MikaStars39/nano-eval huggingface dataset:
A plane wave is incident on a lens of R.I. n=1.5 with one flat side (as shown in figure). The radius of the flat side is R. If all incoming rays are to be focused at the point P at a distance 3R from the flat side, calculate the thickness of the curved side of the middle of the lens. (Do not assume the lens is thin)

Options:
A. $$2R(\sqrt{2} - 2)$$
B. $$2R(\sqrt{5} - 2)$$
C. $$2R(\sqrt{4} - 2)$$
D. $$2R(\sqrt{10} - 2)$$
E. $$2R(\sqrt{7} - 2)$$
F. $$2R(\sqrt{3} + 2)$$
G. $$2R(\sqrt{9} - 2)$$
H. $$2R(\sqrt{8} - 2)$$
I. $$2R(\sqrt{6} - 2)$$
J. 2R
Answer: J
In a political rally, a guy from audience throws a flower at leader. The flower travels a horizontal distance 20 m in 1 s before hitting the leader's from 2.0 m above the horizontal floor with initial velocity, which is making an angle ${ 37 }^{ o }$ above the horizontal. The height of stage above the horizontal floor is ($ g= 10 m{ s }^{ -2 }$)

Options:
A. 10.15 m
B. 11.25 m
C. 10.0 m
D. 9.5 m
E. 10.5 m
F. 10.25 m
G. 12 m
H. 9.75 m
I. 15 m
J. 10.75 m
Answer: F
Dipole moment of $HCl =1.03\ D, HI = 0.38\ D$. Bond length of $HCl = 1.3\overset {\circ}{A}$ and $HI = 1.6\overset {\circ}{A}$. The ratio of fraction of electric charge, $\delta$, existing on each atom in $HCl$ and $HI$ is

Options:
A. $$12 : 1$$
B. $$1 : 2.9$$
C. $$1 : 2.7$$
D. $$3.0 : 1$$
E. $$3.3 : 1$$
F. $$3.6 : 1$$
G. $$1 : 3.3$$
H. $$2.7 : 1$$
I. $$11 : 1$$
J. $$1 : 3.6$$
Answer: A
A 20 litre container at 400 K contains $CO_2(g)$ at pressure $0.4$ atm and an excess of SrO(neglect the volume of solid SrO). The volume of the container is now decreased by moving the movable piston fitted in the container. The maximum volume of the container, when pressure of $CO_2$ attains its maximum value, will be:-
(Given that: $SrCO_3(s) \rightleftharpoons SrO(s)+CO_2(g),)
Kp=1.6atm)$

Options:
A. $$10 \quad litre$$
B. $$7 \quad litre$$
C. $$4 \quad litre$$
D. $$9 \quad litre$$
E. $$8 \quad litre$$
F. $$2 \quad litre$$
G. $$12 \quad litre$$
H. $$3 \quad litre$$
I. $$6 \quad litre$$
J. $$5 \quad litre$$
Answer: C
One mole of an ideal gas at 300 $ \mathrm{K}  $ in thermal contact with surroundings expands isothermally from 1.0 $ \mathrm{L} $ to 2.0 L against a constant pressure of 3.0 atm. In this process, the change in entropy of surroundings $ \left(\Delta S_{s u r r}\right) $ in $  \mathrm{JK}^{-1}  $ is $  :(1 \mathrm{L}  $ atm $ =101.3 \mathrm{J}) $

Options:
A. $$ -1.013 $$
B. $$-5.763$$
C. $$6.763$$
D. $$1.763$$
E. $$3.763$$
F. $$2.563$$
G. $$5.763$$
H. $$ 1.013 $$
I. $$4.763$$
Answer: G
The $lpha$-particles with initial kinetic energy $4.8$ mev are shot at gold atoms $(z=79)$. The potential of $lpha$-particle at the closest distance to the gold nucleus is

Options:
A. 1.2 MV
B. 1.8 MV
C. 3.2 MV
D. 3.6 MV
E. 3.0 MV
F. $$zero$$
G. $$9.6 MV$$
H. $$4.8 MV$$
I. $$2.4 MV$$
J. 1.6 MV
Answer: I
A series resonant circuit contains $L=\dfrac{5}{\pi}mH,C=\dfrac{200}{\pi}\mu F$  and $R=100\Omega$. If a source of e.m.f, e$=200{\,\,}sin 1000\pi t$ is applied,then power delivered to the resistance of the circuit will be:

Options:
A. 250W
B. $300$W
C. 320W
D. 100W
E. 150W
F. $200$W
G. $400$W
Answer: G
A whistle of frequency $500Hz$ tied to the end of a string of length $1.2m$ revolves at $400rev/min$. A listener standing some distance away in the plane of rotation of whistle hears frequencies in the range (speed of sound = $340m/s$)

Options:
A. $426$ to $584$
B. $426$ to $574$
C. 434 to 574
D. 436 to 574
E. $435$ to $586$
F. 426 to 674
Answer: E
AB is a light rigid rod, which is rotating about a vertical axis passing through

Options:
A. The maximum velocity attained by the bead w.r.t the rod is given by $V_{max}=\sqrt{(\dfrac{m\omega ^{4}+K}{m\omega ^{2}-K})\omega ^{2}l ^{2}}$
B. The maximum value of contact force between the bead and the rod is greater than $mg$.
C. X_{max}=\dfrac{2m\omega ^{2}l}{K-2m\omega ^{2}}
D. X_{max}=\dfrac{m\omega ^{2}l}{K-2m\omega ^{2}}
E. X_{max}=\dfrac{m\omega ^{2}l}{K+3m\omega ^{2}}
F. X_{max}=\dfrac{m\omega ^{2}l}{K-m\omega ^{2}}
G. The maximum extension in the spring is given by $X_{max}=\dfrac{2m\omega ^{2}l}{K-m\omega ^{2}}$
H. The maximum extension in the spring is given by $X_{max}=\dfrac{m\omega ^{2}l}{2K-m\omega ^{2}}$
I. The maximum extension in the spring is given by $X_{max}=\dfrac{m\omega ^{2}l}{K+2m\omega ^{2}}$
J. The maximum velocity attained by the bead w.r.t the rod is given by $V_{max}=\sqrt{\dfrac{m\omega ^{4}l ^{2}}{K-m\omega ^{2}}}$
Answer: G
A standing wave $y = A   \sin \left ( \displaystyle \dfrac {20}{3}\pi x \right ) \cos (1000\pi t)$ is maintained in a taut string where $y$ and $x$ are expressed in meters. The distance between the successive points oscillating with the amplitude $A/2$ across a node is equal to :

