MikaStars39/nano-eval huggingface dataset:
There are three points $A, B$ and $C$ on a horizontal plane; such that $AB = AC = 100m$. A vertical tower is placed on the midpoint of $BC$; such that angle of elevation of the top of the tower from $A$ is $\cot^{-1}(3\sqrt{2})$. and that from $B$ is $cosec^{-1}(2\sqrt{2})$. then the height of the tower is

Options:
A. \dfrac{100}{4}
B. \dfrac{100}{5}
C. \dfrac{100}{2}
D. \dfrac{100}{3}
E. $$10$$
F. $$20$$
G. $$25$$
H. $$\dfrac{100}{\sqrt{3}}$$
I. \dfrac{100}{7}
J. \dfrac{100}{6}
Answer: F
The same current is passed through acidulated water and stannous chloride solution. What volume of dry detonating gas at $ 0^{\circ}C $ and 1 atm is evolved from water, when 1.20 g of tin is deposited from the other solution? (Sn = 120)

Options:
A. 504 ml
B. 336 ml
C. 672 ml
D. 560 ml
E. 266 ml
F. 112 ml
G. 168 ml
H. 448 ml
I. 308 ml
J. 224 ml
Answer: B
A Metallic sphere floats in an immiscible mixture of water $\left( { P }_{ w }={ 10 }^{ 3 }{ kg/m }^{ 3 } \right) $and a liquid $\left( { p }_{ l }={ 13.5\times 10 }^{ 3 }{ kg/m }^{ 3 } \right) $ such that its$\frac { 4 }{ 5 } $th portion is in water and $\frac { 1 }{ 5 } $th portion is in liquid. Density of the metal is:

Options:
A. $$1.9\times { 10 }^{ 3 }kg/{ m }^{ 3 }$$
B. $$3.5\times { 10 }^{ 3 }kg/{ m }^{ 3 }$$
C. 3.0 \times { 10 }^{ 3 }kg/{ m }^{ 3 }
D. $$4.0\times { 10 }^{ 3 }kg/{ m }^{ 3 }$$
E. 4.2\times { 10 }^{ 3 }kg/{ m }^{ 3 }
F. 5.0 \times { 10 }^{ 3 }kg/{ m }^{ 3 }
G. 4.7 \times { 10 }^{ 3 }kg/{ m }^{ 3 }
H. 4.6 \times { 10 }^{ 3 }kg/{ m }^{ 3 }
I. 4.3 \times { 10 }^{ 3 }kg/{ m }^{ 3 }
J. $$4.5\times { 10 }^{ 3 }kg/{ m }^{ 3 }$$
Answer: J
A sample LR circuit is connected to a battery at a time $t=0$. The energy stored in the inductor reaches half its maximum value at time

Options:
A. \cfrac { R }{ L } \ln { \left[ \cfrac { \sqrt { 2 } +1 }{ \sqrt { 2 } } \right] }
B. $$\cfrac { L }{ R } \ln { \left[ \cfrac { \sqrt { 2 } }{ \sqrt { 2 } -1 } \right] } $$
C. \cfrac { R }{ L } \ln { \left[ \cfrac { \sqrt { 2 } - 1 }{ \sqrt { 2 } + 1 } \right] }
D. \cfrac { R }{ L } \ln { \left[ \cfrac { \sqrt { 2 } }{ \sqrt { 2 } +1 } \right] }
E. \cfrac { R }{ L } \ln { \left[ \cfrac { \sqrt { 2 } - 1 }{ 2 } \right] }
F. \cfrac { R }{ L } \ln { \left[ \cfrac { \sqrt { 2 } + 1 }{ 2 } \right] }
G. $$\cfrac { R }{ L } \ln { \left[ \cfrac { \sqrt { 2 } -1 }{ \sqrt { 2 } } \right] } $$
H. $$\cfrac { L }{ R } \ln { \left[ \cfrac { \sqrt { 2 } -1 }{ \sqrt { 2 } } \right] } $$
I. $$\cfrac { R }{ L } \ln { \left[ \cfrac { \sqrt { 2 } }{ \sqrt { 2 } -1 } \right] } $$
Answer: I
If potential at the surface of earth is assigned zero value, then potential at centre of earth will be (Mass = M, Radius = R)

Options:
A. -\dfrac{ GM }{ 3R }
B. -\dfrac{ GM }{ 6R }
C. $$0$$
D. -\dfrac{ 3GM }{ 4R }
E. -\dfrac{ GM }{ 4R }
F. -\dfrac{ GM }{ 5R }
G. $$-\dfrac { 2GM }{ 3R } $$
H. $$\dfrac { 3GM }{ 2R } $$
I. -\dfrac{ GM }{ 7R }
J. $$-\dfrac { GM }{ 2R } $$
Answer: J
A particle moving with kinetic energy $=3J$ makes an elastic head-on collision with a stationary particle which has twice its mass. During the impact:

Options:
A. the maximum elastic potential energy of the system is $1.75J$
B. the maximum elastic potential energy of the system is $2J$
C. the maximum elastic potential energy of the system is $1.25J$
D. the maximum elastic potential energy of the system is $1.8J$
E. the maximum elastic potential energy of the system is $1J$
F. the ratio of kinetic energy to potential energy of the system first decreases and then increases
G. the minimum elastic potential energy of the system is $1J$
H. the maximum elastic potential energy of the system is $1.5J$
I. the minimum kinetic energy of the system is $1J$
J. momentum and total energy are conserved at every instant
Answer: B
An electric motor is a device to convert electrical energy into mechanical energy. The motor shown below has a rectangular coil $(15\ cm \times 10\ cm)$ with 100 turns placed in a uniform magnetic fleld $B=2.5\ T$. When a current is passed through the coil, lt completes $50$ revolutions in one second. the power output of the motor is $1.5\ kW$. The current rating of the coil should be

Options:
A. 2.5 A
B. 2.75 A
C. 1.75 A
D. 3.5 A
E. 2 A
F. 3 A
G. 1.5 A
H. 2.25 A
I. 1 A
J. 4 A
Answer: E
A ray of light travelling in a medium of refractive index $\mu $ is incident at an angle $\theta$ on a composite transparent plate consisting of 50 plates of $R.I.1.01 \mu, 1.02\mu, 1.03 \mu, ...... 1.50 \mu$ The ray emerges from the composite plate into a medium of refractive index $1.6 \mu$ at angle 'x' then,

Options:
A. sin x=$\left ( \frac{1.01}{1.5} \right )^{50}sin \theta $
B. sin x=$\frac{5}{8} sin \theta $
C. sin x=$(\frac{5}{9}) sin \theta $
D. sin x=\left(\frac{1.01}{1.5}\right)^{49} \sin \theta
E. sin x=\left(\frac{1.01}{1.5}\right)^{51} \sin \theta
F. sin x=$\frac{101}{160} sin \theta $
G. sin x=\left(\frac{1.01}{1.6}\right)^{49} \sin \theta
H. sin x=\left(\frac{1.01}{1.6}\right)^{51} \sin \theta
I. sin x=\left(\frac{1.01}{1.6}\right)^{50} \sin \theta
J. sin x=\left(\frac{1.01}{1.5}\right)^{50} \sin \theta
Answer: B
A thin prism of glass is placed in air and water respectively. If $n_g=\cfrac 3 2 \quad and \quad n_w=\cfrac 4 3 $, then the ratio of deviation produced by the prism for a small angle of incidence when placed in air and water seperately is:

Options:
A. \cfrac 3 4
B. $$ 4$$
C. \cfrac 3 8
D. \cfrac 3 6
E. \cfrac 5 4
F. $$\cfrac 1 4$$
G. $$ 2$$
H. \cfrac 3 7
I. \cfrac 3 5
J. $$\cfrac 1 2$$
Answer: J
Two blocks of masses $5 \ kg$ and $2\ kg$ (see fig) are initially at rest on the floor. They are connected by a light string, passing over a light frictionless pulley. An upward force $F$ is applied on the pulley and maintained constant. Calculate the acceleration $a_{1}$ and $a_{2}$ of the $5 \ kg$ and $2 \ kg$ masses, respectively, when $F$ is $110 \ N (g = 10 \ m/s^2$)

