MikaStars39/nano-eval huggingface dataset:
A charged particle of mass 'm' and charge 'q' moving under the influence of uniform electric field $Eec{i}$ and a uniform magnetic field $Bec{k}$ follows a trajectory from point P to Q as shown in figure. The velocities at P and Q are respectively, $vec{i}$ and $-2vec{j}$. Then which of the following statements (A, B, C, D) are the correct? (Trajectory shown in schematic and not to scale)
(A) $E=\dfrac{3}{4}\left(\dfrac{mv^2}{qa}\right)$
(B) Rate of work done by the electric field at P is $\dfrac{3}{4}\left(\dfrac{mv^3}{a}\right)$
(C) Rate of work done by both the fields at Q is zero
(D) The difference between the magnitude of angular momentum of the particle at P and Q is $2$ mav.

Options:
A. (B), (C), (D)
B. (A), (C), (D)
C. (A), (C), (B)
D. (A), (C), (E)
E. (A), (D), (C)
F. (B), (C), (A)
G. (A), (B), (E)
H. (A), (B), (C), (D)
I. (A), (B), (D)
J. (A), (B), (C)
Answer: J
Two moles of an ideal mono-atomic gas is taken through a cyclic process as shown in the $P-T$ diagram. In the process $BC, PT^2$ = constant. Then the ratio of heat absorbed and heat released by the gas during the process $AB$ and process $BC$ respectively is:-

Options:
A. $$\frac{3}{2}$$
B. 8
C. $$\frac{6}{5}$$
D. $$4$$
E. 1
F. $$6$$
G. $$3$$
H. $$5$$
Answer: C
A rectangular loop PQRS, is being pulled with constant speed into a uniform transverse magnetic fieldby a force F (as shown). E.m.f. induced in side PS and potential difference between points P and S respectively are (Resistance of the loop = r)

Options:
A. $$Zero , \displaystyle \frac {2Fr} {3 B \ell}$$
B. $$Zero, \displaystyle \frac {Fr} {5 B \ell}$$
C. $$Zero, \displaystyle \frac {Fr} {B \ell} $$
D. $$Zero , \displaystyle \frac {3Fr} {10 B \ell}$$
E. $$Zero , \displaystyle \frac {Fr} {7 B \ell}$$
F. $$\displaystyle \frac {Fr} {6 B \ell} , \frac {Fr} {6 B \ell} $$
G. $$Zero , \displaystyle \frac {Fr} {4 B \ell}$$
H. $$Zero , \displaystyle \frac {Fr} {8 B \ell}$$
I. $$Zero, zero$$
J. $$Zero , \displaystyle \frac {Fr} {6 B \ell} $$
Answer: J
$(IE)_{1} \space and \space (IE)_{2}$ of $Mg_{(g)}$ are 740, 1540 kJ $mol^{-1}$. Calculate percentage of $Mg_{(g)}^{+}$ and $Mg_{(g)}^{2+}$, if 1 g of $Mg_{(g)}$ absorbs 50.0 k J of energy.

Options:
A. %$Mg^{+} = 68.00$ and %$Mg^{+2} = 32.00$
B. %$Mg^{+} = 70.13$ and %$Mg^{+2} = 29.87$
C. %$Mg^{+} = 67.00$ and %$Mg^{+2} = 32.00$
D. %$Mg^{+} = 69.50$ and %$Mg^{+2} = 30.50$
E. %$Mg^{+} = 75$ and %$Mg^{+2} = 25$
F. %$Mg^{+} = 68.65$ and %$Mg^{+2} = 31.65$
G. %$Mg^{+} = 50$ and %$Mg^{+2} = 50$
H. %$Mg^{+} = 70$ and %$Mg^{+2} = 30$
I. %$Mg^{+} = 71.35$ and %$Mg^{+2} = 28.65$
J. %$Mg^{+} = 67.00$ and %$Mg^{+2} = 33.00$
Answer: F
A particle of mass $m$ carrying charge $q$ is lying at the origin at the uniform magnetic field directed along $Y-$ axis. At time $t=0$, it is given velocity $v_{0}$ at angle $\alpha $ with the $X-$ axis as shown in the figure. The coordinates of the particle after one revolution is

Options:
A. $$(0, -({ v }_{ 0 }\sin {\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
B. $$(0, ({ v }_{ 0 }\cos { 2\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
C. $$(0, ({ v }_{ 0 }\cos {\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
D. $$(0, -({ v }_{ 0 }\cos {\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
E. $$(0, ({ v }_{ 0 }\sin { 2\alpha - \pi }) \cfrac { 2\pi m }{ qB }, 0)$$
F. $$(0, ({ v }_{ 0 }\sin { 4\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
G. $$(0, ({ v }_{ 0 }\sin { 3\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
H. $$(0, ({ v }_{ 0 }\sin {\alpha + \pi }) \cfrac { 2\pi m }{ qB }, 0)$$
I. $$(0, ({ v }_{ 0 }\sin {\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
J. $$(0, ({ v }_{ 0 }\sin { 2\alpha }) \cfrac { 2\pi m }{ qB }, 0)$$
Answer: I
At t=o, a block of mass m is projeted with speed v on a rough long plank of mass 3m placed on smooth ground as shown. Work done by friction on the block between t=0 until slipping ceases equals

Options:
A. $$\dfrac { 15 }{ 64 } { mv }^{ 2 }$$
B. $$\dfrac { { mv }^{ 2 } }{ 8 } $$
C. $$\dfrac { 45 }{ 64 } { mv }^{ 2 }$$
D. $$\dfrac { 15 }{ 40 } { mv }^{ 2 }$$
E. $$\dfrac { 15 }{ 48 } { mv }^{ 2 }$$
F. $$\dfrac { -15 }{ 32 } { mv }^{ 2 }$$
G. $$\dfrac { 15 }{ 32 } { mv }^{ 2 }$$
H. $$\dfrac { 15 }{ 16 } { mv }^{ 2 }$$
I. $$\dfrac { { mv }^{ 2 } }{ 16 } $$
J. $$\dfrac { 3 }{ 16 } { mv }^{ 2 }$$
Answer: G
In resonating series LCR circuit containing a resistance of $R= 240 \Omega$, and a source of angular frequency $90 \ rad/s$. If voltage across resistance is $80 \ volt$ and across inductor is $60 \ volts$, then

Options:
A. Angular frequency at which current lags the voltage by $45^\circ$ is $\omega'= 90+ 10 \sqrt{3} \ rad/s$
B. Angular frequency at which current lags the voltage by $45^\circ$ is $\omega'= 90+ 30 \sqrt{3} \ rad/s$
C. Angular frequency at which current lags the voltage by $45^\circ$ is $\omega'= 90+ 10 \sqrt{37} \ rad/s$
D. $$\omega'= 150 \ rad/s$$
E. Angular frequency at which current lags the voltage by $45^\circ$ is $\omega'= 180 \ rad/s$
F. Value of $L= 2H$ and $C= \frac{1}{21600} F$
G. Angular frequency at which current lags the voltage by $45^\circ$ is $\omega'= 90+ 10 \sqrt{33} \ rad/s$
H. Value of $L= 2H$ and $C= \frac{1}{16200} F$
I. Angular frequency at which current lags the voltage by $45^\circ$ is $\omega'= 60+ 10 \sqrt{117} \ rad/s$
J. $$\omega'= 120 \ rad/s$$
Answer: E
A police car moving at $22m/s$, chases a motorcyclist. The police man sounds his horn at $176Hz$, while both of them move towards a stationary siren of frequency $165Hz$. Calculate the speed of the motorcycle, if it is given that he does not observe any beats. (velocity of sound $=330m/s$)

Options:
A. $$24m/s$$
B. $$22m/s$$
C. $$21m/s$$
D. $$28m/s$$
E. $$zero$$
F. $$17m/s$$
G. $$20m/s$$
H. $$33m/s$$
I. $$11m/s$$
J. $$19m/s$$
Answer: B
A person is stuck in Antarctica in a ship wreck. He doesn't have any drinking water available. He finds an ethylene ($CH_2 = CH_2$) cylinder containing 4.2 g of ethylene. He decides to burn this ethylene to melt 1kg of ice at $-10^0C$. When a mole of ethylene is combusted it releases 340 kcal of energy. Assuming, no heat is lost to the environment, the amount of water available for drinking is...