Options:
A. $$12.5 cm$$
B. $$10 cm$$
C. $$2.5 cm$$
D. $$8.75 cm$$
E. $$17.5 cm$$
F. $$5 cm$$
G. $$25 cm$$
H. $$7.5 cm$$
I. $$15 cm$$
Answer: F
The equilibrium constant for the following equilibrium is given at $0^\circ C$.
$N{a_2}HP{O_4}.12{H_2}O \rightleftharpoons N{a_2}HP{O_4}.7{H_2}O(s) + 5{H_2}O(g),$  ${K_p} = 31.25 \times {10^{ - 23}}$
At equilibrium what will be partial pressure of water vapour:

Options:
A. $2 \times 10^{-3}$ atm
B. $\frac {1}{5} \times 10^{-3}$ atm
C. $0.5 \times 10^{-3}$ atm
D. $5 \times 10^{-3}$ atm
E. $$\frac{1}{10} \times 10^{-3}$$
F. $$\frac{2}{5} \times 10^{-3} \text{ atm}$$
G. $$\frac{1}{15} \times 10^{-3} \text{ atm}$$
H. $$\frac{1}{2} \times 10^{-3} \text{ atm}$$
I. $$\frac{1}{4} \times 10^{-3} \text{ atm}$$
J. $5 \times 10^{-2}$ atm
Answer: B
$82$ litres of carbon dioxide are produced at a pressure of $1$ atm and $25^\circ C$ by the action of acid on metal carbonate. The work done by gas (in calories) in pushing back the atmosphere is ($R = 0.082$ litre-atm $\text{deg}^{-1} \text{mol}^{-1}$)

Options:
A. 1460
B. 1260
C. $$1640$$
D. $$820$$
E. 1200
F. $$1000$$
G. $$2000$$
H. 1520
I. 1100
J. 1230
Answer: G
For the reaction, $A(g)\rightarrow B(g)+C(g)$ rate law is $- \cfrac{d[A]}{dt}=K[A]$
At the start pressure is 100 mm and after 10 min, pressure is 120 mm, hence, rate constant (min)

Options:
A. $$\cfrac {2.303}{10}log \cfrac 5 3$$
B. $$\cfrac {2.303}{10}log \cfrac 6 5$$
C. $$\cfrac {2.303}{10}log \cfrac 4 6$$
D. $$\cfrac {2.303}{10}log \cfrac 3 5$$
E. $$\cfrac {2.303}{10}log \cfrac 6 7$$
F. $$\cfrac {2.303}{10}log \cfrac 4 5$$
G. $$\cfrac {2.303}{10}log \cfrac 5 6$$
H. $$\cfrac {2.303}{10}log 5$$
I. $$\cfrac {2.303}{10}log \cfrac 5 4$$
Answer: G
An $lpha$ - particle is projected towards the following nucleus with same kinetic energy in different experiment the distance of closet approach is maximum for

Options:
A. Au (Z=79)
B. Na(Z=11)
C. Ca (Z=21)
D. Ca (Z=20)
E. Ag (Z=47)
F. Cu (Z=29)
G. Cu (Z=28)
H. Ag (Z=48)
Answer: A
10 litre of alkane X require 35 litre of ${O_2}$ for complete combustion. X forms only one monochloro derivative Y. The action of alcoholic KOH on Y yields

Options:
A. $$C{H_3}C{H_2}C{H_2}O{H}$$
B. $$C{H_3}C{H_2}C{H_2}OH$$
C. $$C{H_3}C{H_2}C{H_2}CH_2OH$$
D. $$C{H_2}C{H_2}C{H_2}OH$$
E. $$C{H_3}C{H_2}C{H_2}CH{OH}$$
F. $$C{H_3}C{H_2}C{H_2}CH_3$$
G. $$C{H_2}CH = C{H_2}$$
H. $$C{H_3} - \mathop {CH - C{H_3}}\limits_{\mathop {OH}\limits^1 } $$
I. $$C{H_2} = C{H_2}$$
J. $$C{H_3} - \mathop {CH - C{H_2}}\limits_{\mathop {OH}\limits^1 }$$
Answer: D
A fire cracker is thrown with velocity of 30$ ms^{-1}$ in a direction which makes an angle of $75^o$ with the vertical axis. At some point on its trajectory, the fire cracker split into two identical pieces in such a way that one piece falls 27 m far from the shooting point. Assuming that all trajectories are contained in the same plane, how far will the other piece fall from the shooting point ? (Take $g\, =\, 10 ms^{-2}$ and neglect air resistance)

Options:
A. 63 m or 99 m
B. 63 m or 108 m
C. 63 m or 135 m
D. 63 m or 120 m
E. 63 m or 105 m
F. 63 m or 144 m
G. 63 m or 126 m
H. 28 m or 72 m
I. 72 m or 99 m
J. 63 m or 117 m
Answer: J
Calculate the equilibrium constant for the reaction,
$$Zn(s) + Ag_{2}O(s) + H_{2}O(l) \rightarrow 2Ag(s) + Zn^{2+}(aq.) + 2OH^{-}(aq.)$$
when $E_{cell}^{\circ} = 1.11$ at $298\ K$.

Options:
A. $$2.499\times 10^{37}$$
B. $$3.599 \times 10^{37}$$
C. $$5.599 \times 10^{37}$$
D. $$3.499\times 10^{35}$$
E. $$2.599 \times 10^{36}$$
F. $$2.499\times 10^{35}$$
G. $$4.499 \times 10^{36}$$
H. $$3.599 \times 10^{36}$$
I. $$4.599\times 10^{37}$$
J. $$4.499 \times 10^{37}$$
Answer: I
Compare the rates of a reaction at 273 K than at 289 K:(Given : ${ E }_{ a }=65\ kJ{ mol }^{ -1 }$)

Options:
A. 1/11 times
B. 1/22 times
C. 1/66 times
D. 1/23 times
E. 1/55 times
F. 1/42 times
G. 1/34 times
H. 1/33 times
I. 1/44 times
J. 1/21 times
Answer: B
Calculate $\Delta H$ for the following reaction at $298\ K$:
$$Fe_{2}O_{3}(s)+3CO(g)\rightarrow 2Fe(s)+3CO_{2}(g)$$
from the following thermochemical equation
(i) $3Fe_{2}O_{3}(s)+3CO(g)\rightarrow 2Fe_{3}O_{4}(s)+CO_{2}(g); \Delta H=-46.4\ kJ$
(ii) $FeO(s)+CO(g)\rightarrow Fe(s)+CO_{2}(g); \Delta H=9.0\ kJ$
(iii) $Fe_{3}O_{4}(s)+CO(g)\rightarrow 3FeO(s)+CO_{2}(g); \Delta H=-41.0\ kJ$