Options:
A. a_{1} = 1.5 \ ms^{-2} , a_{2} = 15 \ ms^{-2}
B. $$a_{1} = 1 \ ms^{-2} , a_{2} = 0 \ ms^{-2}$$
C. a_{1} = 1.5 \ ms^{-2} , a_{2} = 16.5 \ ms^{-2}
D. $$a_{1} = 1 \ ms^{-2} , a_{2} = 17.5 \ ms^{-2}$$
E. a_{1} = 1.5 \ ms^{-2} , a_{2} = 17 \ ms^{-2}
F. a_{1} = 1.5 \ ms^{-2} , a_{2} = 16 \ ms^{-2}
G. a_{1} = 1.5 \ ms^{-2} , a_{2} = 18 \ ms^{-2}
H. $$a_{1} = 2 \ ms^{-2} , a_{2} = 17.5 \ ms^{-2}$$
I. $$a_{1} = 2\ ms^{-2} , a_{2} = 0 \ ms^{-2}$$
Answer: D
The van der Waal's equation for a real gas may be rearranged to give $V^3_m-\left(b+\dfrac{RT}{P}\right)V^2_m-\dfrac{a}{P}V_m-\dfrac{ab}{P}=0$ where $V_m$ is the molar volume of the gas. Indicate the correct statement(s) amongst the following.

Options:
A. At temperature less than $T_C$ there are two real values of $V_m$ and one imaginary value
B. At temperature less than $T_C$ there are three real values of $V_m$
C. At temperature less than $T_C$, there are two real values of $V_m$ and one imaginary value.
D. At any temperature, there are three values of $V_m$, one real and two imaginary
E. At temperature greater than $T_C$, there are two real values of $V_m$ and one imaginary value
F. At temperature greater than $T_C$, there are three values of $V_m$, two real and one imaginary
G. At temperature greater than $T_C$, the three real values of $V_m$ are identical
H. At temperature less than $T_C$ there are two real values of $V_m$ and one imaginary value.
Answer: B
A conducting wire bent in the form of a parabola $y^2 = 2x$ carries a current $i = 2A$ as shown in figure. This wire is placed in a uniform magnetic field $\overrightarrow B = 4\hat{k}$ Tesla. The magnetic force on the wire in region $0 < x < 2m$ is (in Newton )

Options:
A. $$-32\hat{i}$$
B. $$16\hat{i}$$
C. $$-16\hat{i}$$
D. -24\hat{i}
E. -8\hat{i}
F. $$32\hat{i}$$
G. 8\hat{i}
H. -28\hat{i}
Answer: A
Two circular coils P and Q are made from similar wire but radius of Q is twice that of P. Relation between the values of potential difference across them so that the magnetic induction at their centers may be the same is

Options:
A. $$ V_q = 1/4V_p $$
B. $$ V_q = 3V_p $$
C. V_q = 1/3V_p
D. V_q = 5V_p
E. V_q = 8V_p
F. V_q = 6V_p
G. V_q = 16V_p
H. $$ V_q = 2V_p $$
I. V_q = 1/2V_p
J. $$ V_q = 4V_p $$
Answer: H
Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see figure).Electrons accelerated by potential V are diffracted from a crystal. If $d =1A^o$ and $i = 30^{\circ}$ , V should be about :$(h=6.6 \times10^{-34}js, m=9.1\times10^{-31}kg, e=1.6\times10^{-19}c)$

Options:
A. $2000$ V
B. $50$ V
C. $500$ V
D. 150 V
E. $1000$ V
F. $75$ V
G. 25 V
H. 100 V
Answer: B
A circular wire loop of radius R is placed in the x-y plane centred at the origin O. A square loop of side a(a << R) having two turns is placed with its centre at $z = \sqrt{3}R$ along the axis of the circular wire loop, as shown in figure. The place of the square loop makes an angle of 45$^o$ with respect to the z-axis. If the mutual inductance between the loops is given by $\displaystyle \frac{\mu_0 a^2}{2^{p/2} R'}$, then the value of p is

Options:
A. 5
B. \frac{\mu_0 a^2}{2^{5/2} R'}
C. 6
D. 7
E. 9
F. \frac{\mu_0 a^2}{2^{3/2} R'}
G. \frac{\mu_0 a^2}{2^{8/2} R'}
H. 8
I. \frac{\mu_0 a^2}{2^{4/2} R'}
J. \frac{\mu_0 a^2}{2^{6/2} R'}
Answer: D
In Fig. two cells have equal e.m.f. $ E $ but internal resistances are $ r_{1} $ and $ r_{2} $. If the reading of the voltmeter is zero, then relation between $ R, r_{1} $ and $ r_{2} $ is

Options:
A. R=r_{2}-r_{1}
B. R=r_{2}+r_{1}
C. $$ R=r_{1}+r_{2} $$
D. R=r_{2}-2r_{1}
E. $$ R=r_{1}-r_{2} $$
F. R=2r_{1}+r_{2}
G. R=2r_{1}+2r_{2}
H. R=2r_{1}-r_{2}
I. $$ 2 r_{1}-r_{2} $$
J. $$ r_{1} r_{2} $$
Answer: E
A horizontally oriented tube AB of length $l$ rotates with a constant angular velocity $\omega$ about a stationary vertical axis OO' passing through the end A (fig). The tube is filled with an ideal fluid. The end A of the tube is open, the closed end B has a very small orifice. Find the velocity of the fluid relative to the tube as a function of the column 'height' $h$.

Options:
A. v=\omega \sqrt{h(3l-2h)}
B. $$v=\omega h$$
C. v=\omega \sqrt{h(3l-h)}
D. v=\omega \sqrt{h(l+h)}
E. v=\omega \sqrt{h(2l+h)}
F. $$v=\omega\sqrt{h(l-h)}$$
G. $$ v=\omega \sqrt{2h(l-h)} $$
H. v=\omega \sqrt{2h(l+h)}
I. v=\omega \sqrt{h(3l-3h)}
J. $$ v=\omega \sqrt{h(2l-h)} $$
Answer: J
Radio waves coming at angle $lpha$ to vertical are received by a ladder after reflection from a nearby  water surface and also directly. What can be height of antenna from water surface so that it records a maximum intensity (a maxima) (wavelength = $\lambda$)

Options:
A. \frac {5\lambda} { 8 \cos \alpha}
B. \frac {3\lambda} { 4 \cos \alpha}
C. \frac {7\lambda} { 8 \cos \alpha}
D. $$\frac {\lambda} { 2 \sin lpha}$$
E. $$\frac {\lambda} { 2 \cos lpha}$$
F. \frac {3\lambda} { 2 \cos \alpha}
G. $$\frac {\lambda} { 4 \cos lpha}$$
H. $$\frac {\lambda} { 4 \sin lpha}$$
I. \frac {5\lambda} { 4 \cos \alpha}
J. \frac {3\lambda} { 4 \sin \alpha}
Answer: G
Due to inter electronic repulsion, an electron experiences less nuclear charge. It is, therefore, said that inner electronic shells electrons shield the nucleus. The actual nuclear charge experienced by an electron is called effective nuclear charge. If the electrons in K-shell shields the nucleus to the extent of 0.5 (say) each. What will be the wave number of the spectral line arising out of n=4 to n = 2 transition in He atom?