Options:
A. 625 ml
B. 575 ml
C. 600 ml
D. 850 ml
E. 900 ml
F. 1 litre
G. It is not sufficient to melt the ice
H. 550 ml
I. 750 ml
J. 525 ml
Answer: B
Three moles of an ideal gas $\left (C_{P} = \dfrac {7R}{2}\right )$ at pressure $P_{A}$ and temperature $T_{A}$ is isothermally expanded to twice the initial volume. The gas is then compressed at constant pressure to its original volume. Finally the gas is heated at constant volume to its original pressure $P_{A}$.
Calculate the net work done by the gas and the net heat supplied to the gas during the complete process.

Options:
A. $0.9\ RT_{A}, \triangle Q = 0.779\ RT_{A}$.
B. $$0.479\ RT_{A}, \triangle Q = 0.579\ RT_{A}$$
C. $$0.679\ RT_{A}, \triangle Q = 0.779\ RT_{A}$$
D. $79\ RT_{A}, \triangle Q = 0.679\ RT_{A}$.
E. $0.7\ RT_{A}, \triangle Q = 0.979\ RT_{A}$.
F. $$0.679\ RT_{A}, \triangle Q = 0.579\ RT_{A}$$
G. $$0.679\ RT_{A}, \triangle Q = 0.879\ RT_{A}$$
H. $$0.579\ RT_{A}, \triangle Q = 0.779\ RT_{A}$$
I. $0.579\ RT_{A}, \triangle Q = 0.579\ RT_{A}$.
J. $$0.679\ RT_{A}, \triangle Q = 0.679\ RT_{A}$$
Answer: I
A perfectly elastic uniform string is suspended vertically with its upper end fixed to the ceiling and the lower end loaded with the weight. If a transverse wave is imparted to the lower end of the string, the pulse will:

Options:
A. travelled upwards with decreasing speed
B. not travel along the length of the string
C. travelled upwards with constant acceleration
D. travel upwards with decreasing speed
E. travel upwards with increasing acceleration
F. travelled upwards with decreasing acceleration
G. travel upwards with constant speed
H. travel upwards with decreasing acceleration
I. travel upwards with constant velocity
Answer: C
A cork cube of side $ 2 \ cm $ floats on water. Determine its depth of immersion. Densiy of cork=$240 \ kg \ m^{-3}$. Surface tension of water =$0.07 \ Nm^{-1}$, angle of contact=$0^o$

Options:
A. $$0.52 \ cm$$
B. $$0.57 \ cm$$
C. $$0.62 \ cm$$
D. $$0.72 \ cm$$
E. $$0.32 \ cm$$
F. $$0.77 \ cm$$
G. $$0.42 \ cm$$
H. $$0.47 \ cm$$
I. $$0.67 \ cm$$
Answer: C
A car of mass $1000\ kg$ moving on a circular track of radius $100\ m$ with increase in speed at a rate of $8\ m/s^{2}$, if at an instant its speed is $30\ m/s$ then the acceleration of the car is (at this instant only)

Options:
A. $$<12$$
B. $$( <10 )$$
C. $$( =10 )$$
D. $$>12$$
E. None of these
F. $$=12$$
G. $$( =9 )$$
H. $$=11$$
I. $$<10$$
J. $$10$$
Answer: F
From the following data :
Enthalpy of formation of $\displaystyle CH_{3}CN=87.86$ kJ / mol
Enthalpy of formation of $\displaystyle C_{2}H_{6}=-83.68$ kJ / mol
Enthalpy of sublimation of graphite $\displaystyle =719.65$ kJ / mol
Enthalpy of dissociation of nitrogen $\displaystyle =945.58$ kJ / mol
Enthalpy of dissociation of $\displaystyle H_{2}=435.14$ kJ / mol
$\displaystyle C-H$ bond enthalpy $\displaystyle =414.22$ kJ / mol
Calculate the bond enthalpy of ( i ) $\displaystyle C-C$ ; ( ii ) $\displaystyle C\equiv N$

Options:
A. ( i ) 343.58 kJ / $mol^{-1}$ ; ( ii ) 870.2 kJ / $mol^{-1}$
B. ( i ) 370.58 kJ / $\displaystyle mol^{-1}$ ; ( ii ) 891.2 kJ / $\displaystyle mol^{-1}$
C. ( i ) 370.58 kJ / $\displaystyle mol^{-1}$ ; ( ii ) 860.2 kJ / $\displaystyle mol^{-1}$
D. ( i ) 343.58 kJ / $\displaystyle mol^{-1}$ ; ( ii ) 891.2 kJ / $\displaystyle mol^{-1}$
E. ( i ) 360.58 kJ / $\displaystyle mol^{-1}$ ; ( ii ) 891.2 kJ / $\displaystyle mol^{-1}$
F. ( i ) 370.58 kJ / $mol^{-1}$ ; ( ii ) 840.2 kJ / $mol^{-1}$
G. ( i ) 370.58 kJ / $\displaystyle mol^{-1}$ ; ( ii ) 850.2 kJ / $\displaystyle mol^{-1}$
H. ( i ) 343.58 kJ / $\displaystyle mol^{-1}$ ; ( ii ) 850.2 kJ / $\displaystyle mol^{-1}$
I. ( i ) 370.58 kJ / $mol^{-1}$ ; ( ii ) 870.2 kJ / $mol^{-1}$
J. ( i ) 370.58 kJ / $mol^{-1}$ ; ( ii ) 880.2 kJ / $mol^{-1}$
Answer: B
A car staring from rest acceleration at the rate f through a distance s, then continues at constant speed for time t and then deceleration at the rate f/2 to come to rest. I f the metal distance traversed is 5s, then

Options:
A. $$s=\dfrac { 1 }{ 2 } { ft }^{ 2 }$$
B. $$s=\dfrac { 1 }{ 4 } { ft }^{ 2 }$$
C. $$s=\dfrac { 1 }{ 3 } { ft }^{ 2 }$$
D. $$s=\dfrac { 1 }{ 8 } { ft }^{ 2 }$$
E. $$s=\dfrac{1}{7} ft^2$$
F. $$s=\dfrac { 1 }{ 72 } { ft }^{ 2 }$$
G. $$s=\dfrac { 1 }{ 6 } { ft }^{ 2 }$$
H. $$s=\dfrac { 1 }{ 10 } { ft }^{ 2 }$$
I. $$s = \dfrac{1}{9} ft^2$$
J. $$s=\dfrac { 1 }{ 5 } { ft }^{ 2 }$$
Answer: F
For which metal $'M'$ complex
$[M(n^{1}-C_{5}H_{5})(n^{3}-C_{5}H_{5})(n^{5}-C_{5}H_{5})]$ 
follows EAN rule ($n^{x}$, where x=Hapticity=number of atoms participtate in donation)

Options:
A. Pt
B. Ru
C. $$Co$$
D. $$Zn$$
E. $$Cd$$
F. $$Cu$$
G. $$Ag$$
H. $$Ni$$
I. $$Fe$$
Answer: D
A bird is flying at a height of $3.6m$ above the surface of water and a fish is in the water at a depth of $1.2m$. The apparent height of bird to the fish is ($\mu_{water}= \dfrac{4}{3})$

Options:
A. $$4.5m$$
B. $$4.2m$$
C. $$6.0m$$
D. $$5.4m$$
E. $$3.9m$$
F. $$4.4m$$
G. $$5.6m$$
H. $$4.8m$$
I. $$5.2m$$
J. $$5.8m$$
Answer: C
In the dissociation of $PCl_5$ as:$PCl_5 (g)\rightleftharpoons PCl_3 (g)+ Cl_2(g)$ if the degree of dissociation is $\alpha$ at equilibrium pressure P, then the the equilibrium constant for the reaction is:

Options:
A. $$K_p = \dfrac {lpha^2P}{1+lpha^2}$$
B. $$K_p = \dfrac {\alpha P^2}{1 - \alpha P^2}$$
C. $$K_p = \dfrac {\alpha^2P^2}{1-\alpha^2P}$$
D. $$K_p = \dfrac {\alpha^2}{1+\alpha^2P}$$
E. $$K_p = \dfrac {\alpha P^2}{1+\alpha^2P}$$
F. $$K_p = \dfrac {\alpha P^2}{1+\alpha^2}$$
G. $$K_p = \dfrac {\alpha P^2}{1-\alpha^2P}$$
H. $$K_p = \dfrac {\alpha^2P}{1-\alpha^2}$$
I. $$K_p = \dfrac {\alpha P^2}{1-\alpha^2}$$
J. $$K_p = \dfrac {\alpha^2 P}{1 - \alpha P}$$
Answer: I
A Body of mass m, moving along the positive X-direction is subjected to a resistive force $F = K{v^2}$  ( where K is a constant and v the particle velocity). If $m = 10kg,v = 10m/s\;{\text{at}}\;t = 0,\;{\text{and}}\;K = 2N/{m^2} - {s^{ - 2}}$ the velocity when t = 2s is

Options:
A. $$\dfrac{{10}}{3}m/s$$
B. $$\dfrac{10}{6}m/s$$
C. $$2 m/s$$
D. $$-\dfrac{10}{2}m/s$$
E. $$\dfrac{{3}}{10}m/s$$
F. $$\dfrac{{10}}{2}m/s$$
G. $$\dfrac{{10}}{5}m/s$$
H. $$\dfrac{{10}}{4}m/s$$
I. $$-\dfrac{{10}}{3}m/s$$
J. $$\dfrac{{10}}{7}m/s$$
Answer: I
$E^\circ _{Fe^{3+}/Fe} = -0.441\, V$ and $E^\circ _{Fe^{3+}/Fe^{2+}} = 0.771\, V$, The standard emf of the reaction $Fe + 2F{e^{3 + }} \to 3F{e^{2 + }}$ will be

Options:
A. $$0.888\, V$$
B. $$0.330\, V$$
C. $$1.653\, V$$
D. $$0.555\, V$$
E. $$0.440\, V$$
F. $$0.110\, V$$
G. $$0.666\, V$$
H. $$1.212\, V$$
I. $$0.222\, V$$
J. $$0.111\, V$$
Answer: H
The difference between heat of reaction at pressure and constant volume for the reaction, ${ CH }_{ 2 }={ CH }_{ 2 }(g)+{ 3O }_{ 2 }(g)\rightarrow { 2CO }_{ 2 }(g)+{ 2H }_{ 2 }O(l)$ at 300 K is?

Options:
A. -7.25 kJ
B. -4.99 kJ
C. -5.87 kJ
D. 6.89 kJ
E. 6.37 kJ
F. 6.41 kJ
G. 5.87 kJ
H. 6.87 kJ
I. 6.53 kJ
J. 6.35 kJ
Answer: G
Enthalpy of formation of $NH_{3}$ is $-X\ kJ$ and $\triangle H_{H - H}, \triangle H_{N-H}$ are respectively $Y\ kJ\ mol^{-1}$ and $Z\ kJ \ mol^{-1}$. The value of $\triangle H_{N = N}$ is

Options:
A. $$Y - 6Z - \dfrac {X}{3}$$
B. $$Y - 6Z + \dfrac{X}{4}$$
C. $$Y + 6X + Z$$
D. $$3Y + 6Z + X$$
E. $$Y - 6Z + \dfrac {X}{3}$$
F. $$Y - 6Z - \dfrac{X}{2}$$
G. $$-3Y + 6Z - 2X$$
H. $$Y - 6Z + \dfrac{2X}{3}$$
I. $$3Y - 6Z - X$$
Answer: G
A coil of radius R carries a current I. Another concentric coil of radius r ( r < < R) carries current $\displaystyle \frac{I}{2} $. Initially planes of the two coils are mutually perpendicular and both the coils are free to rotate with M and m respectively (m < M). During the subsequent motion, let $\displaystyle K_{1}\: and\: K_{2} $ be the maximum kinetic energies of the two coils respectively and let $U$ be the magnitude of maximum potential energy of magnetic interaction of the system of the coils. Choose the correct options :

Options:
A. \(\displaystyle K_{1} \gg K_{2} \)
B. \(\displaystyle K_{2} > K_{1} \)
C. $$\displaystyle K_{1}=\frac{Umr^{2}}{mr^{2}+MR^{2}},K_{2}=\frac{UMR^{2}}{mr^{2}+MR^{2}}$$
D. $$\displaystyle K_{1} > K_{2}$$
E. \(\displaystyle K_{2} \approx K_{1}\)
F. $$\displaystyle K_{2}> > K_{1}$$
G. $$\displaystyle \frac{K_{1}}{K_{2}}=\frac{M}{m}\left ( \frac{R}{r} \right )^{2} $$
Answer: F
A uniform rod AB of mass 2kg and length 1m is placed on a sharp support O such that AO=0.4 m and OB=0.6 m. A spring of force constant k=600 N/m is attached to end B. To keep the rod horizontal, its end A is tied with a thread such that spring is elongated by 1 cm. If reaction of support O on the rod when thread is burnt is R, then the value of $\dfrac { R }{ 10 } $ is:

Options:
A. 3
B. 6
C. 4
D. 1
E. 3.5
F. 5
G. 2
H. 3.2
I. 8
J. 7
Answer: C
The resistance of a copper wire and an iron wire at 20C are 4.1$\Omega $ and 3.9$\Omega $ respectively. Neglecting any thermal expansion, find the temperature at which resistances of bolh are equal. 
$\alpha _{ cu }=4.0\times { 10 }^{ -3 }{ K }^{ -1 }$. and $\alpha _{ fe }=5.0\times { 10 }^{ -3 }{ K }^{ -1 }$.

Options:
A. $${ 95 }^{ o }C$$
B. $${ 75 }^{ o }C$$
C. $${ 280 }^{ o }C$$
D. $${ 268 }^{ o }C$$
E. $${ 220 }^{ o }C$$
F. $${ 255 }^{ o }C$$
G. $${ 180 }^{ o }C$$
H. $${ 265 }^{ o }C$$
I. $${ 285 }^{ o }C$$
J. $${ 65 }^{ o }C$$
Answer: E
A specific gravity bottle contains 100gm of water at 4$^{0}C$. Calculate the mass of water in the bottle at $40^{0}C$ $(\gamma _{R}=2.7\times 10^{-4/0}C, \gamma _{g}=4.0\times  0^{-4/0}C)$

Options:
A. 100gm
B. 98.43gm
C. 99.04gm
D. 99.14gm
E. 98.14gm
F. 98.41gm
G. 9.914gm
H. 98.42gm
I. 0.9914gm
J. 98.40gm
Answer: A
$9.0\, g $ water is added into oleum sample labelled as $112\ % $ $H_2SO_4 $ then the amount free $SO_3 $ remaining in the solution is: (STP = $1$ atm and $273\,K $)

Options:
A. 10.24\,L
B. $3.73\,L $ at STP
C. $11.2\,L $ at STP
D. 4.73\,L at STP
E. 5.46\,L at STP
F. $14.93 \,L $ at STP
G. $5.92\,L $ at STP
H. $2.73\,L $ at STP
I. $7.46\,L $ at STP
J. $4.26\,L $ at STP
Answer: B
The end $B$ of the rod $AB$ which makes angle $\theta$ with the floor is being pulled with a constant velocity $v_{0}$ as shown. The length of the rod is $l$

Options:
A. At $\theta= 37^{\circ}$ angular velocity of rod is $\displaystyle \frac{5v_{0}}{3l}$
B. Velocity of end $A$ is constant
C. $$\frac{3v_{0}}{5l}$$
D. At $\theta= 37^{\circ}$ angular velocity of rod is $\displaystyle \frac{5v_{0}}{4l}$
E. $$\frac{5v_{0}}{4l}$$
F. At $\theta= 37^{\circ}$ angular velocity of rod is $\displaystyle \frac{3v_{0}}{4l}$
G. At $\theta=37^{\circ}$ velocity of end A is $\displaystyle \frac{4}{3}v_{0}$ downwards
H. $$\frac{3v_{0}}{4l}$$
I. Angular velocity of rod is constant
J. $$\frac{4v_{0}}{3l}$$
Answer: A
A ball falls vertically downward and bounces off a horizontal floor. The speed of the ball just before reaching the floor $(u_1)$ is equal to the speed just after leaving contact with the floor $(u_2);\, u_1\, =\, u_2$. The corresponding magnitudes of accelerations are denoted respectively by $a_1$ and $a_2$. The air resistance during motion is proportional to speed and is not negligible. If $g$ is acceleration due to gravity, then