Options:
A. $$-165.97\ kJ$$
B. $$-24.8\ kJ$$
C. $$-155.97\ kJ$$
D. $$+24.8\ kJ$$
E. $$+145.97\ kJ$$
F. $$-17.97\ kJ$$
G. $$-175.97\ kJ$$
H. $$+155.97\ kJ$$
I. $$+165.97\ kJ$$
J. $$+17.97\ kJ$$
Answer: H
A man of mass 60 kg is standing on a platform which is oscillating up and down with frequency 2 oscillations/s and amplitude 50 cm.A machine on the plateform indicates weights of the man with respect to time, then the maximum reading of the machine will be-$\left( {g = 10\;m/{s^2}} \right)$

Options:
A. $${10^4}\;kg$$
B. 534 kg
C. 620 kg
D. 600 kg
E. 618 kg
F. 628 kg
G. $${10^3}\;kg$$
H. 630 kg
I. 610 kg
J. 532.8 kg
Answer: B
For the reaction :
$$C{H}_{4}\left(g\right) + {H}_{2}O\left(g\right) \rightleftharpoons CO\left(g\right) + 3{H}_{2}\left(g\right),              \Delta H = +ve$$
Which statement is not correct?

Options:
A. Addition of an inert gas constant pressure shifts the reaction towards products
B. Addition of an inert gas at constant pressure does not shift the reaction towards reactants
C. Addition of an inert gas at constant volume shifts the reaction towards reactants
D. A rise in temperature of equilibrium mixture shifts the reaction towards products
E. Addition of an inert gas at constant volume does not shift the reaction towards products
F. The compression of equilibrium mixture shifts the reaction toward products
G. Addition of an inert gas at constant volume does not affect the position of equilibrium
H. Addition of an inert gas at constant volume shifts the reaction towards products
I. Addition of an inert gas at constant pressure shifts the reaction towards reactants
J. Addition of an inert gas at constant volume shifts the reaction towards the products
Answer: C
As per VSEPR theory in which molecular species nitrogen has $N{X_3}L$ type arrangement but lone pair is NOT localised on nitrogen  $(N = nitrogen,X = \sigma \,bond\,L = Lone\,pair)$

Options:
A. $$C{H_3}N{H_2}$$
B. $$C{H_3}NH{CH_3}$$
C. $$C{H_3}N{(C{H_3})_2}$$
D. $$N{(C{H_3})_3}$$
E. $$N{(C{H_3})_4}^ + $$
F. $$N{(C{H_3})_2}OH$$
G. $$C{H_3}N{(Si{H_3})_2}$$
H. $$ N{(C{H_3})_2} $$
I. $$(N = {(C{H_2})_3}$$
J. $$C{H_3}N{H_3}$$
Answer: A
In figure $\displaystyle m_{1}=1$ kg and $\displaystyle m_{2}=4$ kg. Find the mass $M$ (in nearest integer) of the hanging block which will prevent the smaller block from slipping over the triangular block (angle of wedge= $30^o$). All the surfaces are frictionless and the strings and the pulleys are light :

Options:
A. $$3 kg$$
B. $$2 kg$$
C. $$4 kg$$
D. $$1 kg$$
E. $$9 kg$$
F. $$6 kg$$
G. $$5 kg$$
H. $$10 kg$$
I. $$8 kg$$
J. $$7 kg$$
Answer: J
One mole of an ideal gas undergoes a process $ P=\frac { { P }_{ 0 } }{ 1+\left( \frac { { V }_{ 0 } }{ V }  \right) ^{ 2 } }  $ Here $ P_0 $ and $ V_0 $ are constants.
Change in temperature of the gas when volume is changed from $ V = V_0 $ to $ V = 2V_0 $ Is :

Options:
A. $$- \frac {4P_0V_0 }{5R}$$
B. $$\frac {3P_0V_0 }{8R}$$
C. $$ - \frac {5P_0V_0 }{4R} $$
D. $$ \frac {11P_0V_0 }{10R} $$
E. $$ - \frac {2P_0V_0 }{5R} $$
F. $$ P_0V_0 $$
G. $$- \frac {3P_0V_0 }{10R}$$
H. $$- \frac {3P_0V_0 }{5R}$$
I. $$\frac {3P_0V_0 }{5R}$$
J. $$- \frac {2P_0V_0 }{4R}$$
Answer: E
A string has a mass per unit length of $10^{-6} kgcm^{-1}$. The equation of simple harmonic wave produced in it is $y=0.2\sin(2x+80t)$. The tension in the string is

Options:
A. $$16 N$$
B. $$0.016 N$$
C. $$0.00016 N$$
D. $$0.0016 N$$
E. $$0.16 N$$
F. $$1.60 N$$
G. $$1.6 N$$
H. $$0.064 N$$
Answer: E
The solubility of  $CaF_ \left( \mathrm { K } _ { \mathrm { sp } } = 5.3 \times 10 ^ { - 9 } \right)$  in  $0.1\mathrm { M }$  solution of  $NaF $ would be : Asscation/anion.

Options:
A. $$5.3 \times 10 ^ { - 15 } \mathrm { M }$$
B. $$5.3 \times 10 ^ { - 6 } \mathrm { M }$$
C. $$5.3 \times 10 ^ { - 9 } \mathrm { M }$$
D. $$5.3 \times 10 ^ { - 12 } \mathrm { M }$$
E. $$5.3 \times 10 ^ { - 8 } \mathrm { M }$$
F. $$5.3 \times 10 ^ { - 14 } \mathrm { M }$$
G. $$5.3 \times 10 ^ { - 10 } \mathrm { M }$$
H. $$5.3 \times 10 ^ { - 11 } \mathrm { M }$$
I. $$5.3 \times 10 ^ { - 13 } \mathrm { M }$$
J. $$5.3 \times 10 ^ { - 7 } \mathrm { M }$$
Answer: G
0.2 mole of $NH_{4}Cl$ are introduced into an empty container of 10 litre and heated to 327$^{\circ}C$ attain equilibrium as:
$NH_{4}Cl_{(g)} \rightleftharpoons NH_{3(g)} + HCl_{g} ; K_{p} = 0.36 atm^{2}.$ The quantity of solid $NH_{4}Cl$ left is -

Options:
A. 0.055 mole
B. 0.07 mole
C. 0.025 mole
D. 0.03 mole
E. 0.05 mole
F. 0.02 mole
G. 0.035 mole
H. 0.095 mole
I. 0.078 mole
J. 0.04 mole
Answer: F
If the vertical wall is removed and the force $F$ is applied then choose the correct option(s)