Options:
A. $$\displaystyle \frac{3}{4}R_{H}$$
B. \displaystyle \frac{7}{8}R_{H}
C. \displaystyle \frac{1}{2}R_{H}
D. \frac{1}{8}R_{H}
E. $$\displaystyle \frac{3}{8}R_{H}$$
F. \frac{3}{8}R_{H}
G. \displaystyle \frac{5}{8}R_{H}
H. $$\displaystyle \frac{R_{H}}{4}$$
I. $$\displaystyle 8R_{H}$$
J. \displaystyle \frac{1}{4}R_{H}
Answer: A
You are given several identical resistance each of value $R=10$ and each capable of carrying a maximum current of one ampere. It is required to make a suitable combination of these resistance of $5$ which can carry a current of $4$ ampere. The minimum number of resistance's of the type $R$ that will be required for this job is

Options:
A. $$20$$
B. 16
C. 6
D. $$4$$
E. 18
F. $$10$$
G. $$8$$
H. 14
I. 12
Answer: G
Consider air to be a diatomic gas with average molecular mass of 29g/mole. A mass 1.45kg of air is contained in a cylinder with a piston at $27^{0}C$ and pressure $1.5\times10^{5} N/m^{2}$. Energy is given to the system as heat and the system is allowed to expand till the final pressure $3.5\times10^{5} N/m^{2}$. As the gas expands, pressure and volume follow the relation $ V= KP^{2}$ where K is constant (R =8.3J/(mole-K).Increase of internal energy of the system is nearly :

Options:
A. $$5.4\times10^{4}J$$
B. 3.5 \times 10^{6}J
C. $$86\times10^{3}J$$
D. $$6.3\times10^{4}J$$
E. 4.6 \times 10^{6}J
F. $$4.2\times10^{4}J$$
G. $$3.6\times10^{6}J$$
H. 3.9 \times 10^{6}J
I. 7.2 \times 10^{5}J
J. 4.8 \times 10^{6}J
Answer: G
A heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross-section $A$ as shown in diagram. The cylinder contains one mole of an ideal diatomic gas. The current flows through the element such that the temperatures rises with time $t$ as $\Delta T=lpha t+\dfrac { 1 }{ 2 } \beta { t }^{ 2 }$ ($lpha $ and $\beta $ are constants), while pressure remains constant. The atmospheric pressure above the piston is ${ P }_{ 0 }$. Then:

Options:
A. the piston moves upwards with constant speed
B. the piston moves upwards with a linearly decreasing speed
C. the piston moves upwards with a linearly decreasing acceleration
D. the piston moves upwards with a quadratic speed increase
E. the piston moves upwards with a linearly increasing speed
F. the piston moves upwards with constant acceleration
G. the rate of increase in internal energy is $\dfrac { 5 }{ 2 } R\left( lpha +\beta t \right) $
H. the piston moves upwards with a linearly increasing acceleration
I. the piston moves upwards with constant velocity
J. the current flowing in the element is $\sqrt { \dfrac { 5 }{ 2r } R\left( lpha +\beta t \right) } $
Answer: F
In a basic aqueous solution chloromethane undergoes a substitution reaction in which $Cl^-$ is replaced by $OH^-$ as.$CH_3Cl (aq) + OH^-  \rightleftharpoons CH_3 OH(aq) + Cl^- (aq)$The equilibrium constant of above reaction $K_c = 1 \times 10^{15}$. If a solution is prepared by mixing equal volumes of $0.1 M CH_3Cl$ and $0.2 M NaOH$(100% dissociated) then [$OH^-$] concentration at equilibrium in mixture will be ______.

Options:
A. $$0.2 M$$
B. 0.025 M
C. 0.075 M
D. 0.06 M
E. 0.125 M
F. $$0.1 M$$
G. 0.15 M
H. 0.175 M
I. $$0.05 M$$
J. $$0.5 M$$
Answer: I
If the volume of the tetrahedron with edges $2i + j - k,\,\,i + aj + k$  and $i + 2j - k$ is one cubic unit then $a$=

Options:
A. $$1$$
B. -4
C. $$2$$
D. $$-2$$
E. 4
F. $$-1$$
G. $$3$$
H. -5
I. -3
Answer: D
An adiabatic cylinder closed at both ends consists of a freely moving non-conducting piston which divides the cylinder into two equal parts and each part contains $28\ g$ of $N_{2}$. Initially $1/3rd$ molecules of nitrogen in the right part are dissociated into atoms. The length of the cylinder is $1\ m$ and area of cross-section is $10^{-2}m^{2}$. The natural length of the spring connected to the piston and right wall of the cylinder is $l=50\ cm$ and $k=\sqrt{2}\times  10^{3}\ N/m$. If the initial pressure in each part is $P_{0}=\sqrt{2}\times 10^{5}N/m^{2}$ then what work must be done by the gas in the right part?

Options:
A. -1818\ J
B. $$1414\ J$$
C. $$-1414\ J$$
D. $$1515\ J$$
E. -1616\ J
F. $$-1717\ J$$
G. -1212\ J
H. -1313\ J
I. $$-1515\ J$$
Answer: C
If $A(2, 1), B(0, 5)$ are limiting points of a coaxial system then the radical axis of the conjugate coaxial system is

Options:
A. 2x + y - 6 = 0
B. $$x - 2y + 5 = 0$$
C. 2x + y - 3 = 0
D. 2x + y - 8 = 0
E. $$2x - y - 3 = 0$$
F. 2x - y - 5 = 0
G. 2x + y - 4 = 0
H. 2x + y - 7 = 0
I. $$x + 2y - 7 = 0$$
J. $$2x + y - 5 = 0$$
Answer: J
As per bohr model, the minimum energy (in eV) required to remove an electron from the ground state of double ionized Li atom(Z=3) is:

Options:
A. 1.51
B. 1.54
C. 1.50
D. 13.6
E. 4.53
F. 50.8
G. 122.4
H. 1.52
I. 1.53
J. 1.55
Answer: A
Sea water at frequency $
u= 4\times10^8\ Hz$ has permittivity $\epsilon pprox80\ \epsilon_0$, permeability $\mu pprox \mu_0$ and resistivity $\rho= 0.25\ \Omega$. Imagine a parallel plate capacitor immersed in sea water and driven by an alternating voltage source $V(t) =V_0\ sin(2\pi
u t$). The ratio of amplitude of the conduction current density to the displacement current density is:

Options:
A. $$2$$
B. 9/8
C. 8/3
D. 9/2
E. 3/4
F. 8/9
G. $$2/3$$
H. $$4/9$$
I. $$9/4$$
J. 3/2
Answer: H
In YDSE, a thin film $(\mu = 1.6)$ of thickness $0.01mm$ is introduced in the path of one of the interfering beams. The central fringe moves to a position occupied by the $10^{th}$ bright fringe earlier. The wave length of wave is

Options:
A. 540\mathring{A}
B. 560\mathring{A}
C. 660\mathring{A}
D. $$60\mathring{A}$$
E. $$6000\mathring{A}$$
F. 6600\mathring{A}
G. $$600\mathring{A}$$
Answer: G
A solenoid of inductance $L$ and resistance $r$ is connected in parallel to a resistance $R$. A battery of emf $E$ and of negligible internal resistance is connected across this parallel combination. At $t=0$, battery is removed then which of the following is possible :

Options:
A. the amount of heat generated in the solenoid due to removing the battery is $\dfrac {E^2L}{2r(r+2R)}$
B. the amount of heat generated in the solenoid due to removing the battery is $\dfrac {E^2L}{2R(r+R)}$
C. total energy dissipated in the solenoid and the resistor long time after removing the battery is $\dfrac {1}{2}L\dfrac {E^2(R+r)^2}{r^2R^2}$
D. the amount of heat generated in the solenoid due to removing the battery is \dfrac {E^2L}{2r(2R+r)}
E. the amount of heat generated in the solenoid due to removing the battery is \dfrac {E^2L}{2(r+2R)}
F. the amount of heat generated in the solenoid due to removing the battery is $\dfrac {E^2L}{2rR}$
G. the amount of heat generated in the solenoid due to removing the battery is $\dfrac {E^2L}{2(r+R)}$
H. current in the inductor just after moving the battery is $\dfrac {E(r+R)}{rR}$
I. the amount of heat generated in the solenoid due to removing the battery is $\dfrac {E^2L}{2r^2R}$
J. the amount of heat generated in the solenoid due to removing the battery is $\dfrac {E^2L}{2r(r+R)}$
Answer: J
What are the energy $(kj/mol)$ associated with the de-excitation of an electron from $n=6$ to $n=2$ in He ion?

Options:
A. 1.16 \times 10^{-22}
B. 1.17 \times 10^{-20}
C. $$1.78 \times 10^{-22}$$
D. 1.78 \times 10^{-21}
E. $$1.36 \times 10^3$$
F. 1.18 \times 10^{-22}
G. 1.17 \times 10^{-21}
H. 1.17 \times 10^{-23}
I. $$1.17 \times 10^{-22}$$
J. $$1.16 \times 10^3$$
Answer: I
In an isolated $Li^{2+}$, electron jumps from n $=2$ to n$= 1$. What is the recoil momentum of the ion (in kg- m/s)?