Options:
A. $$a_1\, <\, a_2\, -\, g$$
B. $$a_1\, >\, a_2\, +\, g$$
C. $$a_1\, =\, a_2\, +\, g$$
D. $a_1\, =\, a_2\,
eq\, g$
E. $$a_1\, =\, a_2\, =\, 2g$$
F. $$a_1\, <\, a_2\, +\, g$$
G. $$a_1\, <\, a_2$$
H. $$a_1\, =\, a_2\, =\, g$$
I. $$a_1\, >\, a_2$$
J. $$a_1\, >\, a_2\, +\, 2g$$
Answer: D
A block of mass $m_1$ = 150 kg is at rest on a very long frictionless table, one end of which is terminated in a wall. Another block of mass $m_2$ is placed between the first block and the wall, and set in motion towards $m_1$ with constant speed $u_2$. Assuming that all collisions are completely elastic, find the value of $m_2$ for which both blocks move with the same velocity after $m_2$ has collided once with $m_1$ and once with the wall (The wall has effectively infinite mass)-

Options:
A. 20 kg
B. 12 kg
C. 10 kg
D. 45 kg
E. 40 kg
F. 15 kg
G. 50 kg
H. 35 kg
I. 25 kg
J. 30 kg
Answer: C
A $\alpha -$ particle moving in a circular orbit of radius a makes $n$ revolutions per second. The magnetic field produced at the centre has a value $B$

Options:
A. $$B = \mu_0 n e / 9a \pi$$
B. $$B = zero$$
C. B = \mu_0 n e / 7a \pi
D. $$B = \mu_0 ne/a$$
E. $$B = \mu_0 n e / 5a$$
F. $$B = \mu_0 ne/2a \pi$$
G. $$B = \mu_0 ne/4a \pi$$
H. $$B = \mu_0 n^2 e/a$$
I. $$B = \mu_0 n e / 6a \pi$$
J. $$( B = \mu_0 n e / 3a )$$
Answer: B
A heavy particle slides under gravity down the inside of a smooth vertical tube held in vertical plane. It starts from the highest point with velocity $\displaystyle \sqrt{2ag}$ where a is the radius of the circle. Find the angular position $\displaystyle \theta $ (as shown in figure) at which the vertical acceleration of the particle is maximum.

Options:
A. $$\displaystyle \theta = \cos ^{-1}\left ( \frac{3}{4} \right )$$
B. $$\displaystyle \theta = \cos^{-1}\left( \frac{2}{7} \right)$$
C. $$\displaystyle \theta = \cos^{-1}\left( \frac{1}{6} \right)$$
D. $$\displaystyle \theta = \cos^{-1}\left( \frac{1}{2} \right)$$
E. \displaystyle \theta = \cos^{-1}\left( \frac{1}{5} \right)
F. $$\displaystyle \theta = \cos ^{-1}\left ( \frac{1}{3} \right )$$
G. $$\displaystyle \theta = \cos ^{-1}\left ( \frac{2}{3} \right )$$
H. $$\displaystyle \theta = \cos^{-1}\left( \frac{1}{4} \right)$$
I. $$\displaystyle \theta = \cos ^{-1}\left ( \frac{2}{5} \right )$$
J. $$\displaystyle \theta = \cos^{-1}\left( \frac{3}{5} \right)$$
Answer: G
If two solution of strong acids of pH =2 and pH=4 be mixed in volume ratio 1:z, the pH of resulting solution was founds to be 2.75, then z is (log 1.75=0.25)

Options:
A. 10
B. 8
C. 6
D. 3.5
E. 2
F. 3
G. 7
H. 4
I. 4.5
J. 5
Answer: E
The circuitshown in Fig. consists of a battery of e.m.f. $ arepsilon=10 \mathrm{V} ; $ a capacitor of capacitance $ C=1.0 \mu \mathrm{F} $ and three resistors of values $ R_{1}=2 \Omega, R_{2}=2 \Omega $ and $ R_{3}=1 \Omega $. Initially, the capacitor is completely uncharged and the switch $ S $ is open. The switch $ S $ is closed at $ t=0 . $ Then

Options:
A. the current through resistor \( R_{3} \) a long time after the switch closed is 3 A
B. the current through resistor \( R_{3} \) a long time after the switch closed is 2 A
C. the current through resistor \( R_{3} \) a long time after the switch closed is 1 A
D. the current through resistor $ R_{3} $ a long time after the switch closed is 5 A
E. all of these
F. the current through resistor $ R_{3} $ a long time after the switch closed is 1.5 A
G. the ratio of current through $ R_{1} $ and $ R_{2} $ is always constant
H. the current through resistor \( R_{3} \) a long time after the switch closed is 4 A
I. the current through resistor $ R_{3} $ a long time after the switch closed is 2.5 A
J. the current through resistor $ R $, at the moment the switch closed is zero
Answer: E
Suppose a reaction has $\Delta H = -40 \, \text{KJ}$ and $\Delta S = -50 \, \text{J/K}$. At what temperature range will the reaction become spontaneous from non-spontaneous?

Options:
A. 800 K to 798 K
B. 800 K to 803 K
C. 800 K to 799 K
D. 801 K to 799 K
E. 800 K to 802 K
F. 800 K to 801 K
G. 799 K to 803 K
H. 799 K to 801 K
I. 799 K to 800 K
J. 799 K to 802 K
Answer: D
A block of mass m is hung on a spring of spring constant k and of negligible mass. The spring extends by x on the surface of the earth. When taken to a height h = R/2, the spring will extend by (R = radius of earth)

Options:
A. $$\frac{x}{2}$$
B. $$\frac{5x}{8}$$
C. $$\frac{3x}{8}$$
D. $$\frac{3x}{4}$$
E. $$\frac{2x}{3}$$
F. $$\frac{4x}{9}$$
G. $$x$$
H. $$\frac{3x}{5}$$
I. $$\frac{5x}{12}$$
J. $$\frac{7x}{12}$$
Answer: G
A Voltaic cell consists of an electrode of solid silver immersed in 0.10M ${ AgNO }_{ 3 }$ Solution and an electrode of unknown metal X immersed in a 0.10M solution ${ X(NO }_{ 3 })_{ 2 }$. A porous barrier separates the two half of the cell. Also given:
$E\left( { Ag^{ + } }/{ Ag } \right)$ = 0.80V and ${ E }_{ cell }$ = 1.05V and at 25 degree Celsius
$${ X_{ (s) } }/{ { X }^{ 2+ } }(1.0M)\parallel { Ag }^{ + }(0.1M)/{ Ag }_{ (s) }$$
Which of the following statements regarding the cell and X is correct?