Options:
A. If $F = 9 0 N$ the normal reaction between $A \text { and } B$ will be $70 \mathrm { N }$
B. If $F = 9 0 N$ the normal reaction between $A \text { and } B$ will be $40 \mathrm { N }$
C. If $F = 9 0 N$ the normal reaction between $A \text { and } B$ will be $60 \mathrm { N }$
D. The minimum value of $F$ so that $A \text { and } B$ just start moving together is $60 \mathrm { N }$
E. If $F = 90 N$ the normal reaction between $A \text { and } B$ will be $65 \mathrm { N }$
F. If $F = 90 N$ the normal reaction between $A \text { and } B$ will be $55 \mathrm { N }$
G. If $F = 1 2 0 N$ the accelerations of the blocks will be $4 m / s ^ { 2 }$
H. If $F = 9 0 N$ the normal reaction between $A \text { and } B$ will be $30 \mathrm { N }$
I. If $F = 1 2 0 N$ the accelerations of the blocks will be $\frac { 80 } { 15 } \mathrm { m } / \mathrm { s } ^ { 2 }$
J. If $F = 9 0 N$ the normal reaction between $A \text { and } B$ will be $50 \mathrm { N }$
Answer: J
A man weighing W stands on a horizontal beam (hinged on a wall) of negligible weight at point C and holds a massless rope passing over two smooth pulleys. The rope is attached to point B on the beam. If the system is in equilibrium, then:

Options:
A. Hinge force at A is always in vertically upward direction, but varies with the value of W
B. If W= 600 N hinge force at A is 200 N
C. If W=600 N , tension in string is 400 N
D. Hinge force at A is always in vertically upward direction, irrespective of value of W
E. Hinge force at A is always in vertically upward direction, but varies quadratically with the value of W
F. Hinge force at A is always in vertically downward direction, irrespective of value of W
G. Hinge force at A is always in vertically upward direction, but does not vary with the value of W
H. Hinge force at A is always in vertically downward direction, but varies with the value of W
I. Hinge force at A is always in vertically downward direction, but varies linearly with the value of W
J. Hinge force at A is always in vertically downward direction, but varies inversely with the value of W
Answer: F
The conductivity of a saturated solution $\text{CaF}_2$ after deducting the contribution from water is $4.0 \times 10^{-5} \, \text{S cm}^{-1}$. If molar conductivity of $\text{CaF}_2$ at infinite dilution is $200 \, \text{S cm}^2 \text{mol}^{-1}$, then the solubility product of $\text{CaF}_2$ is:

Options:
A. $$3.2 \times 10 ^ { - 9 } \mathrm { M } ^ { 3 }$$
B. $$3.2 \times 10 ^ { - 13 } \mathrm { M } ^ { 3 }$$
C. $$3.2 \times 10 ^ { - 14 } \mathrm { M } ^ { 3 }$$
D. $$3.2 \times 10 ^ { - 11 }\ \mathrm { M } ^ { 3 }$$
E. $$3.2 \times 10 ^ { - 15 } \mathrm { M } ^ { 3 }$$
F. $$3.2 \times 10 ^ { - 10 }\ \mathrm{ M } ^ { 3 }$$
G. $$3.2 \times 10 ^ { - 8 } \mathrm { M } ^ { 3 }$$
H. $$3.2 \times 10 ^ { - 12 } \mathrm { M } ^ { 3 }$$
I. $$3.2 \times 10 ^ { - 16 } \mathrm { M } ^ { 3 }$$
Answer: F
For an ideal gas during an adiabatic process $\left ( \frac{T^{1}}{P^{2}} \right )^{\frac{1}{5}}$  = constant.  The molar heat capacity at constant volume of the gas is

Options:
A. $$3.5 R$$
B. $$0.5 R$$
C. $$2.7 R$$
D. $$1.5 R$$
E. $$3 R$$
F. $$\frac{7}{2}R$$
G. $$2.5 R$$
H. $$2.25 R$$
I. $$1.75 R$$
J. $$2.75 R$$
Answer: G
A ball is thrown vertically downwards from a height of $20\, m$ with an initial velocity of $v_o$. It collides with the ground, loses $50$% of its energy in the collision and rebounds to the same height. The initial velocity $v_o$ is (Take $g=10\, m/s^2$)

Options:
A. $$28\, m/s$$
B. $$20\, m/s$$
C. $$8\, m/s$$
D. $$14\, m/s$$
E. $$12\, m/s$$
F. $$18\, m/s$$
G. $$16\, m/s$$
H. $$10\, m/s$$
I. $$13\, m/s$$
J. $$15\, m/s$$
Answer: B
two solid spheres with uniformly distributed charges throughout their volumes are shown. Each has a radius $R$. Point $P$ lies on a line connecting the centers of the spheres, at radial distance $\dfrac{R}{2}$ from the center of sphere $1$. If the net electric field at point $P$ is zero, and $Q_1$ is $64\ \mu C$ and $Q_2$ = $8x\ \mu C$, what is the value of $x$?

Options:
A. $$6$$
B. $$11$$
C. $$12$$
D. $$10$$
E. $$7$$
F. $$9$$
G. $$8$$
Answer: F
$H_2S$ behaves as a weak diprotic acid in aqueous solution. Which of the following is the correct ex-planation for pH of a solution of $H_2S$ in terms of $pK_1,pK_2,[H_2S]$ and $[S^{2-}]$

Options:
A. $$pH=\left( { pK }_{ 1 }+{ pK }_{ 2 }-\log { \frac { \left[ { S }^{ 2- } \right] }{ \left[ { H }_{ 2 }S \right] } } \right) $$
B. $$pH=\left( pK_1 + pK_2 - \log \frac{[H_2S]}{[S^{2-}]} \right)$$
C. $$pH=\left( pK_1 + pK_2 + \log \frac{[S^{2-}]}{[H_2S]} \right)$$
D. $$pH=\left( pK_1 - pK_2 - \log \frac{[H_2S]}{[S^{2-}]} \right)$$
E. $$pH=\left( pK_1 + pK_2 - \log \frac{[S^{2-}]}{[H_2S]} \right)$$
F. $$pH=\left( { pK }_{ 1 }-{ pK }_{ 2 }+\log { \frac { \left[ { S }^{ 2- } \right] }{ \left[ { H }_{ 2 }S \right] } } \right) $$
G. $$pH=\left( pK_1 + pK_2 + \log \frac{[H_2S]}{[S^{2-}]} \right)$$
H. $$pH=\frac 12(pK_1+pK_2)$$
I. $$pH=\left( { pK }_{ 1 }+{ pK }_{ 2 }+\log { \frac { \left[ { S }^{ 2- } \right] }{ 2\times \left[ { H }_{ 2 }S \right] } } \right) $$
J. $$pH=\left( pK_1 - pK_2 + \log \frac{[H_2S]}{[S^{2-}]} \right)$$
Answer: H
For the following reaction $K_{ c }$ = 4 at a certain temperature $2NOCL(G)\ \rightleftharpoons \ 2NO(g)\ +\ { Cl }_{ 2 }(g)$. What concentration of NOCl must be put into a 4 it reaction vessel at the start, in order that the equilibrium concentration of NOCl will be 1 M ?