Options:
A. 13.6 x 10-27
B. 10.88 x 10-27
C. $$5.44 x 10-27$$
D. 21.76 x 10-27
E. 5.44 x 10-28
F. $$8.16 x 10-27$$
G. 2.72 x 10-27
H. 10.88 x 10-28
I. $$ 16.32 x 10-27$$
J. $$48.96 x 10-27$$
Answer: C
A train is moving towards a platform with the speed of 54 km/h and the wind is blowing in the direction of train's motion, what is observed frequency of the horn of train by the passenger who is standing on the platform? Original frequency of horn $f_0$ = 300 Hz and $v_{air}$ = 108 km/h, $v_{sound}$ in still air = 330 m/s.

Options:
A. 315 Hz
B. 325 Hz
C. 285 Hz
D. 300 Hz
E. 335 Hz
F. 354 Hz
G. 334 Hz
H. 338 Hz
I. 345 Hz
J. 344 Hz
Answer: J
A certain metal when irradiated with light (v = 3.2*$10^{ 16 }$ Hz) emits photo electrons with twice kinetic energy as did photo electron when the same metal is irradiated by light (v = 2.0*$10^{ 16 }$ Hz). Calculate ${ v }_{ 0 }$ of electron?

Options:
A. 2.4*$10^{ 14 }$ Hz
B. 4*$10^{ 12 }$ Hz
C. 1.2*$10^{ 14 }$ Hz
D. 8*$10^{ 15 }$ Hz
E. 1.8*$10^{ 14 }$ Hz
F. 1.2*$10^{ 16 }$ Hz
G. 3.2*$10^{ 14 }$ Hz
H. 1.6*$10^{ 14 }$ Hz
I. 2.0*$10^{ 14 }$ Hz
J. 2.4*$10^{ 15 }$ Hz
Answer: C
Radiation of frequency $v = 9{v_o}$ is incident on a metal surface, having threshold frequency ${v_o}$. The maximum velocity of the ejected photoelectrons is $6 \times {10^6}m/s$. What will be the maximum velocity of the ejected photoelectrons, if v is reduced to $3{v_o}$ ?

Options:
A. 1 \times {10^6}{\text{m/s}}
B. $$3.5 \times {10^6}{\text{m/s}}$$
C. $$3 \times {10^6}{\text{m/s}}$$
D. $$1.5 \times {10^6}{\text{m/s}}$$
E. 5 \times {10^6}{\text{m/s}}
F. $$2 \times {10^6}{\text{m/s}}$$
G. 2.5 \times {10^6}{\text{m/s}}
H. 1.75 \times {10^6}{\text{m/s}}
I. $$4 \times {10^6}{\text{m/s}}$$
Answer: F
$ 4 m^3 $ of water is to be pumped to a height of $ 20 $ m and forced into a reservoir at a pressure of $ 2 \times 10^5 N/m^2 $. The work done by the motor is (External pressure = $ 10^5 N/m^2 ) $

Options:
A. 4 \times 10^5 J
B. $$ 8 \times 10^5 J $$
C. 10 \times 10^5 J
D. 20 \times 10^5 J
E. 28 \times 10^5 J
F. $$ 32 \times 10^5 J $$
G. 6 \times 10^5 J
H. 24 \times 10^5 J
I. $$ 16 \times 10^5 J $$
J. $$ 12 \times 10^5 J $$
Answer: B
A solid body rotates about a stationary axis so that its angular velocity depends on rotational angle $\emptyset $  as $\omega={omega}_{0}-k \emptyset $ where ${\omega}_{0}$ and k are positive constants. At the moment $t=0$, $ \emptyset = 0$, the time dependence of rotation angle is

Options:
A. \frac{\omega_{0}}{2k}(1-e^{-kt})
B. \frac{\omega_{0}}{k}(1+e^{-kt})
C. \frac{\omega_{0}}{k}(1+e^{kt})
D. $$\frac{k}{{\omega}_{0}}({e}^{-kt}-{1})$$
E. $$ k{\omega}_{0} {e}^{-kt}$$
F. \frac{\omega_{0}}{k}(1-e^{kt})
G. $$\frac{{\omega}_{0}}{k}(1-{e}^{-kt})$$
H. $$ \frac{\omega_{0}}{2k}(1-e^{-2kt})$$
I. $\frac{{\omega}^{0}}{k} {e}^{-kt}
J. \frac{k}{\omega_{0}}(1-e^{-kt})
Answer: E
A parallel beam of light ray parallel to the X-axis is incident on a parabolic reflecting surface $x = 2by^{2}$ as shown in the figure. After reflecting it passes through focal point $F$. The focal length of the reflecting surface is:

Options:
A. \dfrac{1}{6b}
B. \dfrac{1}{12b}
C. \dfrac{1}{16b}
D. \dfrac{1}{10b}
E. $$\dfrac {1}{b}$$
F. $$\dfrac {1}{4b}$$
G. \dfrac{1}{2b}
H. \dfrac{1}{3b}
I. $$\dfrac {1}{8b}$$
J. None of these
Answer: I
The velocity  vector $v$ and displacement vector $x$ of a particle executing SHM are related as $\dfrac{v dv}{dx} ={\omega}^2 x$ with the initial condition $v = v_0$ at  $x = 0$. The velocity $v$, when displacement is $x$, is:

Options:
A. v = \sqrt{v^2_0 - {\omega}^2x^2/2}
B. $$v = {3}\sqrt{v^3_0 + {\omega}^3x^3}$$
C. $$v = \sqrt{v^2_0 - 3{\omega}^2x^2}$$
D. v = \sqrt{v^2_0 + 2{\omega}^2x^2}
E. v = \sqrt{v^2_0 - 4{\omega}^2x^2}
F. v = \sqrt{v^2_0 - 2{\omega}^2x^2}
G. $$v = \sqrt{v^2_0 + {\omega}^2x^2}$$
H. $$v = v_0 - ({\omega}^3 x^3 e^{x^3})^{1/3}$$
I. $$v = \sqrt{v^2_0 - {\omega}^2x^2}$$
J. v = \sqrt{v^2_0 - 2\omega^2x^2}
Answer: I
A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet ?
[Given : Mass of planet = $8 \times 10^{22} kg$;
Radius of planet = $2 \times 10^6 m$, 
Gravitational constant $G = 6.67 \times 10^{-11} Nm^2 / kg^2$]

Options:
A. $$13$$
B. 19
C. $$17$$
D. $$11$$
E. 12
F. 15
G. $$9$$
H. $$10$$
I. 16
J. 14
Answer: D
An ideal gas expands according to the law ${  PV}^{3/2  }$ = constant. We conclude

Options:
A. The molar heat capacity $C = C_v-2R$
B. The adiabatic exponent of the gas $K = 1.5$
C. The molar heat capacity \( C = C_v + \frac{5R}{2} \)
D. Temperature increases during the process
E. Such a process is not feasible
F. The molar heat capacity \( C = C_v + 3R \)
G. The molar heat capacity \( C = C_v + 2R \)
H. The molar heat capacity $C = C_v - R$
I. The molar heat capacity \( C = C_v + \frac{3R}{2} \)
J. The molar heat capacity \( C = C_v + R \)
Answer: A
Slit 1 of Young's double-slit experiment is wider than slit 2, so that the light from slits are given as ${A}_{1}={A}_{0}\sin { \omega t } $ and ${A}_{2}={A}_{0}\sin { \left( \omega t+\cfrac { \pi  }{ 3 }  \right)  } $. The resultant amplitude and intensity, at a point where the path difference between them is zero, are $A$ and $I$ respectively. Then

Options:
A. I \propto 5{ A }_{ 0 }^{ 2 }
B. I \propto 4{ A }_{ 0 }^{ 2 }
C. $$I \propto 3{ A }_{ 0 }^{ 2 }$$
D. I \propto 9{ A }_{ 0 }^{ 2 }
E. $$A=4{A}_{0}$$
F. $$I \propto 7{ A }_{ 0 }^{ 2 }$$
G. $$A=\sqrt 3 {A}_{0}$$
H. I \propto 2{ A }_{ 0 }^{ 2 }
I. I \propto 6{ A }_{ 0 }^{ 2 }
J. $$I \propto16{ A }_{ 0 }^{ 2 }$$
Answer: C
A road of radius $20m$ is banked for vehicles going at speed of $10m/s$. The road has sufficient friction to prevent slipping. A vehicle of $200Kg$ moves on this road. Take $g=10m/{s}^{2}$.
What will the magnitude and direction of friction on the vehicle if it negotiates the curve at a constant speed of $15m/s$.