Options:
A. Standard Emf (E) of ${ { X }^{ 2+ } }/{ X }$ is -0.25V at 25 degree Celsius
B. As cell operate, concentrations of both ${ X }^{ 2+ }and{ Ag }^{ + }$ increases in their respective half cells
C. As cell operates, concentration of ${ X }^{ 2+ }$ decreases in anode chamber while concentration of ${ Ag }^{ + }$ increases in cathode chamber
D. As cell operate, concentration of ${ X }^{ 2+ }$ increases in anode chamber while concentration of ${ Ag }^{ + }$ increases in cathode chamber
E. As cell operate, concentration of ${ X }^{ 2+ }$ decreases in anode chamber while concentration of ${ Ag }^{ + }$ decreases in cathode chamber
F. As cell operate, concentration of ${ X }^{ 2+ }$ decreases in anode chamber while concentration of ${ Ag }^{ + }$ remains constant in cathode chamber
G. As cell operate, concentration of ${ X }^{ 2+ }$ increases in anode chamber while concentration of ${ Ag }^{ + }$ decreases in cathode chamber
H. As cell operates, concentration of ${ X }^{ 2+ }$ increases in anode chamber while concentration of ${ Ag }^{ + }$ remains constant in cathode chamber
I. As cell operate, concentration of ${ X }^{ 2+ }$ decreases in anode chamber while concentration of ${ Ag }^{ + }$ increases in cathode chamber
J. X is a stronger reducing agent than ${ H }_{ 2{ (g) } }$
Answer: B
A parallel plate capacitor having plates of area $S$ and plate separation $d$, has capacitance ${C}_{1}$ in air. When two dielectrics of different relative permittivities (${ \epsilon }_{ 1 }=1$ and ${ \epsilon }_{ 2 }=4$) are introduced between the two plates as shown in the figure, the capacitance become ${C}_{2}$. The ratio $\dfrac { { C }_{ 2 } }{ { C }_{ 1 } }$ is

Options:
A. $$8/3$$
B. $$7/5$$
C. $$7/4$$
D. $$8/5$$
E. $$5/2$$
F. $$6/5$$
G. $$7/3$$
H. $$3/2$$
I. $$5/3$$
Answer: G
$A(aq)\longrightarrow B(aq)+C(aq)$ is a first order reaction.
Time
$$t$$
$$\infty$$
moles of reagent
$${ n }_{ 1 }$$
$${ n }_{ 2 }$$
Reaction progress is measure with help of titration '$R$'. If all $A,B$ and $C$ reacted with reagent and have '$n$' factors [$n$ factor; $eq.wt=\cfrac {mol.wt.}{n}$] in the ratio of $1:2:3$ with the reagent. The $k$ in terms of $t,{ n }_{ 1 }$ and ${ n }_{ 2 }$ is :

Options:
A. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac {6 { n }_{ 2 } }{5( { n }_{ 2 }-{ { n }_{ 1 }) } } \right) }$$
B. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac { 2{ n }_{ 2 } }{ { n }_{ 2 }-{ { n }_{ 1 } } } \right) } $$
C. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac {3 { n }_{ 2 } }{4( { n }_{ 2 }-{ { n }_{ 1 }) } } \right) }$$
D. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac {4 { n }_{ 2 } }{ { n }_{ 2 }-{ { n }_{ 1 } } } \right) } $$
E. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac { { n }_{ 2 } }{ { n }_{ 2 }-{ { n }_{ 1 } } } \right) } $$
F. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac { 7 { n }_{ 2 } }{ 5 ( { n }_{ 2 } - { n }_{ 1 } ) } \right) }$$
G. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac {6 { n }_{ 2 } }{4( { n }_{ 2 }-{ { n }_{ 1 }) } } \right) }$$
H. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac {4 { n }_{ 2 } }{5( { n }_{ 2 }-{ { n }_{ 1 }) } } \right) } $$
I. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac { 5 { n }_{ 2 } }{ 3 ( { n }_{ 2 } - { n }_{ 1 } ) } \right) }$$
J. $$k=\cfrac { 1 }{ t } \ln { \left( \cfrac {3 { n }_{ 2 } }{ { n }_{ 2 }-{ { n }_{ 1 } } } \right) }$$
Answer: H
Two concentric coils each of radius equal to 2 $\pi$ cm are placed at right angles to each other. 3 ampere and 4 ampere are the currents flowing in each coil respectively. The magnetic induction in weber/m$^2$ at the centre of the coils will be
$$(\mu_0 = 4 \pi \times 10^{-7} Wb/ A.m)$$

Options:
A. $$10^{-5}$$
B. $$13 \times 10^{-5}$$
C. $$8 \times 10^{-5}$$
D. $$14 \times 10^{-5}$$
E. $$12 \times 10^{-5}$$
F. $$9 \times 10^{-5}$$
G. $$7 \times 10^{-5}$$
H. $$ 6\times 10^{-5}$$
I. $$11 \times 10^{-5}$$
J. $$5 \times 10^{-5}$$
Answer: G
If the sum of the intensities of two component waves are 5I units and upon superposition with a phase difference of $\pi $ radians, their resultant is 2I, what are the intensities of component waves

Options:
A. $$I_1=2I, I_2=3I$$
B. $$I_1=3.5I, I_2=1.5I$$
C. $$I_1=I, I_2=4I$$
D. $$I_1=4I, I_2=1I$$
E. $$I_1=2I, I_2=2I$$
F. $$I_1=4.5I, I_2=0.5I$$
G. $$I_1=2.5I, I_2=2.5I$$
H. $$I_1=3I, I_2=2I$$
I. $$I_1=3I, I_2=I$$
Answer: B
One mole of an ideal monoatomic gas is heated in a process $P{V}^{5/2}=$ constant. By what amount heat is absorbed in the process in ${26}^{o}C$ rise in temperature?

Options:
A. 205J
B. $$100J$$
C. 190J
D. 195J
E. $$208J$$
F. $$200J$$
G. 175J
H. 170J
I. $$180J$$
J. 210J
Answer: I
A student constructs a series RLC circuit. While operating the circuit at a frequency f, he uses an AC voltmeter and measures the potential difference across each device as $(\Delta V_R)_{max}$ = 4.8 V.  $(\Delta V_C)_{max}$ = 29 V, and $(\Delta V_L)_{max}$ = 20 V.
How should be frequency of this circuit be changed to increase the current $i_m$ through the circuit?

Options:
A. Decrease the inductance in the circuit
B. The current is already at a maximum
C. Increase f
D. Decrease the amplitude of the AC source
E. Increase the inductance in the circuit
F. Increase the amplitude of the AC source
G. The current is already at a minimum
H. There is not enough information to answer the question
I. Increase the resistance in the circuit
J. Decrease f
Answer: H
A small body of mass's' is placed at the top of a smooth radius '$R$' placed on a horizontal surface as shown. Now the sphere is given a constant horizontal acceleration of $a=g$. The velocity of the body relative to the sphere at the moment when it loses contact with the sphere is

Options:
A. $$\sqrt { \dfrac { 2gR }{ 3 } } $$
B. $$\sqrt { \dfrac { 10gR }{ 3 } }$$
C. $$\sqrt { \dfrac { 4gR }{ 3 } }$$
D. $$\sqrt { \dfrac { 6gR }{ 3 } }$$
E. $$\sqrt { \dfrac { 7gR }{ 3 } } $$
F. $$\sqrt { \dfrac { gR }{ 3 } } $$
G. $$\sqrt { \dfrac { 9gR }{ 3 } }$$
H. $$\sqrt { \dfrac { 5gR }{ 3 } } $$
I. $$\sqrt { \dfrac { 3gR }{ 3 } }$$
J. $$\sqrt { \dfrac { 8gR }{ 3 } }$$
Answer: F
A sample of $U^{238} ((half\!-\! life = 4.5\times 10^9 \: yr)$ ore is found to contain 23.8 g of $U^{238}$ and 20.6 g of $Pb^{206}$. The age of the ore is :

Options:
A. $$4.00 \times 10^9 \:year$$
B. $$3.5\times 10^9 \:year$$
C. $$5.50 \times 10^9 \:year$$
D. None of these
E. $$3.75\times 10^9 \:year$$
F. $$6.25\times 10^9 \:year$$
G. $$4.5\times 10^9 \:year$$
H. $$5.25 \times 10^9 \:year$$
I. $$5.75 \times 10^9 \:year$$
J. $$4.25 \times 10^9 \:year$$
Answer: G
Two blocks of copper metal are of same size (heat capacity $=C$) but at different temperatures ${T}_{1}$ and ${T}_{2}$. These blocks are brought together and allowed to attain thermal equilibrium. The entropy change of system is

Options:
A. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }+{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } \right] }$$
B. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }-{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } +1\right] } $$
C. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }+{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } + \frac{1}{2} \right] }$$
D. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }-{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } - \frac{1}{2} \right] }$$
E. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }-{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } \right] } $$
F. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }+{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } - \frac{1}{2} \right] }$$
G. C\ln { \left[ \cfrac { { \left( { T }_{ 2 }-{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } -1 \right] }
H. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }+{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } -1 \right] } $$
I. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }+{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } +1 \right] } $$
J. $$C\ln { \left[ \cfrac { { \left( { T }_{ 2 }-{ T }_{ 1 } \right) }^{ 2 } }{ 4{ T }_{ 1 }{ T }_{ 2 } } + \frac{1}{2} \right] }$$
Answer: B
$20$ mL of $0.1M\ H_2SO_4$ solution is added to $30$ mL of $0.2 M\, NH_4OH$ solution. The pH of the resultant mixture is: $[ \ pK_{b (NH_4OH)}=4.7]$.