Options:
A. 3.80 M
B. 3.55 M
C. 3.25 M
D. 3.50 M
E. 3.85 M
F. 3.45 M
G. 3.60 M
H. 3.65 M
I. 3.75 M
J. 3 M
Answer: I
The standard reduction potential for two reactions are given below:
$$\begin{array}  {\text{AgCl}}\left( {\text{s}} \right){\text{ + }}{{\text{e}}^ - } \to {\text{Ag}}\left( {\text{s}} \right){\text{ + C}}{{\text{l}}^ - }\left( {{\text{aq}}} \right);{\text{ }}{{\text{E}}^{\text{o}}} = 0.22{\text{ V}} \hfill \  {\text{A}}{{\text{g}}^{\text{ + }}}\left( {{\text{aq}}} \right){\text{ + }}{{\text{e}}^ - } \to {\text{Ag}}\left( {\text{s}} \right);{\text{ }}{{\text{E}}^{\text{o}}} = 0.80{\text{ V}} \hfill \ \end{array} $$
The solubility product of $\rm{AgCl}$ under standard conditions of temperature ($298\, K$) is given by

Options:
A. $$2.4 \times {10^{ - 5}}$$
B. $$3.0 \times {10^{ - 8}}$$
C. 1.5 \times {10^{ - 5}}
D. $$1.6 \times {10^{ - 5}}$$
E. $$3.2 \times {10^{ - 5}}$$
F. $$3.2 \times {10^{ - 10}}$$
G. $$3.5 \times {10^{ - 5}}$$
H. $$1.5 \times {10^{ - 8}}$$
I. $$1.5 \times {10^{ - 10}}$$
J. $$2.0 \times {10^{ - 5}}$$
Answer: D
A particle executing S.H.M has velocities $v_{1}$ and $v_{2}$, acceleration $a_{1}$ and $a_{2}$ when the displacements of the particle from the mean position are $x_{1}$ and $x_{2}$ respectively. Select the correct option:

Options:
A. period = $2\pi \sqrt { \frac { { x }_{ 2 }^{ 2 }-{ x }_{ 1 }^{ 2 } }{ { v }_{ 1 }^{ 2 }+{ v }_{ 2 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }= \frac { { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } $
B. period = $2\pi \sqrt { \frac { { x }_{ 2 }^{ 2 }-{ x }_{ 1 }^{ 2 } }{ { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }=\frac { { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } }{ { a }_{ 1 }-{ a }_{ 2 } } $
C. period = \(2\pi \sqrt { \frac { { x }_{ 1 }^{ 2 }-{ x }_{ 2 }^{ 2 } }{ { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } } } \) , \( { x }_{ 1 }-{ x }_{ 2 }= \frac { { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } \)
D. period = $2\pi \sqrt { \frac { { x }_{ 2 }^{ 2 }-{ x }_{ 1 }^{ 2 } }{ { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }=\frac { { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } $
E. period = \(2\pi \sqrt { \frac { { x }_{ 1 }^{ 2 }-{ x }_{ 2 }^{ 2 } }{ { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } } } \) , \( { x }_{ 1 }-{ x }_{ 2 }= \frac { { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } }{ { a }_{ 1 }-{ a }_{ 2 } } \)
F. period = $2\pi \sqrt { \frac { { x }_{ 1 }^{ 2 }-{ x }_{ 2 }^{ 2 } }{ { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }=\frac { { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } $
G. period = $2\pi \sqrt { \frac { { x }_{ 1 }^{ 2 }-{ x }_{ 2 }^{ 2 } }{ { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }=\frac { { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } $
H. period = $2\pi \sqrt { \frac { { x }_{ 1 }^{ 2 }-{ x }_{ 2 }^{ 2 } }{ { v }_{ 1 }^{ 2 }+{ v }_{ 2 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }=\frac { { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } $
I. period = $2\pi \sqrt { \frac { { x }_{ 1 }^{ 2 }-{ x }_{ 2 }^{ 2 } }{ { v }_{ 1 }^{ 2 }-{ v }_{ 2 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }=\frac { { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } $
J. period = $2\pi \sqrt { \frac { { x }_{ 2 }^{ 2 }-{ x }_{ 1 }^{ 2 } }{ { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } } } ,{ x }_{ 1 }-{ x }_{ 2 }=\frac { { v }_{ 2 }^{ 2 }-{ v }_{ 1 }^{ 2 } }{ { a }_{ 1 }+{ a }_{ 2 } } $
Answer: I
The half life period of a first order chemical reaction is 6.93 minutes. Time required for the completion of 99% of the chemical reaction will be $(log 2 = 0.301) :$

Options:
A. 230.6 minutes
B. 11.515 minutes
C. 230.30 minutes
D. 23.33 minutes
E. 46.06 minutes
F. 23.06 minutes
G. 23.03 minutes
H. 460.6 minutes
I. 23.30 minutes
J. 230.3 minutes
Answer: G
Three highly conducting concentric spherical shells (acting as black body) of radii a, b and c are kept in vacuum. The region between the shells is evacuated and they can exchange energy only by radiation. Inside the innermost shell there is a heater of constant power P. In steady state, ($\sigma$ = Stefan's constant)

Options:
A. If both outer and middle shell were removed, the temperature of innermost shell would decrease by a factor of \(\left( \dfrac{a^2}{c^2} \right)^{1/4}\)
B. If both outer and middle shell were removed, the temperature of innermost shell would increase
C. If both outer and middle shell were removed, the temperature of innermost shell would stay the same
D. If both outer and middle shell were removed, the temperature of innermost shell would decrease
E. If both outer and middle shell were removed, the temperature of innermost shell would decrease by a factor of \(\left( \dfrac{a^2}{b^2} \right)^{1/4}\)
F. Temperature of innermost shell is $\left ( \dfrac{P}{4\pi\sigma} \right)^{1/4} \left ( \dfrac{1}{a^2} - \dfrac{1}{b^2}\right )^{1/4}$
G. If both outer and middle shell were removed, the temperature of innermost shell would increase by a factor of \(\left( \dfrac{b^2}{a^2} \right)^{1/4}\)
H. If both outer and middle shell were removed, the temperature of innermost shell would remain unchanged
Answer: D
Consider the following statements
(a) The $pH$ of a mixture containing $400 mL$ of $0.1 M \ H_2SO_4$ and $400mL$ of $0.1 M \ NaOH$ will be approximately $1.3$
(b) Ionic product of water is temperature dependent.
(c) A monobasic acid with $K_a = 10^{-5}$ has a $pH = 5$. The degree of dissociation of this acid is $50\%$.
(d) The Le Chatelier's principle is not applicable to common-ion effect.
The correct statement are :