Options:
A. 400\sqrt{5}N \text{ downwards}
B. $400\sqrt{5}N$ upwards
C. 400\sqrt{5}N \text{ upwards}
D. $300\sqrt{5}N$ downwards
E. 300\sqrt{5}N \text{ downwards}
F. 300\sqrt{5}N \text{ upwards}
G. $600\sqrt{5}N$ upwards
H. $500\sqrt{5}N$ downwards
I. $500\sqrt{5}N$ upwards
J. $300\sqrt{5}N$ upwards
Answer: I
Following are the observations made for a black body at 3000 K:
Amount of energy emitted emitted per secondRange of wavelength of radiation${ U }_{ 1 }$490 nm to 500 nm${ U }_{ 2 }$990 nm to 1000 nm${ U }_{ 3 }$1490 nm to 1500 nmWien's constant=$3\times { 10 }^{ 6 }$nm K. Mark the correct option.

Options:
A. $${ U }_{ 1 }>{ U }_{ 2 }$$
B. ${ U }_{ 2 }>{ U }_{ 1 } $
C. ${ U }_{ 1 }<{ U }_{ 2 }$
D. ${ U }_{ 3 }>{ U }_{ 1 }$
E. $${ U }_{ 3 }>{ U }_{ 2 }$$
F. ${ U }_{ 2 }={ U }_{ 3 }$
G. $${ U }_{ 2 }>{ U }_{ 3 }$$
H. ${ U }_{ 2 }<{ U }_{ 1 }$
I. $${ U }_{ 1 }={ U }_{ 2 }={ U }_{ 3 }$$
Answer: A
Consider a particle of mass m moving in two dimensions between two perfectly reflecting walls which intersect at an angle x at the origin (see figure). Assume that when the particle is reflected, its speed is unchanged and its angle of incidence equals its angle of reflection.
The particle is attracted to the origin by a potential $U(r) = -\dfrac{C}{r^3},$ where C is some constant.
Now start the particle at a distance R from the origin on the x-axis with a velocity vector $V = (V_x, V_y).$ Assume $V_{y}
eq 0, V_{x} < 0.$

Options:
A. The equation for distance of closest approach to the origin is $\dfrac{1}{2}mv^2 - \dfrac{C}{R^4} = \dfrac{m^2v_y^2R^2}{2mr_m^2} - \dfrac{C}{r_m^2}$ where $r_{m}$ is distance of closest approach.
B. The particle will reach the origin if $U > \dfrac{m^3R^6V_y^6}{54C^2}$ (Here U is total energy)
C. The equation for distance of closest approach to the origin is \dfrac{1}{2}mv^2 - \dfrac{C}{R^4} = \dfrac{m^2v_y^2R^2}{2mr_m^2} - \dfrac{C}{r_m^3} where \( r_{m} \) is distance of closest approach.
D. \dfrac{1}{2}mv^2 - \dfrac{C}{R^2} = \dfrac{m^2v_y^2R^2}{2mr_m^2} - \dfrac{C}{r^2_m} \text{ where } r_{m} \text{ is distance of closest approach.}
E. \dfrac{1}{2}mv^2 - \dfrac{C}{R^4} = \dfrac{m^2v_y^2R^2}{2mr_m^2} - \dfrac{C}{r_m^4} \text{ where } r_{m} \text{ is distance of closest approach.}
F. The particle will never reach the origin for any value of R or U. (Here U is total energy)
G. The equation for distance of closest approach to the origin is $\dfrac{1}{2}mv^2 - \dfrac{C}{R^2} = \dfrac{m^2v_y^2R^2}{2mr_m^2} - \dfrac{C}{r_m^3}$ where $r_{m}$ is distance of closest approach.
H. The equation for distance of closest approach to the origin is $\dfrac{1}{2}mv^2 - \dfrac{C}{R^3} = \dfrac{m^2v_y^2R^2}{2mr_m^2} - \dfrac{C}{r^2_m}$ where $r_{m}$ is distance of closest approach.
I. \dfrac{1}{2}mv^2 - \dfrac{C}{R^3} = \dfrac{m^2v_y^2R^2}{2mr_m^2} - \dfrac{C}{r_m^4} \text{ where } r_{m} \text{ is distance of closest approach.}
J. The particle will reach the origin if $R < \left ( \dfrac{3C}{mv_y^2} \right)^{1/3}$
Answer: H
Water stands to a depth H behind the vertical face of dam and exerts a certain resultant horizontal force on the dam and a certain torque tending to overturn the dam about the point O. What is the total torque about O(p=density of water), per unit width of dam?

Options:
A. $$\dfrac { 1 }{ 2 } { pgH }^{ 3 }$$
B. $$\dfrac { 1 }{ 3 } { pgH }^{ 3 }$$
C. \dfrac { 1 }{ 8 } { pgH }^{ 3 }
D. $${ pgH }^{ 3 }$$
E. $$\dfrac { 1 }{ 6 } { pgH }^{ 3 }$$
F. \dfrac { 1 }{ 5 } { pgH }^{ 3 }
G. \dfrac { 1 }{ 4 } { pgH }^{ 3 }
H. \dfrac { 1 }{ 9 } { pgH }^{ 3 }
I. \dfrac { 1 }{ 7 } { pgH }^{ 3 }
J. \dfrac { 1 }{ 10 } { pgH }^{ 3 }
Answer: E
A racing car has a siren and is speeding with the speed u=306 km/h. The frequency of the siren is ${v_0} = 40Hz.$.Ahead of it there is another car with exactly the same siren (this car is at rest).Each of the drivers  distinguishes the sound of siren of another car, because it is of higher frequency than the sound of his own siren.(Take speed of sound=340 m/s)

Options:
A. The beat frequency heard by the speeding driver is 13 Hz
B. The beat frequency heard by the speeding driver is 10 Hz
C. The beat frequency heard by the speeding driver is 11.33 Hz
D. The beat frequency heard by the driver at rest is 13.33 Hz
E. The beat frequency heard by the speeding driver is 12 Hz
F. The beat frequency heard by the speeding driver is 13.33 Hz
G. The beat frequency heard by the driver at rest is 12 Hz
H. The beat frequency heard by the speeding driver is 11 Hz
I. The beat frequency heard by the speeding driver is 14 Hz
J. The beat frequency heard by the driver at rest is 10Hz.
Answer: F
A circular race track of radius 30 m is baked at an angle of $ 15^o $. If the coefficient of friction between the wheels of  race-car and the road is 0.2 choose the correct option $ (g = 9.8 m/s^2) (\sqrt{3} = 1.732) $

Options:
A. The optimum speed of the race-car to avoid wear and tear on its tyres is 40.1 m/s
B. The optimum speed of the race-car to avoid wear and tear on its tyres is 25.1 m/s
C. Maximum permissible speed to avoid slipping is 28.1 m/s
D. The optimum speed of the race-car to avoid wear and tear on its tyres is 35.1 m/s
E. The optium speed of the race-car to avoid wear and tear on its tyres is 38.1 m/s
F. The optimum speed of the race-car to avoid wear and tear on its tyres is 33.1 m/s
G. The optimum speed of the race-car to avoid wear and tear on its tyres is 30.1 m/s
H. The optimum speed of the race-car to avoid wear and tear on its tyres is 32.1 m/s
I. The optimum speed of the race-car to avoid wear and tear on its tyres is 28.1 m/s
J. Maximum permissible speed to avoid is 38.1 m/s
Answer: E
Three slabs of thicknesses ${x}_{1}, {x}_{2} $ and ${x}_{3}$ and coefficients of thermal conductivity ${K}_{1},{K}_{2}$ and ${K}_{3}$ are placed in contact in steady state the combination behaves as a single slab of material of conductivity $K$. given by