Options:
A. $$9.4$$
B. $$5.0$$
C. 8.7
D. $$5.2$$
E. $$9.0$$
F. 8.8
G. $$4.9$$
H. 8.9
I. 9.3
J. 5.1
Answer: E
A particle of mass $m$ attached to the end of string of length $\ell$ is released from the initial position $A$ as shown in the figure. The particle moves in a vertical circular path about $O$. When it is vertically below $O$, the string makes contact with nail $N$ placed directly below $O$ at distance $h$ and rotates around it. If the particle just complete the vertical circle about $N$, then

Options:
A. $$h=\displaystyle\frac{5l}{6}$$
B. $$h=\displaystyle\frac{3l}{10}$$
C. $$h=\displaystyle\frac{6l}{5}$$
D. h=\displaystyle\frac{3l}{4}
E. $$h=\displaystyle\frac{4l}{5}$$
F. $$h = \dfrac{3l}{5}$$
G. $$h=\displaystyle\frac{3l}{5}$$
H. $$h=\displaystyle\frac{l}{5}$$
I. $$h=\displaystyle\frac{2l}{5}$$
Answer: E
If a body is moving in a circular path with decreasing speed, then (symbols have their usual meaning)

Options:
A. $$\vec{a} \cdot \vec{v} < 0$$
B. $$\vec{r} . \vec{\omega} \neq 0$$
C. $$\vec{r} . \vec{\omega} = -1$$
D. $$( \vec{a} . \vec{v} = 0 )$$
E. $$\vec{r} . \vec{\omega} = 1$$
F. $$ ( \vec{r} . \vec{\omega} = 1 ) $$
G. $$\vec{r} . \vec{\omega}=0$$
H. All of these
I. $$\vec{\tau}.\vec{v}=0$$
Answer: G
The rate law of reaction between the substance $A$ and $B$ is given by $rate={K}{[A]}^{n}{[B]}^{m}$. On doubling the concentration of $A$ and making the volume of $B$ half the ratio of new rate to the earlier rate of reaction will be:

Options:
A. $$\dfrac{1}{2^{m-n}}$$
B. $$\dfrac{1}{2^{n+m-1}}$$
C. $${2}^{n-m}$$
D. $$m+n$$
E. $$\dfrac{1}{2^{n+m}}$$
F. $${2}^{m-n}$$
G. $${2}^{n+m}$$
H. $$\dfrac{1}{2^{n-m}}$$
I. $$\dfrac{1}{2^{m+n}}$$
Answer: C
A first-order reaction: $A\rightarrow B$, activation energy is $4.8kcal/mol$. When a $20$% solution of $A$ was kept at ${ 27 }^{ o }C$ for $21.6min$, $75$% decomposition took place. What will be the percent decomposition in $8.0min$ in a $30$% solution maintained at ${ 47 }^{ o }C$? Assume that activation energy remains constant in this range of temperature.
$$(e=2.7)$$

Options:
A. 45%
B. 55%
C. $50$%
D. 52.5%
E. 65%
F. 62.5%
G. $87.5$%
H. $75$%
I. 60%
J. $25$%
Answer: C
A plane elctromagnetic wave of wavelength $ \lambda  $ has an intensity I. It is propagating along the positive x-direction. The allowed expressions for the electri and magnetic fields are given by

Options:
A. $$ \overrightarrow { E } =\sqrt { \frac { 2I }{ { { \epsilon}_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x+ct \right) \right] \hat { j } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { k } $$
B. $$\overrightarrow { E } =\sqrt { \frac { I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x+ct \right) \right] \hat { i } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { k }$$
C. $$\overrightarrow { E } =\sqrt { \frac { 2I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x+ct \right) \right] \hat { i } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { k }$$
D. $$ \overrightarrow { E } =\sqrt { \frac { I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x-ct \right) \right] \hat { i } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { k } $$
E. $$\overrightarrow { E } =\sqrt { \frac { I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x-ct \right) \right] \hat { k } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { i }$$
F. $$ \overrightarrow { E } =\sqrt { \frac { 2I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x-ct \right) \right] \hat { j } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { k } $$
G. $$\overrightarrow { E } =\sqrt { \frac { I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x+ct \right) \right] \hat { j } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { k }$$
H. $$ \overrightarrow { E } =\sqrt { \frac { I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x-ct \right) \right] \hat { j } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { k } $$
I. $$\overrightarrow { E } =\sqrt { \frac { 2I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x+ct \right) \right] \hat { j } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { i }$$
J. $$\overrightarrow { E } =\sqrt { \frac { I }{ { { \epsilon }_{ 0 }C } } } cos\left[ \frac { 2\pi }{ \lambda } \left( x-ct \right) \right] \hat { j } ;\overrightarrow { B } =\frac { 1 }{ c } E\hat { i }$$
Answer: F
A small block of mass m is rigidly attached at P to a ring of mass 3m and radius r. the system is released from rest at $\theta = 90^0$ and rolls without sliding. The angular acceleration of ring just after release is:

Options:
A. $$\dfrac{g}{2r}$$
B. $$\dfrac{g}{8r}$$
C. $$\dfrac{g}{7r}$$
D. $$\dfrac{g}{4r}$$
E. $$\dfrac{g}{5r}$$
F. $$\dfrac{g}{9r}$$
G. $$\dfrac{g}{6r}$$
H. $$\dfrac{g}{3r}$$
I. none of these
J. $$\dfrac{g}{10r}$$
Answer: B
The electric field inside a sphere which carries a charge density proportional to the distance from the centre $\rho = \alpha r \quad (\alpha \text{ is a constant})$

Options:
A. $$\dfrac { \alpha { r }^{ 2 } }{ 2{ \epsilon }_{ 0 } }$$
B. $$\dfrac { \alpha { r }^{ 2 } }{ 6{ \epsilon }_{ 0 } }$$
C. $$\dfrac { lpha { r }^{ 2 } }{ 8{ \epsilon }_{ 0 } }$$
D. $$\dfrac { \alpha { r }^{ 3 } }{ 4{ \epsilon }_{ 0 } }$$
E. $$\dfrac { \alpha { r }^{ 2 } }{ 5{ \epsilon }_{ 0 } }$$
F. $$none\ of\ these$$
G. $$\dfrac { \alpha { r }^{ 3 } }{ 3{ \epsilon }_{ 0 } }$$
H. $$\dfrac { \alpha { r }^{ 2 } }{ 3{ \epsilon }_{ 0 } }$$
I. $$\dfrac { \alpha { r }^{ 2 } }{ 4{ \epsilon }_{ 0 } }$$
J. $$\dfrac { \alpha { r }^{ 2 } }{ 7{ \epsilon }_{ 0 } }$$
Answer: H
For a binary liquid solution obeying Raoult's law at $300 K$, $X_A = 0.75$, Given: $P_A^o = 400 mm Hg ; P_B^o = 600 mm Hg$ Then select correct statement

Options:
A. Normal boiling point of liquid solution should be more than $300K$
B. Vapour pressure of solution will be $475 mm$ $Hg$
C. Normal boiling point of liquid solution should be less than $300K$
D. In distillate mol fration of $B$ will be $2/3$
E. Normal boiling point of liquid solution should be exactly $300K$
F. Normal boiling point of liquid solution should be exactly 305K
G. Vapour pressure of solution will be $500 mm$ $Hg$
H. Vapour pressure of solution will be $550 mm$ $Hg$
Answer: A
$3.175$ g of Cu is deposited at the cathode in the electrolysis of Cu(NO$_3$)$_2$ when $9.65$ A current is passed for $2000$ second. The current efficiency of the process is
[M.W. of Cu is $63.5$ g / mol, $1 $ F$=96500$ C]

Options:
A. $$65%$$
B. $$55\%$$
C. $$40\%$$
D. $$60%$$
E. $$25\%$$
F. $$30\%$$
G. $$50\%$$
H. $$75\%$$
I. $$35\%$$
J. $$70\%$$
Answer: E
A capacitor $C_1 = 1.0 \mu F$ is charged up to a voltage V = 60 V 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 switch (2), as shown in the figure. The sum of final charges on $C_2$ and $C_3$ is:

Options:
A. $$48 \mu C$$
B. $$42 \mu C$$
C. $$38 \mu C$$
D. $$54 \mu C$$
E. $$20 \mu C$$
F. $$30 \mu C$$
G. $$40 \mu C$$
H. $$36 \mu C$$
I. $$44 \mu C$$
J. $$32 \mu C$$
Answer: G
A current of dry air was bubbled through two bulbs, first containing $26.66$g of an organic substance in $200$g of water, second containing fused $CaCl_2$. The loss of weight of water bulb $=0.0870$g and the gain of the weight of $CaCl_2$ tube $=2.036$g. The molecular weight of the organic substance is?