Options:
A. (a), (b) and (d) with a slight modification in one statement: (a) The $pH$ of a mixture containing $400 mL$ of $0.1 M \ H_2SO_4$ and $400mL$ of $0.1 M \ NaOH$ will be approximately $1.1$.
B. (b) and (c)
C. (a), (b), and (d)
D. (a), (b) and (c)
E. (a) and (c)
F. (a) and (b) and (d) with a slight modification in one statement: (a) The $pH$ of a mixture containing $400 mL$ of $0.1 M \ H_2SO_4$ and $400mL$ of $0.1 M \ NaOH$ will be approximately $1.2$.
G. (a), (b), and (c) with a slight modification in one statement: (a) The $pH$ of a mixture containing $400 mL$ of $0.1 M \ H_2SO_4$ and $400mL$ of $0.1 M \ NaOH$ will be approximately $1.4$.
H. (a) and (b)
I. (a), (c), and (d)
J. (a), (b) and (d)
Answer: D
Calculate the amount (in milligrams) of $\displaystyle SeO_{3}^{2-}$ in solution on the basis of following data:$20$ mL of $\dfrac{M}{60}$ solution of $\displaystyle KBrO_{3}$ was added to a definite volume of $\displaystyle SeO_{3}^{2-}$ solution. The bromine evolved was removed by boiling and excess of $\displaystyle KBrO_{3}$ was back titrated with $5$ mL of $\dfrac{M}{25}$ solution of $\displaystyle NaAsO_{2}$. The reactions are given below.$1)\,\,\displaystyle SeO{_{3}}^{2-}+BrO{_{3}}^{-}+H^{-}\rightarrow SeO{_{4}}^{2-}+Br_{2}+H_{2}O$$2) \,\,BrO{_{3}}^{-}+AsO_{2}+H_{2}O\rightarrow Br^{-}+AsO{_{4}}^{3-}+H^{+}$(Atomic mass of $K = 39, Br = 80, As = 75, Na = 23, O = 16, Se = 79$)

Options:
A. 105
B. None of the above
C. $$95$$
D. 98
E. $$84$$
F. $$42$$
G. 110
H. 96
I. 93
J. 100
Answer: C
Charges $Q_1$ and $Q_2$ lie inside and outside respectively of a closed surface S. Let E be the field at any point on S and $\psi$ be the flux of E over S.

Options:
A. If $Q_1
B. If $Q_1 = 0$ and $Q_2 \neq 0$ then $E = 0$ but $\psi = 0$
C. If $Q_1 = 0$ and $Q_2 \neq 0$ then $E \neq 0$ but $\psi = Q_2$
D. If $Q_2$ changes, E will change but $\psi$ will not change.
E. If $Q_1 = 0$ and $Q_2 \neq 0$ then $E = 0$ but $\psi \neq 0$
F. If $Q_1 = 0$ and $Q_2 \neq 0$ then $E \neq 0$ but $\psi = 0$.
G. If $Q_1 = 0$ and $Q_2 \neq 0$ then $E \neq 0$ but $\psi = 0$
H. If $Q_1 = 0$ and $Q_2 \neq 0$ then $E \neq 0$ but $\psi \neq 0$.
I. If $Q_1 = 0$ and $Q_2 \neq 0$ then $E = 0$ but $\psi = 0$.
J. If $Q_1$ changes, both E and $\psi$ will change.
Answer: F
A particle moves along a circle of a fixed radius with a variable centripetal acceleration ${a}_{n}=k{t}^{n}$, where $k$ is a constant and $t$ is time. If slope of $P-t$ graph on log-log scale is $2$, then value of $n$ is (Here, $P$ is instantaneous power)

Options:
A. $$-1$$
B. $$6$$
C. $$4$$
D. $$3$$
E. $$1$$
F. $$0$$
G. $$2$$
H. $$5$$
Answer: D
Consider the reaction  : $2H_2(g) + 2NO(g) \rightarrow\  N_2(g) + 2H_2O(g)$The rate law for this reaction is :$Rate = k[H_2][NO]^2$Under what conditions could these steps represent the mechanism?Step 1 : $2NO(g) \rightleftharpoons  N_2O_2(g)$Step 2 : $N_2O_2  + H_2 \rightarrow\ N_2O + H_2O$Step 3 : $N_2O + H_2 \rightarrow\ H_2O + N_2$

Options:
A. Step 2 should be the slowest step
B. Step 1 should be the rate-determining step
C. Step 3 should be the slowest step
D. Step 2 should be the fastest step and Step 3 should be the rate-determining step
E. These steps can never satisfy the rate law
F. Step 2 should be the rate-determining step
G. Step 1 should be the fastest step
H. Step 2 should be the fastest step
I. Step 1 should be the slowest step
J. Step 3 should be the rate-determining step
Answer: E
A certain mixture of HCI and $ CH_3 - COOH $ is 0.1 M in each of the acid =s . 20 ml of this solution is titrated against 0.1 M NaOH. by how many units does the pH change from the start to the stage when the HCI is alomost completely neutralised ? $ K_a $ for acetic acid $ = 1.8 \times 10^{-5} $.

Options:
A. 1.03
B. 0.775
C. 3.172
D. 2.04
E. 1.775
F. 2.335
G. 1.335
H. 1.825
I. 2.325
J. 2.03
Answer: J
In a lake, a fish rising vertically to the surface of water uniformly at the rate of $3\ m/s$, observes a bird diving vertically towards the water at a rate of $9\ m/s$ vertically above it. the actual velocity of the dive of the bird is : (Given refractive index water $=4/3$)

Options:
A. $$4.5\ m/s$$
B. $$9.0\ m/s$$
C. $$9.2\ m/s$$
D. $$3.6\ m/s$$
E. $$6.0\ m/s$$
F. $$3.0\ m/s$$
G. $$5.4\ m/s$$
H. $$7.2\ m/s$$
I. $$8.1\ m/s$$
J. $$3.2\ m/s$$
Answer: A
Four light waves are represented by 
$(i)y=a_1\sin\omega t$           $(ii)y=a_2\sin(\omega t+\varepsilon)$
$(iii)y=a_1\sin2\omega t$      $(iv)y=a_2\sin2(\omega t+\varepsilon)$
Interference fringes may be observed due to superposition of

Options:
A. (i) and (iii)
B. $(ii)$ and $(iii)$
C. (i) and (iv)
D. (i) and (ii)
E. $(ii)$ and $(iv)$
F. $(i)$ and $(ii)$ and $(iii)$ and $(iv)$
Answer: F
A uniform solid sphere is lying at rest between a vertical wall and a fixed inclined plane as shown. There is no friction between sphere and the vertical wall out but coefficient of friction between sphere and the Fixed inclined plane is $\mu=\frac{1}{2}$. Then the magnitude of frictional force exerted by fixed inclined plane on sphere is