Options:
A. $$ \dfrac { k }{ { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } } =\dfrac { ^{ x }{ 1 } }{ { K }_{ 1 } } +\dfrac { ^{ x }{ 2 } }{ { K }_{ 2 } } +\dfrac { ^{ x }{ 3 } }{ { K }_{ 3 } }$$
B. \dfrac { { x }_{ 1 }+{ x }_{ 2 }+2{ x }_{ 3 } }{ K } =\dfrac { { K }_{ 1 } }{ ^{ x }{ 1 } } +\dfrac { { K }_{ 2 } }{ ^{ x }{ 2 } } +\dfrac { { K }_{ 3 } }{ ^{ x }{ 3 } }
C. \left( \dfrac { 1 }{ ^{ x }{ 1 } } +\dfrac { 1 }{ ^{ x }{ 2 } } +\dfrac { 1 }{ ^{ x }{ 3 } } \right) =\dfrac { { K }_{ 1 }x_{ 1 }+{ K }_{ 2 }{ x }_{ 2 }+{ K }_{ 3 }{ x }_{ 3 } }{ { x }_{ 1 }+{ x }_{ 2 }-{ x }_{ 3 } }
D. $$\dfrac { { x }_{ 1 } }{ { K }_{ 1 } } +\dfrac { { x }_{ 2 } }{ { K }_{ 2 } } +\dfrac { { x }_{ 3 } }{ { K }_{ 3 } } =\dfrac { { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } }{ K }$$
E. $$ \dfrac { { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } }{ k } =\dfrac { { K }_{ 1 } }{ ^{ x }{ 1 } } +\dfrac { { K }_{ 2 } }{ ^{ x }{ 2 } } +\dfrac { { K }_{ 3 } }{ ^{ x }{ 3 } }$$
F. \left( \dfrac { 1 }{ { x }{ 1 } } +\dfrac { 1 }{ { x }{ 2 } } +\dfrac { 1 }{ { x }{ 3 } } \right) =\dfrac { { K }_{ 1 }x_{ 1 }+{ K }_{ 2 }{ x }_{ 2 }+{ K }_{ 3 }{ x }_{ 3 } }{ { x }_{ 1 }+{ x }_{ 2 }-{ x }_{ 3 } }
G. \left( \dfrac { 1 }{ ^{ x }{ 1 } } +\dfrac { 1 }{ ^{ x }{ 2 } } +\dfrac { 1 }{ ^{ x }{ 3 } } \right) =\dfrac { { K }_{ 1 }x_{ 1 }+{ K }_{ 2 }{ x }_{ 2 }+{ K }_{ 3 }{ x }_{ 3 } }{ { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } }
H. \left( \dfrac { 1 }{ ^{ x }{ 1 } } +\dfrac { 1 }{ ^{ x }{ 2 } } +\dfrac { 1 }{ ^{ x }{ 3 } } \right) =\dfrac { { K }_{ 1 }x_{ 1 }+{ K }_{ 2 }{ x }_{ 2 }+{ K }_{ 3 }{ x }_{ 3 } }{ { x }_{ 1 }+{ x }_{ 2 }+2{x }_{ 3 } }
I. $$\left( \dfrac { 1 }{ ^{ x }{ 1 } } +\dfrac { 1 }{ ^{ x }{ 2 } } +\dfrac { 1 }{ ^{ x }{ 3 } } \right) ={ K }_{ 1 }x_{ 1 }+{ K }_{ 2 }{ x }_{ 2 }+{ K }_{ 3 }{ x }_{ 3 }$$
J. \left( \dfrac { 1 }{ ^{ x }{ 1 } } +\dfrac { 1 }{ ^{ x }{ 2 } } +\dfrac { 1 }{ ^{ x }{ 3 } } \right) ={ K }_{ 1 }+{ K }_{ 2 }+{ K }_{ 3 }
Answer: I
An object of mass 4 kg is moving along X - axis and its potential energy as function of X can be expressed as 
$U(x)= 4(1-cos2x)$  joule then time period for small oscillations is:

Options:
A. \( \frac{5\pi}{2} \) sec
B. \( 2\pi \) sec
C. $$\pi sec$$
D. \frac{7\pi}{4} \text{sec}
E. \frac{5\pi}{4} \text{sec}
F. $$\frac{\pi }{2}sec$$
G. \( \frac{3\pi}{4} \) sec
H. $$3\frac{\pi }{2}sec$$
I. $$\frac{\pi }{4}sec$$
J. \frac{3\pi}{4} sec
Answer: F
It is desired to make a long cylindrical conductor whose temperature coefficient of resistivity at $20^{\circ}C$ will be close to zero. If such a conductor is made by assembling alternate disks of iron and carbon, Find the ratio of the thickness of a carbon disk to that an iron disk. (For carbon, $\rho = 3500\times 10^{-8}\Omega m$ and $lpha = -0.50 \times 10^{-3}\ ^{\circ}C^{-1}$ for iron, $\rho = 9.68\times 10^{-8}\Omega$ and $lpha = 6.5\times 10^{-3}\ ^{\circ}C^{-1})$.

Options:
A. $$2.0$$
B. $$0.036$$
C. 0.37
D. $$0.36$$
E. 0.072
F. $$1.0$$
G. 0.048
H. 0.046
I. 0.18
J. 0.72
Answer: B
$A$ and $B$ are two capacitors having air as the dielectric medium between the plates. Both the capacitors have same separation between their plates but area of plates for capacitor $A$ is double that of capacitor $B$. Without removing the battery a dielectric of constant $(K = 6)$ is uniformly filled between the plates of capacitor $B$ completely. As a result the energy stored in the two capacitors changes by $1.2\times 10^{-4} J$.
Energy stored in capacitor $B$ before dielectric is added is

Options:
A. 32\mu J
B. 128\ mJ
C. 32\ mJ
D. $$64\mu J$$
E. 128\mu J
F. $$16\ mJ$$
G. $$64\ mJ$$
H. $$16\mu J$$
Answer: D
Determine ${ \Delta H }/{ kJ }$ for the following reaction using the listed enthalpies of reaction:$4CO\left( g \right) +8{ H }_{ 2 }\left( g \right) \longrightarrow 3C{ H }_{ 4 }\left( g \right) +C{ O }_{ 2 }\left( g \right) +2{ H }_{ 2 }O\left( l \right) $$C\left( graphite \right) +{ 1 }/{ 2 }{ O }_{ 2 }\left( g \right) \longrightarrow CO\left( g \right) ;                           { \Delta H }/{ kJ }=-110.5kJ$$CO\left( g \right) +{ 1 }/{ 2 }{ O }_{ 2 }\left( g \right) \longrightarrow C{ O }_{ 2 }\left( g \right)         ;                 { \Delta H }/{ kJ }=-282.9kJ$${ H }_{ 2 }\left( g \right) +{ 1 }/{ 2 }{ O }_{ 2 }\left( g \right) \longrightarrow { H }_{ 2 }O\left( l \right)        ;                   { \Delta H }/{ kJ }=-285.8kJ$$C\left( graphite \right) +2{ H }_{ 2 }\left( g \right) \longrightarrow C{ H }_{ 4 }\left( g \right)           ;                       { \Delta H }/{ kJ }=-74.8kJ$

Options:
A. -622.4
B. -679.8
C. -699.3
D. -686.2
E. -754.5
F. -709.3
G. -712.3
H. -747.5
I. -653.5
J. -658.5
Answer: H
A uniform wire ABC has a mass of $10$ $gm$. A current of $2$ $A$ flows through it. The wire is kept in a uniform magnetic field of $2$ $T$. The acceleration of the wire will be :[Given $AB$=$4$ $cm$, $BC$=$5$ $cm$]

Options:
A. 12 $m/s^{2}$ along z-direction
B. 18 $m/s^{2}$ along y-direction
C. zero
D. 32 $m/s^{2}$ along y-direction
E. 14 $m/s^{2}$ along y-direction
F. 14 $m/s^{2}$ along x-direction
G. 36 $m/s^{2}$ along x-direction
H. 10 $m/s^{2}$ along y-direction
I. 12 $m/s^{2}$ along y-direction
J. 16 $m/s^{2}$ along y-direction
Answer: I
The wavelength of the radiation emitted, when in a hydrogen atom electron falls from infinity to stationary state 1, would be (Rydberg constant = 1.097x $ 10^{7}m^{-1}$)