Options:
A. $$67$$
B. 53
C. $$54$$
D. $$78$$
E. $$56$$
F. $$47$$
G. $$41$$
H. $$55$$
I. $$58$$
J. $$57$$
Answer: E
A body of mass m = ${ m=10 }^{ -2 }$ kg is moving in a medium and experiences a frictional force ${F =kv }^{ -2 }.$ Its initial speed is ${ v }_{ 0 }=10{ ms }^{ -1 }$. If after 10 s,its energy is $\dfrac { 1 }{ 8 } { mv }_{ 0 }^{ 2 }$, the value of k will be

Options:
A. $${ 10 }^{ -1 }kg{ m }^{ -1 } { s }^{ -1 }$$
B. $${ 10 }^{ -2 }kg{ m }^{ -1 }$$
C. $${ 10 }^{ -5 } {kgs}^{ -1 }$$
D. $${ 10 }^{ -5 }kg{ m }^{ -1 }$$
E. $${ 10 }^{ -4 }kg{ m }^{ -1 }$$
F. $${ 10 }^{ -3 } {kgs}^{ -1 }$$
G. $${ 10 }^{ -3 }kg{ m }^{ -1 }$$
H. $${ 10 }^{ -6 }kg{ m }^{ -1 }$$
Answer: E
An acid -base indicator which is a weak acid has a $ pK_4 $ value = 5.35. At what concentration ratio of sodium acetate to acetic acid would the indicator show a colour half-way between those of its acid and conjugate base forms ? $ pK_a $ of acetic acid  = 4.75 . [ log 2 = 0.3 ]

Options:
A. 7 : 1
B. 5 : 1
C. 4 : 1
D. 2 : 1
E. 10 : 1
F. 9 : 1
G. 6 : 1
H. 8 : 1
I. 3 : 1
J. 7 : 2
Answer: B
A uniform line charge with linear density $\lambda$ lies along the y-axis. What flux crosses a spherical surface centred at he origin with $r\ =\ R$

Options:
A. $$\large{\frac{\lambda}{\epsilon_0}}$$
B. $$\large{\frac{R\lambda}{2\epsilon_0}}$$
C. $$\large{\frac{R\lambda}{3\epsilon_0}}$$
D. $$\large{\frac{R\lambda}{4\epsilon_0}}$$
E. $$\large{\frac{2\lambda}{\epsilon_0}}$$
F. $$\large{\frac{2R\lambda}{\epsilon_0}}$$
G. $$\large{\frac{4R\lambda}{3\epsilon_0}}$$
H. $$\large{\frac{3R\lambda}{2\epsilon_0}}$$
I. none of the above
J. $$\large{\frac{R\lambda}{\epsilon_0}}$$
Answer: J
The atomic mass of $B^{10}$ is $10.811\ amu$. The binding energy of $B^{10}$ nucleus is [ Given: The mass of electron is $0.0005498\ amu$, the mass of proton is $m_p=1.007276\ amu$ and the mass of neutron is $m_n=1.008665\ amu$]:

Options:
A. $$378.432\ MeV$$
B. $$678.932\ MeV$$
C. $$377.272\ MeV$$
D. $$378.332\ MeV$$
E. $$none\ of\ these$$
F. $$378.272\ MeV$$
G. $$377.932\ MeV$$
H. $$378.932\ MeV$$
I. $$-678.272\ MeV$$
J. $$377.172\ MeV$$
Answer: I
A block of mass $m$ moving with a velocity $v_0$ on a smooth horizontal floor collides with a light spring stiffness $k$ that is rigidly fixed horizontally with a vertically wall. If the maximum force imparted by the spring on the block is $F$ then prove that $F=v_0\sqrt{km}$

Options:
A. $$8\sqrt{km v_0}$$
B. $$\sqrt{km}$$
C. $2\sqrt{km v_0}$
D. $$6\sqrt{km v_0}$$
E. $$7\sqrt{km v_0}$$
F. $4\sqrt{km v_0}$
G. $$3\sqrt{km v_0}$$
H. $$5\sqrt{km v_0}$$
I. $$\sqrt{km v_0}$$
J. $$9\sqrt{km v_0}$$
Answer: F
A solid cube of mass $5$ kg is placed on a rough horizontal surface, in xy-plane as shown. The friction coefficient between the surface and the cube is $0.4$. An external force $\vec{F} = 6\hat{i} + 8\hat{j} + 20\hat{k}$N is applied on the cube.(use $g=10m/s^2$)

Options:
A. The contact force exerted by the surface on the cube is \(10\sqrt{9}\)N
B. The contact force exerted by the surface on the cube is \(10\sqrt{5}\)N
C. The block starts moving at an angle of $45^o$ with the positive x-axis
D. The contact force exerted by the surface on the cube is $10\sqrt{10}$N
E. The block starts slipping over the surface
F. The friction force acts in xy-plane at angle $127^o$ with the positive x-axis in clockwise direction
G. The contact force exerted by the surface on the cube is \(10\sqrt{8}\)N
H. The friction force on the cube by the surface is $10$N
I. The contact force exerted by the surface on the cube is \(10\sqrt{7}\)N
J. The contact force exerted by the surface on the cube is \(10\sqrt{6}\)N
Answer: E
Ends of a thin uniform inextensible rope of mass m=1.0 kg and length 2 m are fastened to ceiling close to each other. If one of the end is released and allowed to fall freely as shown in figure, find the largest force that the fastening at the other end must be capable of bearing so that rope remains fastened there (g=10 $ms^{-2}$)

Options:
A. 18 N
B. 30 N
C. 40 N
D. 15 N
E. 20 N
F. 12 N
G. 16 N
H. 10 N
I. 25 N
J. 5 N
Answer: H
An aluminium rod (Young's modulus $= 7\times 10^{9} N/m^{2})$ has a breaking strain of $0.2$%. The minimum cross-sectional area of the rod in order to support a load of $10^{4}$ Newton's is:

Options:
A. $$1\times 10^{-2}m^{2}$$
B. $$3.5\times 10^{-3}m^{2}$$
C. $$7.1\times 10^{-4}m^{2}$$
D. $$2.8 \times 10^{-3}m^{2}$$
E. $$3.4 \times 10^{-3}m^{2}$$
F. $$1.4 \times 10^{-4}m^{2}$$
G. $$1.4\times 10^{-3}m^{2}$$
H. $$3.6 \times 10^{-3}m^{2}$$
I. $$7.0 \times 10^{-3}m^{2}$$
J. $$7.0 \times 10^{-4}m^{2}$$
Answer: B
An electric dipole has a fixed dipole moment \(\vec{p}\), which makes an angle \(\theta\) with respect to the x-axis. When subjected to an electric field \(\vec{E_1} = E \hat{i}\), it experiences a torque \(\vec{T_1} = \tau \hat{k}\). When subjected to another electric field \(\vec{E_2} = \sqrt{3} E_1 \hat{j}\), it experiences a torque \(\vec{T_2} = -\vec{T_1}\). The angle \(\theta\) is:

Options:
A. $$150^\circ$$
B. $$105^\circ$$
C. $$165^\circ$$
D. $$120^\circ$$
E. $$45^\circ$$
F. $$135^\circ$$
G. $$75^\circ$$
H. $$30^\circ$$
I. $$90^\circ$$
J. $$60^\circ$$
Answer: I
In the arrangement shown the two blocks attached to the two ends of a string passing over a pulley, resting on an incline plane $\left (\theta = \dfrac {\pi}{4}\right )$. The plane is moved with an acceleration $g/2$ towards left $(m_{2} > m_{1})$.