Options:
A. $$0$$
B. $$\dfrac{\sqrt{3}}{7}mg$$
C. $$\dfrac{\sqrt{3}}{4}mg$$
D. $$\dfrac{\sqrt{3}}{6}mg$$
E. $$\dfrac{mg}{2}$$
F. $$\dfrac{\sqrt{3}}{3}mg$$
G. $$\dfrac{\sqrt{3}}{2}mg$$
H. $$\dfrac{\sqrt{3}}{8}mg$$
I. $$\dfrac{\sqrt{3}}{5}mg$$
J. $$mg$$
Answer: A
A spherically symmetric change distribution is characterized by a change density having the following variation:
$p(r)=p_{0}\left( 1-\dfrac { r }{ R }  \right)$ for $r\lt R$,
$ p(r)=0$ for $r \ge R $. The radial electric field is given by:

Options:
A. $$\dfrac { { p }_{ 0 } }{ { 6buat }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
B. $$\dfrac { { p }_{ 0 } }{ { 2\varepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
C. $$\dfrac { { p }_{ 0 } }{ { 8arepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
D. $$ \dfrac { { p }_{ 0 } }{ { 3\varepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
E. $$\dfrac { { p }_{ 0 } }{ { 9\varepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
F. $$ \dfrac { { p }_{ 0 } }{ { \varepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
G. $$ \dfrac { { p }_{ 0 } }{ { 4\varepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
H. $$\dfrac { { p }_{ 0 } }{ { 12\varepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
I. $$\dfrac { { p }_{ 0 } }{ { 7zk }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
J. $$\dfrac { { p }_{ 0 } }{ { 5\varepsilon }_{ 0 } } \left( \dfrac { r }{ 3 } -\dfrac { { r }^{ 2 } }{ 4R } \right)$$
Answer: F
A block of mass $1\ kg$ resting on a rough inclined plane of inclination $60^{o} (\mu =0.5)$. What minimum force $F$ applied perpendicular to the plane on the block , so that the block does not slip $(g=10\ ms^{-2})$

Options:
A. $$6.32N$$
B. $$4.32N$$
C. $$12.32N$$
D. $$11.32N$$
E. $$8.32N$$
F. $$5N$$
G. $$10N$$
H. $$9.32N$$
I. 7.32N
J. $$6N$$
Answer: F
Two particle each of mass $'m'$ are placed at A and C are such $AB=BC=L.$. The gravitational force on the third particle placed at D at a distance Lon the perpendicular bisector of the line AC is

Options:
A. $$\cfrac{{G{m^2}}}{{3\sqrt 2 {L^2}}}\,along\,DB$$
B. $$\cfrac{{G{m^2}}}{{\sqrt 3 {L^2}}}\,along\,DB$$
C. $$\cfrac{{G{m^2}}}{{2\sqrt 3 {L^2}}}\,along\,DB$$
D. $$\cfrac{{G{m^2}}}{{\sqrt 2 {L^2}}}\,along\,AC$$
E. $$\cfrac{{G{m^2}}}{{{L^2}}}\,along\,AC$$
F. $$\cfrac{{G{m^2}}}{{\sqrt 2 {L^2}}}\,along\,DB$$
G. $$\cfrac{{G{m^2}}}{{2{L^2}}}\,along\,DB$$
H. $$\cfrac{{G{m^2}}}{{{L^2}}}\,along\,BD$$
I. $$\cfrac{{G{m^2}}}{{2\sqrt 2 {L^2}}}\,along\,DB$$
J. $$\cfrac{{G{m^2}}}{{\sqrt 2 {L^2}}}\,along\,BD$$
Answer: F
Two blocks of mass m each are connected by a string passing over a pulley. The table is smooth. Moment of inertia of pulley is $1$. Find out acceleration of the hanging mass.

Options:
A. $$a = \dfrac{mg}{2m+\dfrac{I}{r^2}}$$
B. $$a = \dfrac{mg}{2m+2\dfrac{I}{r^2}}$$
C. $$a = \dfrac{2mg}{m+\dfrac{I}{r^2}}$$
D. $$a = \dfrac{mg}{m+2\dfrac{I}{r^2}}$$
E. $$a = \dfrac{2mg}{2m+\dfrac{I}{r^2}}$$
F. $$a = \dfrac{2mg}{2m+2\dfrac{I}{r^2}}$$
G. $$a = \dfrac{2mg}{m+2\dfrac{I}{r^2}}$$
H. $$a = \dfrac{mg}{m+\dfrac{I}{r^2}}$$
I. $$a = \dfrac{2mg}{2m+\dfrac{2I}{r^2}}$$
J. $$a = \dfrac{2mg}{m+\dfrac{2I}{r^2}}$$
Answer: A
A square loop of side 5 cm being moved towards right at a constant speed of 1 cm/s. the front edge entres the 20 cm wide magnetic field at t=0 . find the emf induced in the loop at

Options:
A. $$t= 30s, 6 \times 10^{-4} V$$
B. $t=10 s,$    $3\times 10^{-4}$ V
C. $t= 30s,$    $9 \times 10^{-4}$ V
D. $t= 30s,$    $4.5 \times 10^{-4}$ V
E. $$t= 30s, 3 \times 10^{-4} V$$
F. $t=2s,$   $3\times 10^{-4}$ V and $t= 10s,$    $0$ V
G. $$t= 30s, 1.5 \times 10^{-4} V$$
H. $t= 22 s,$    $0$ V
Answer: F
Diborane tetrachloride was treated with $NaOH$ and the following reaction occured:
$$B_2Cl_4+NaOH\to NaBO_2+H_2O+H_2+NaCl$$
If $1362\ ml$ of hydrogen gas is formed at $STP$, how much $B_2Cl_4$ was consumed?
$$(B=11)$$

Options:
A. $$9.78\ g$$
B. $$9.80\ g$$
C. $$9.64\ g$$
D. $$9.70\ g$$
E. $$9.88\ g$$
F. $$9.97\ g$$
G. $$9.84\ g$$
H. $$0.0968\ g$$
I. $$9.75\ g$$
J. $$23.57\ g$$
Answer: G
For the reaction: $A\rightarrow Products$ (order $=n$) the theoritical time of completion of reaction is