Options:
A. 407 nm
B. 408 nm
C. 91.5 nm
D. 192 nm
E. 91 nm
F. 412 nm
G. 9.1x $ 10^{-8}$
H. 406 nm
I. 404 nm
J. 410 nm
Answer: H
A circuit is comprised of eight identical batteries and a resistor $R=0.8\Omega$. Each battery has an emf of $1.0V$ and internal resistance of $0.2\Omega$. The voltage difference across any of the battery is

Options:
A. 1.25V
B. $$1.0V$$
C. 0.4V
D. 0.6V
E. $$0.25V$$
F. 0.5V
G. 0.75V
H. $$0V$$
I. 1.5V
J. $$2V$$
Answer: H
A uniform sphere of radius $R$ is placed on a rough horizontal surface and given a linear velocity $v_0$ and angular velocity $w_0$ as shown. The sphere comes to rest after moving some distance to the right. It follows that

Options:
A. 2v_0 = 7 w_0 R
B. 2v_0 = 3 w_0 R
C. $$2v_0 = 5 w_0 R$$
D. 5v_0 = 2 w_0 R
E. $$2v_0= w_0R$$
F. 2v_0 = 6 w_0 R
G. 2v_0 = 4 w_0 R
H. 2v_0 = 9 w_0 R
I. $$v_0 = w_0 R$$
Answer: C
Along the X-axis, three charges$\dfrac {q}{2},-q$  and  $\dfrac {q}{2}$ are placed at x = 0, x = a and x = 2a respectively. The resultant electric potential at x = a+r (if, a << r) is :( $\epsilon _{0} $ is the permittivity of free space)

Options:
A. \dfrac{3qa^{2}}{16\pi \epsilon _{0}r^{3}}
B. \dfrac{qa^{2}}{2\pi \epsilon _{0}r^{3}}
C. \dfrac{qa^{2}}{8\pi \epsilon _{0}r^{3}}
D. $$\dfrac{q}{4\pi \epsilon _{0}r}$$
E. $$\dfrac{qa^{2}}{4\pi \epsilon _{0}r^{3}}$$
F. \dfrac{qa^{2}}{4\pi \epsilon _{0}r^{2}}
G. \dfrac{3qa^{2}}{8\pi \epsilon _{0}r^{3}}
H. $$\dfrac{q\left ( a^{2}/4 \right )}{4\pi \epsilon _{0}r^{3}}$$
I. \dfrac{qa}{4\pi \epsilon _{0}r^{2}}
Answer: E
The electrons, identified by quantum numbers n and $l$ can be placed in order of increasing energy, from the lowest to highest will be:i. n = 3 & l = 1ii. n = 3 & l = 0iii. n = 4 & l = 2iv. n = 4 & l = 0

Options:
A. iv < iii < i < ii
B. $$iii < ii < iv < i$$
C. $$i < iii < ii < iv$$
D. $$ii < iv < i < iii$$
E. iii < i < ii < iv
F. $$iii < iv < i < ii$$
G. $$iv < ii < iii < i$$
H. iv < i < ii < iii
I. ii < iii < iv < i
J. $$iv < iii < ii < i$$
Answer: F
A mild steel wire of length 1.0 m and cross sectional area $0.50\times10^{-2}$ $cm^{2}$ is stretched, well within its elastic limit, horizontally between two pillars. A mass of 100 g is suspended from the mid-point of the wire. Calculate the depression at the mid-point.Given:$Y=2\times10^{11}$$Nm^{-2}$

Options:
A. 1.03 cm
B. 1.15 cm
C. 0.07 cm
D. 2.7 cm
E. 1.05 cm
F. 1.17 cm
G. 1.04 cm
H. 1.09 cm
I. 1.07 cm
J. 1.7 cm
Answer: I
Suppose that the potential energy of a hypothetical atom consisting of a proton and an electron is given by U = - Ke$^{2}$ / 3 r$^{3}$.Then if Bhor's postulates are applied to this atom ,then the radius of the n $^{th}$ orbit will be proportional to:

Options:
A. n^{6}
B. n$^{2}$
C. n^{5}
D. n^{4}
E. n^{3}
F. 1 / n$^{3}$
G. 1 / n$^{2}$
Answer: B
Given: $Ag(NH_{3})_{2}^{+} \leftrightharpoons Ag^{+} + 2NH_{3}$, $K_{c} = 7.2 \times 10^{-8}$ and $K_{sp}$ of $AgCl = 1.8 \times 10^{-10}$ at $298 K$. If ammonia is added to a water solution containing excess of $AgCl(s)$ only, calculate the concentration of the complex in $1.0 M$ aqueous ammonia.

Options:
A. $$0.091 M$$
B. $$1.0 M$$
C. $$0.0455 M$$
D. 0.090 M
E. 0.02275 M
F. $$0.023 M$$
G. 0.0226 M
H. 0.045 M
I. 0.0225 M
J. 0.0445 M
Answer: C
In the circuit shown, the power factor of the circuit is $\dfrac { 3 }{ 5 }$ , Power factor of only RC circuit is $\dfrac { 4 }{ 5 }$ , source voltage is $100$ volt and its angular frequency  $\omega =100 rad/ sec$. RMS current in circuit is A> If inductive reactance is greater then capacitive reactance then the value of self inductance L is

Options:
A. \dfrac { 1 }{ 9 } H
B. \dfrac { 1 }{ 7 } H
C. \dfrac { 1 }{ 5 } H
D. $$1 H$$
E. \dfrac { 1 }{ 8 } H
F. \dfrac { 1 }{ 6 } H
G. $$\dfrac { 1 }{ 2 } H$$
H. \dfrac { 1 }{ 10 } H
I. $$\dfrac { 1 }{ 4 } H$$
J. $$\dfrac { 1 }{ 3 } H$$
Answer: I
The switch shown in the figure is closed at $t=0$. The charge on the capacitor as a function of time is given by

Options:
A. $$CV(1-e^{-3t/RC})$$
B. CV(1-e^{-2t/RC})
C. CV(1-e^{-t/6RC})
D. CV(1-e^{-t/4RC})
E. $$3CV(1-e^{-t/RC})$$
F. $$CV(1-e^{-t/3RC})$$
G. CV(1-e^{-t/2RC})
H. CV(1-e^{-t/5RC})
I. $$CV(1-e^{-t/RC})$$
J. CV(1-e^{-t/7RC})
Answer: A
Calculate partial pressure at which the solubility of oxygen gas in water at $20^{\circ}C$ is 0.10 g/L. Assume density of solution is 1 g/mL. $(K_{H}=3.486\times 10^{4}bar)$

Options:
A. 1.92 bar
B. 1.96 bar
C. 0.99 bar
D. 1.69 bar
E. 2.65 bar
F. 3.42 bar
G. 1.86 bar
H. 3.12 bar
I. 2.42 bar
J. 1.48 bar
Answer: B
Two identical semicircular discs of mass 'm' each and radius 'R" are placed in the XY (horizontal) plane and the YZ (vertical) plane, respectively. They are so placed that they have their common diameter along the Y-axis. Then the moment of inertia ($I_n$) of the system about the approximate axis is given by ($I_n$ refers to moment of inertia about axis n-where n is X, Y, Z)

Options:
A. I_Z = \dfrac{1}{8}mR^2
B. I_Z = \dfrac{5}{8}mR^2
C. I_Z = \dfrac{1}{4}mR^2
D. I_Z = \dfrac{3}{8}mR^2
E. $$I_Y = \dfrac{1}{2}mR^2$$
F. I_Z = \dfrac{5}{4}mR^2
G. $$I_Z = \dfrac{3}{4}mR^2$$
H. I_Z = \dfrac{1}{2}mR^2
I. $$I_X = I_Y = I_Z$$
J. $$I_X = \dfrac{1}{2}mR^2$$
Answer: G
A point source of light is placed at a depth $h=0.5 m$ below the surface of a liquid $\left( {\mu  = \frac{5}{4}} \right)$. Then, the fraction of light energy that escape directly from the liquid surface is :-

Options:
A. 0.25
B. $$0.1$$
C. 0.175
D. 0.225
E. $$0.2$$
F. 0.15
G. $$0.4$$
H. $$0.3$$
I. 0.275
J. 0.35
Answer: E
Two electrochemical cells are assembled in which following reactions occur.$V^{2+} + VO^{2+} + 2H^+ \rightarrow 2V^{3+} + H_2O;    \quad \quad    E^o_{cell} = 0.616 V$$V^{3+} + Ag + 2H_2O   \rightarrow    VO_2 + 4H^+  + Ag(s) ; E^o_{cell} = 0.439 V$$E^o$ for half reaction, $V^{3+} + e \rightarrow V^{2+}$ is :Given that $E_{Ag^+/ Ag} = 0.799 \space V.$

Options:
A. -0.276 V
B. -0.236 V
C. -0.216 V
D. -0.256 V
E. -0.356 V
F. -0.346 V
G. -0.316 V
H. None of these
I. -0.246 V
J. -0.286 V
Answer: D
A bomb of mass $M$ at rest explodes into two fragments of masses $m_{1}$ and $m_{2}$. The total energy released in the explosion is $E$. If $E_{1}$ and $E_{2}$ represent the energies carried by masses $m_{1}$ and $m_{2}$ respectively, then which of the following is correct?

