specific heat $S$ J /$k{g}^{o}C$ and viscosity $\eta$. The beaker is filled up to height h. To estimate the rate of heat transfer per unit area $\left(\stackrel{\cdot }{Q}/A\right)$ by convection when beaker is put on a hot place, a student proposed that it should depend on $\eta$, $\left(\frac{S\Delta \theta }{h}\right)$ and $\left(\frac{1}{\rho g}\right)$ when $\Delta \theta$ (in ${}^{o}C)$ is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for $\left(\stackrel{\cdot }{Q}/A\right)$ is

1. $\left(\frac{S\Delta \theta }{\eta h}\right)\left(\frac{1}{\rho g}\right)$
2. $\frac{S\Delta \theta }{\eta h}$
3. $\eta \frac{S\Delta \theta }{h}$
4. $\eta \left(\frac{S\Delta \theta }{h}\right)\left(\frac{1}{\rho g}\right)$

Q. A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is 12 rad/s, the magnitude of its angular momentum about a point on the ground right under the centre of the circle is

1. $14.4kg{m}^2{s}^{-1}$
2. $11.52kg{m}^2{s}^{-1}$
3. $20.16kg{m}^2{s}^{-1}$
4. $8.64kg{m}^2{s}^{-1}$

Q. If electron charge e, electron mass m, speed of light in vacuum c and Planck's constant h are taken as fundamental constant h are taken as fundamental quantities, the permeability of vacuum ${\mu }_0$ can be expressed in units of

1. $\left(\frac{m{c}^2}{h{e}^2}\right)$
2. $\left(\frac{h}{m{e}^2}\right)$
3. $\left(\frac{hc}{m{e}^2}\right)$
4. $\left(\frac{h}{c{e}^2}\right)$

Q. Unpolarized light of intensity $I_0$ is incident on surface of a block of a glass of Brewster's angle. In that case, which one of the following statement is true?

1. transmitted light is partially polarized with intensity $I_0/2$
2. transmitted light is completely polarized with intensity less than $I_0/2$
3. reflected light is partially polarized with intensity $I_0/2$
4. reflected light is completely polarized with intensity less than $I_0/2$

Q. A wire, of length $L(=20cm)$, is bent into a semi-circular arc. If the two equal halves, of the arc, were each to be uniformly charged with charges $\pm Q,[|Q|={10}^3{ϵ}_0$ Coulomb where ${ϵ}_0$ is the permittivity (in SI units) of free space] the net electric field at the centre O of the semi-circular arc would be :

1. $(25\times {10}^{3}N/C)\hat{j}$
2. $(25\times {10}^{3}N/C)\hat{i}$
3. $(50\times {10}^{3}N/C)\hat{j}$
4. $(50\times {10}^{3}N/C)\hat{i}$

Q. In the electric network shown, when no current flows through the $4\Omega$ resistor in the arm $EB$ the potential difference between the points $A$ and $D$ will be

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

Q. The AC voltage across a resistance can be measured using a

1. potentiometer
2. moving coil galvanometer
3. hot wire voltmeter
4. moving magnet galvanometer

Q. A $2V$ battery is connected across AB as shown in the figure. The value of the current supplied by the battery when in one case, battery's positive terminal is connected to A and in other case when positive terminal of battery is connected to B will respectively be

1. $0.2A$ and $0.1A$
2. $0.1A$ and $0.2A$
3. $0.2A$ and $0.4A$
4. $0.4A$ and $0.2A$

Q. A pendulum with time period of $1s$ is losing energy due to damping. At certain time its energy is $45J$. If after completing $15$ oscillations, its energy has become $15J$, its damping constant (in $s^{-1}$) is:

1. $\frac{1}{2}$
2. $\frac{2}{15}ln3$
3. $\frac{1}{30}ln3$
4. $2$

Q. In figure is shown a system of four capacitors connected across a $10V$ battery. Charge that will flow from switch $S$ when it is closed is :

1. zero
2. $20\mu C$ from a to b
3. $5\mu C$ from b to a
4. $5\mu C$ from a to b

Q. If a Young's double slit experiment with light of wavelength $\lambda$ the separation of slits is d and distance of screen is $D$ such that $D>>d>>\lambda$. If the Fringe width is $\beta$, the distance from point of maximum intensity to the point where intensity falls to half of maximum intensity on either side is

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

Q. An electric field $\overrightarrow{E}=(25\hat{i}+30\hat{j})N{C}^{-1}$ exists in a region of space. If the potential at the origin is taken to be zero then the potential at $x=2m,y=2m$ is

1. $-110J$
2. $-140J$
3. $-130J$
4. $-120J$

Q. The value of the resistor, $R_S$, needed in the DC voltage regulator circuit shown here, equals

1. $\frac{(V_i+V_L)}{(n+1)I_L}$
2. $\frac{(V_i+V_L)}{n{I}_L}$
3. $\frac{(V_i-V_L)}{(n+1)I_L}$
4. $\frac{(V_i-V_L)}{n{I}_L}$

Q. Two long straight parallel wires, carrying (adjustable) current $I_1$ and $I_2$, are kept at a distance d apart. If the force 'F' between the two wires is taken as 'positive' when the wires repel each other and 'negative' when the wires attract each other, the graph showing the dependence of 'F', on the product $I_1{I}_2$, would be

1. 
   
2. 
   
3. 
   
4. 
   

Q. For plane electromagnetic waves propagating in the z direction, which one of the following combination gives the correct possible direction for $\overrightarrow{E}$ and $\overrightarrow{B}$ field respectively?

1. $(2\hat{i}+3\hat{j})$ and $(\hat{i}+2\hat{j})$
2. $(3\hat{i}+4\hat{j})$ and $(4\hat{i}-3\hat{j})$
3. $(\hat{i}+2\hat{j})$ and $(2\hat{i}-\hat{j})$
4. $(-2\hat{i}-3\hat{j})$ and $(3\hat{i}-2\hat{j})$

Q. For the LCR circuit, shown here, the current is observed to lead the applied voltage. An additional capacitor C', when joined with the capacitor C present in the circuit, makes the power factor of the circuit unity. The capacitor C', must have been connected in parallel with C and has a magnitued

1. $\frac{1-{\omega }^{2}LC}{{\omega }^{2}L}$ parallel with C
2. $\frac{C}{({\omega }^{2}LC-1)}$
3. $\frac{C}{({\omega }^{2}LC-1)}$ parallel with C
4. $\frac{1-{\omega }^{2}LC}{{\omega }^{2}L}$ series with C