Options:
A. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}}\right )\dfrac {g}{\sqrt {2}}$$
B. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}} \right )\dfrac {g}{\sqrt {5}}$$
C. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}}\right )\dfrac {g}{2}$$
D. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}}\right )g$$
E. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}} \right )\dfrac {g}{\sqrt {6}}$$
F. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}} \right )\dfrac {g}{4}$$
G. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}} \right )\dfrac {g}{3}$$
H. $$\dfrac {3\sqrt {2}}{4}\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}}\right )g$$
I. $$\left (\dfrac {m_{2} - m_{1}}{m_{2} + m_{1}} \right )\dfrac {g}{\sqrt {3}}$$
Answer: A
A bod of mass M is suspended by a massless string of length L.The horizontal velocity V at position A is just sufficient to make real the point B.The angle $\theta$ at which the speed of the bob is half of that it A satisfies:

Options:
A. $$\dfrac{3\pi}{4} < \theta<\pi$$
B. $$\dfrac{\pi}{2} < \theta<\dfrac{3\pi}{4}$$
C. $$\dfrac{\pi}{2} < \theta < \dfrac{5\pi}{6}$$
D. $$\dfrac{\pi}{3} < \theta < \dfrac{5\pi}{6}$$
E. $$\dfrac{\pi}{2} < \theta < \dfrac{5\pi}{4}$$
F. $$\dfrac{\pi}{4} < \theta<\frac{\pi}{2}$$
G. $$\theta=\dfrac{\pi}{4}$$
H. $$\dfrac{\pi}{4} < \theta < \dfrac{3\pi}{4}$$
I. $$0 < \theta < \dfrac{\pi}{4}$$
Answer: A
If 100 mole of $H_2O_2$ decompose at 1 bar and 300 K, the work done (kJ) by one mole of $O_2(g)$ as it expands against 1 bar pressure is 
$$2H_2O_2(l) \rightleftharpoons 2H_2O(l) + O_2(g)$$
($R = 8.3 JK^{-1}mol^{-1}$)

Options:
A. 240.00
B. 497.50
C. 502.00
D. 498.00
E. 512.50
F. 496.00
G. 499.50
H. 62.25
I. 124.50
J. 499.00
Answer: D
Which of the following is the correct relation to convert mole fraction of solute to Molarity?

Options:
A. $$M = \frac{{X_{solute} M_{solvent}}}{{M_{solvent} (X_{solvent} M_{solvent} + X_{solute} M_{solute})}}$$
B. $$M = \frac{{(1 - {m_{solute}}{\rho _{solution}})}}{{\left( {{m_{solute}}\,{M_{solute}}} \right)}}$$
C. $$M = \frac{{X_{solute} M_{solute}}}{{M_{solute} (X_{solvent} M_{solvent} + X_{solute} M_{solute})}}$$
D. $$M = \frac{{\left( {{x_{solvent}}{M_{solvent}} + {x_{solute}}{M_{solute}}} \right)}}{{{x_{solute}}{M_{solute}}}}$$
E. $$M = \frac{{{x_{solute}}\rho }}{{\left( {{x_{solvent}}{M_{solvent}} + {x_{solute}}{M_{solute}}} \right)}}$$
F. $$M = \frac{{{X_{solute}}}}{{{M_{solvent}}\left( {{X_{solvent}}{M_{solvent}} + {X_{solute}}{M_{solute}}} \right)}}$$
G. $$M = \frac{{X_{solute} M_{solute}}}{{M_{solvent} (X_{solvent} M_{solvent} + X_{solute} M_{solute})}}$$
H. $$M = \frac{{X_{solute}}}{{M_{solvent} (X_{solvent} M_{solvent} + X_{solute} M_{solute})}}$$
I. $$M = \frac{{X_{solute} M_{solvent}}}{{M_{solute} (X_{solvent} M_{solvent} + X_{solute} M_{solute})}}$$
J. $$M = \frac{{{X_{solute}}}}{{{M_{solute}}\left( {{X_{solvent}}{M_{solvent}} + {X_{solute}}{M_{solute}}} \right)}}$$
Answer: J
Two square metallic plates of $1$m side are kept $0.01$ m apart, like a parallel plate capacitor, in air in such a way that one of their edges is perpendicular to an oil surface in a tank filled with insulating oil. The plates are connected to a battery of emf $500$V. The plates are then lowered vertically into the oil at a speed of $0.001$ m/s. Calculate the current drawn from the battery during the process. (dielectric constant of oil $=11, arepsilon_o=8.85\times 10^{-12}C^2N^{-1}m^{-2}$)

Options:
A. $4.425\times 10^{-9}A$.
B. $4.425\times 10^{-8}A$.
C. 4.425 \times 10^{-14}A
D. 4.425 \times 10^{-13}A
E. $4.425\times 10^{-5}A$.
F. $4.425\times 10^{-7}A$.
G. 4.425 \times 10^{-11}A
H. 4.425 \times 10^{-12}A
I. 4.425 \times 10^{-6}A
J. 4.425 \times 10^{-10}A
Answer: A
A superconducting rod of mass $m$ is placed on two resistanceless parallel smooth rails connected by a resistor of resistance $R$. Uniform magnetic field $B$ is acting perpendicular to the plane of loop. At $t=0$ velocity $v_0$ is given to the rod. The total heat generated in rod before coming to rest is :

Options:
A. \(\dfrac{1}{2}mv_0^2 - \dfrac{(Bvl)^2}{4R}\)
B. \(\dfrac{1}{2}mv_0^2 - \dfrac{(Bvl)^2}{R}\)
C. \(\dfrac{1}{2}mv_0^2 + (Bvl)^2/R \)
D. none of these
E. zero
F. $$\dfrac{1}{2}mv_0^2$$
G. $$(Bvl)^2/R$$
H. \(\dfrac{1}{2}mv_0^2 - (Bvl)^2/R \)
I. $$\dfrac{1}{2}mv_0^2 - \dfrac{(Bvl)^2}{3R}$$
J. \(\dfrac{1}{2}mv_0^2 - \dfrac{(Bvl)^2}{2R}\)
Answer: E
Two particles 'A' and 'B' are projected in the vertical plane with same initial speed ${u_0}$ from position (0,0) and ($\left( {l, - h} \right)$) towards each other as shown in figure at t=0.The time when separation between A and B is minimum is:

Options:
A. $$\dfrac{2l}{{5{u_0}\cos \theta }}$$
B. $$\sqrt {\dfrac{{2h}}{g}} $$
C. $$\dfrac{l}{{2{u_0}\cos \theta }}$$
D. $$\dfrac{2l}{{7{u_0}\cos \theta }}$$
E. $$\dfrac{l}{{3{u_0}\cos \theta}}$$
F. $$\dfrac{2l}{{2{u_0}\cos \theta }}$$
G. $$\dfrac{x}{{{u_0}\cos \theta }}$$
H. $$\dfrac{l}{{4{u_0}\cos \theta}}$$
I. $$\dfrac{3l}{{4{u_0}\cos \theta}}$$
J. $$\dfrac{2l}{{3{u_0}\cos \theta}}$$
Answer: G
The  copper half of an inclined   plane of inclination 0 is perfectly smooth while  the lower half is rough , A  body starting from the rest at top comes back to rest at the bottom if the coefficient  of friction for the lower half is given

Options:
A. $$\mu = 2 \tan \theta$$
B. $$\mu = \frac{1}{2} \sin \theta$$
C. $$\mu = \frac{1}{2} \tan \theta$$
D. $$ \mu = 2 \cos \theta $$
E. $$ \mu = \sin \theta $$
F. $$\mu = \frac{1}{2} \cos \theta$$
G. $$\mu = 2 \sin \theta$$
H. $$ \mu = \cos \theta $$
I. $$\mu = \tan \theta$$
Answer: E