Options:
A. E_{2} = \dfrac {m_{2}}{M}E
B. E_{1} = \dfrac {m_{1} + m_{2}}{M}E
C. E_{1} = \dfrac {m_{1}}{m_{1} + m_{2}}E
D. $$E_{1} = \dfrac {m_{1}}{m_{2}}E$$
E. E_{1} = \dfrac {m_{2}}{M}E
F. E_{2} = \dfrac {m_{2}}{m_{1}}E
G. $$E_{1} = \dfrac {m_{1}}{M}E$$
H. $$E_{1} = \dfrac {m_{2}}{m_{1}}E$$
I. E_{1} = \dfrac {m_{1} + m_{2}}{m_{2}}E
J. $$E_{2} = \dfrac {m_{1}}{M}E$$
Answer: H
According to a scientist, he applied a force $\displaystyle F=-CX^{1/3}$ on a particle and the particle is performing SHM. No other force acted on the particle. He refuses to tell whether c is a constant or not. For the motion to be in SHM

Options:
A. the motion cannot be SHM
B. as x decreases c increases
C. $$c=0$$
D. as x increases c becomes zero
E. as x increases c remains constant
F. as x decreases c remains constant
G. as x increases c remains constatnt
H. as x increases c increases
I. as x increases c decreases
J. as x decreases c remains constatnt
Answer: G
An inductor $(X_{L}=2\Omega)$ a capacitor $(X_{C}=8\ \Omega)$ and a resistance $(R=8\ \Omega)$ are connected is series with an $AC$ source. The voltage output of $AC$ source is given by $V=10\ \cos(100\pi t)$.The instantaneous potential difference between points $A$ and $B$, when the applied voltage is $3/5th$ of the maximum value of applied voltage is:

Options:
A. $$0\ V$$
B. 2\ V
C. 4\ V
D. 5\ V
E. 10\ V
F. $$None\ of\ these$$
G. 8\ V
H. 12\ V
I. $$6\ V$$
Answer: I
A cannon shell fired breaks into two equal parts at its highest point. One part retraces the path to the cannon with energy $E_1$ and kinetic energy of the second part is $E_2$. Relation between $E_1$ and $E_2$ is:

Options:
A. E_2=12E_1
B. E_2=7E_1
C. $$E_2=15E_1$$
D. E_2=6E_1
E. E_2=10E_1
F. $$E_2=9E_1$$
G. E_2=11E_1
H. $$E_2=4E_1$$
I. $$E_2=E_1$$
J. E_2=8E_1
Answer: F
A clock is calibrated at a temperature of $20^\circ C$. Assume that the pendulum is a thin brass rod of negligible mass with a heavy bob attached to the end $({lpha}_{brass}=19\times 10^{-6}/K)$

Options:
A. On a cold day at $10^\circ C$ the clock loses $8.2\:s$
B. On a cold day at $10^\circ C$ the clock gains $6.2\:s
C. On a hot day at $30^\circ C$ the clock gains $8.2\:s$
D. On a cold day at $10^\circ C$ the clock loses $6.2\:s$
E. On a cold day at $10^\circ C$ the clock loses $7.2\:s$
F. On a cold day at $10^\circ C$ the clock gains $7.2\:s$
G. On a hot day at $30^\circ C$ the clock loses $8.2\:s$
H. On a cold day at $10^\circ C$ the clock gains $8.2\:s$
I. On a cold day at $10^\circ C$ the clock gains $9.2\:s$
Answer: H
A plane wavelength $(\lambda =6\times10^{-7}m)$ falls on a slit $0.4mm$ wide. A convex lens of focal length $0.8m$ placed behind the  slit focusses the length on a screen. What is the linear  diameter of second maximum

Options:
A. 0.96\ mm
B. 0.24\ mm
C. 0.80\ mm
D. 0.32\ mm
E. $$3\ mm$$
F. 0.64\ mm
G. $$9\ mm$$
H. $$12\ mm$$
I. $$0.16\ mm$$
J. 0.48\ mm
Answer: I
De-Broglie wavelength of two particles $A$ and $B$ are plotted against $\left (\dfrac {1}{\sqrt {V}}\right )$; where $V$ is potential on the particle. Which of the following relation is correct about mass of particle.

Options:
A. $$M_{A} \geq M_{B}$$
B. $$M_{A} < M_{B}$$
C. M_{A} = \frac{4}{3}M_{B}
D. $$M_{A} > M_{B}$$
E. M_{A} = \frac{3}{2}M_{B}
F. M_{A} = 2M_{B}
G. M_{A} = \frac{5}{3}M_{B}
H. $$M_{A} \leq M_{B}$$
I. $$M_{A} = M_{B}$$
Answer: D
A small object is enclosed in a sphere of solid glass  $8 cm$  in radius. It is situated  $2 cm$  from centre and is viewed from the side to which it is nearest. Where will it appear to be it  $\mu$  of glass  $= 1.5$ ?

Options:
A. $$6 cm$$
B. 5\dfrac{1}{6} cm
C. 5\dfrac { 1 }{ 2 } { cm }\quad
D. 5\dfrac { 1 }{ 4 } { cm }\quad
E. 5\dfrac{1}{2} cm
F. $$3\dfrac { 2 }{ 3 } { cm }$$
G. $$4 cm$$
H. 7 cm
I. 5\dfrac { 2 }{ 3 } { cm }\quad
J. $$5\dfrac { 1 }{ 3 } { cm }\quad $$
Answer: A
A weightless rod of length $l$ with a small load of mass $m$ at the end is hinged at point $A$ as shown and occupies a strictly vertical position, touching a body of body of mass $M$. A light jerk sets the system in motion.
(a) For what mass ratio $M/m$ will the rod form an angle $lpha=\pi /6$ with the horizontal at the moment of the separation from the body?
(b) What will be the velocity $u$ of the body at this moment? Friction should be neglected.

Options:
A. (a) $4$;  (b) $\cfrac { 1 }{ 2 } \sqrt { \cfrac { gl }{ 4 }  }
B. (a) $5$;  (b) $\cfrac { 1 }{ 3 } \sqrt { \cfrac { gl }{ 2 }  } $
C. (a) $6$;  (b) $\cfrac { 1 }{ 2 } \sqrt { \cfrac { gl }{ 2 }  } $
D. (a) $7$;  (b) $\cfrac { 1 }{ 2 } \sqrt { \cfrac { gl }{ 2 }  }
E. (a) $4$;  (b) $\cfrac { 1 }{ 2 } \sqrt { \cfrac { gl }{ 2 }  } $
F. (a) $6$;  (b) $\cfrac { 1 }{ 4 } \sqrt { \cfrac { gl }{ 2 }  } $
G. (a) $4$;  (b) $\cfrac { 1 }{ 3 } \sqrt { \cfrac { gl }{ 2 }  } $
H. (a) $5$;  (b) $\cfrac { 1 }{ 2 } \sqrt { \cfrac { gl }{ 2 }  }
I. (a) $4$;  (b) $\cfrac { 1 }{ 2 } \sqrt { \cfrac { gl }{ 3 }  }
J. (a) $3$;  (b) $\cfrac { 1 }{ 2 } \sqrt { \cfrac { gl }{ 2 }  }
Answer: E