Options:
A. \cfrac { { \left[ { A }_{ 0 } \right] }^{ 1-n } }{ K(n-1) } \text{ for } n < 1
B. \cfrac { { \left[ { A }_{ 0 } \right] }^{ 1-n } }{ K(1+n) } \text{ for } n < 1
C. $\quad \cfrac { { \left[ { A }_{ 0 } \right] }^{ 1-n } }{ K(1-n) } $ for $n
e 1$
D. Infinite for all value of $n$
E. $\quad \cfrac { { \left[ { A }_{ 0 } \right] }^{ 1-n } }{ K(1-n) } $ for $n< 1$
F. \cfrac { { \left[ { A }_{ 0 } \right] }^{ 1-n } }{ K(1-n) } \text{ for } n \neq 1
G. \cfrac { { \left[ { A }_{ 0 } \right] }^{ 2-n } }{ K(1-n) } \text{ for } n < 1
H. \quad \cfrac { { \left[ { A }_{ 0 } \right] }^{ 2-n } }{ K(2-n) } \text{ for } n < 1
I. \cfrac { { \left[ { A }_{ 0 } \right] }^{ 1-n } }{ K(2-n) } \text{ for } n < 1
Answer: E
A shell, made of material of electric conductivity $\frac{{{{10}^8}}}{\pi }{\left( {\Omega  - m} \right)^{ - 1}},$ has thickness $t=2mm$ and $R=10cm,$ In an arrangement, its inside surface is kept at a lower potential then its outside surface. The resistance offered by the shell is equal to

Options:
A. $$5 - {10^{ - 12}}\Omega $$
B. $$5\pi - {10^{ - 12}}\Omega $$
C. $$5.5\pi - {10^{ - 12}}\Omega$$
D. $$5.5\pi - {10^{ - 11}}\Omega$$
E. $$5.25\pi - {10^{ - 12}}\Omega$$
F. $$6\pi - {10^{ - 12}}\Omega$$
G. $$5\pi - {10^{ - 13}}\Omega$$
H. $$4.5\pi - {10^{ - 12}}\Omega$$
I. $$5 - {10^{ - 11}}\Omega $$
J. $$2.5\pi - {10^{ - 11}}\Omega $$
Answer: B
Two moles of ideal helium gas are in a rubber balloon at $30^o$C. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to $35^o$C. The amount of heat required in raising the temperature is nearly (take $R=6.31$ J/mol.K).

Options:
A. 144 J
B. $208$ J
C. $62$ J
D. $104$ J
E. 186 J
F. 218 J
G. 166 J
H. 216 J
I. 124 J
Answer: B
A metal bob of radius 4cm and mass 4kg is hung from arigid support with the help of athin metal wire  of area of cross section $2x{ 10 }^{ -6 }{ m }^{ 2 }$ and length 1.96 m and Young modulus of the material $2x{ 10 }^{ 11 }N/{ m }^{ 2 }.$ When the bob is at rest the bottom of the bob is 4 mm above ground. What is the maximum velocity in m/s with the bob can be oscillated? $(g={ 10 }{ m }/s^{ 2 })$

Options:
A. 19.67
B. 19.72
C. 19.62
D. 19.52
E. 19.92
F. 1.97
G. 20
H. 19.87
I. 19.82
J. 197.2
Answer: B
$C_6H_6$ freezes at $5.5^oC$. At what temperature will a solution of 10.44g of$C_4H_{10}$ in 200 g of $C_6H_6$ freeze? $K_f(C_6H_6) = 5.12^oC/m$

Options:
A. $$5.308 ^oC$$
B. $$4.808 ^oC$$
C. $$4.408 ^oC$$
D. $$4.508 ^oC$$
E. $$4.708 ^oC$$
F. $$5.5^oC$$
G. $$0.892^oC$$
H. None of the above.
I. $$4.608 ^oC$$
J. $$4.908 ^oC$$
Answer: I
A chain of length $l$ lies in a pile on the floor and its mass per unit length linearly increases from $\lambda$ to $2\lambda$ from end point $A$ to end point $B$. If its end point $A$ is raised vertically by applying force applied force at a constant speed ${V}_{0}$. Then

Options:
A. The graph between the applied force and \( y \) is a parabola (after losing the contact)
B. The graph between the applied force and $y$ is a hyperbola (after losing the contact)
C. The graph between the applied force and $y$ is a hyperbola (before losing the contact)
D. The graoh between the applied force and $y$ is straight line (after losing the contact)
E. The graph between the applied force and \( y \) is a straight line (before losing the contact)
F. The graph between the applied force and $y$ is a parabola (after losing the contact)
Answer: D
The e.m.f. of the cell $\begin{matrix}Pt H_2\ 1atm\end{matrix} \begin{vmatrix}HOCN 1.3 \times 10^{-3} M \end{vmatrix} \begin{vmatrix}Ag^+ 0.8 M \end{vmatrix} Ag(s)$ is 0.982 V. Calculate the $K_a$ for HOCN. Note that $Ag^+  + e \rightarrow Ag(s);         E^o = 0.80 V$

Options:
A. $$3.33 \times 10^{-4} $$
B. $$2.4 \times 10^{-3}$$
C. $$6.74 \times 10^{-4} $$
D. $$4.44 \times 10^{-4} $$
E. $$1.5 \times 10^{-3}$$
F. $$2.22 \times 10^{-4} $$
G. $$1.9 \times 10^{-3}$$
H. $$1.8 \times 10^{-3}$$
I. $$2.2 \times 10^{-3}$$
J. $$2.4 \times 10^{-4}$$
Answer: C
The resistance of an electrical toaster has a temperature dependence given by $R(T) = R_0 [1 + \alpha(T- T_0)]$ in its range of operation. At $T_0 = 300 K, R_0 =100 \Omega$ and at $T= 500 K, R= 120 \Omega$. The toaster is connected to a voltage source at 200 V and its temperature is raised at a constant rate from 300 to 500 K in 30 s. The total work done in raising the temperature is :

Options:
A. $$400\, ln\dfrac{1.5}{1.3}J$$
B. $$400\, ln\dfrac{6}{7}J$$
C. $$200\, ln\dfrac{4}{5}J$$
D. $$200\, ln\dfrac{2}{3}J$$
E. $$300\, ln\dfrac{4}{5}J$$
F. $$400\, ln\dfrac{6}{5}J$$
G. $$300 J$$
H. $$200\, ln\dfrac{5}{6}J$$
I. $$400\, ln\dfrac{5}{6}J$$
J. $$300\, ln\dfrac{5}{6}J$$
Answer: I
A block of mass m is pushed towards a movable wedge of mass nm and height h with a velocity u. All surfaces are smooth. Choose the correct statement from the following.

Options:
A. If the block overshoots P, the angle of projectile is $2\alpha$
B. If the block overshoots P, the angle of projectile is less than $\alpha$
C. If the block overshoots P, the angle of projectile is $\alpha/2$
D. If the block overshoots P, the angle of projectile is $\alpha - 5^\circ$
E. If the block overshoots P, the angle of projectile is $\alpha + 10^\circ$
F. If the block overshoots P, the angle of projectile is greater than $\alpha$
G. Block will reach the top of the wedge if $u=\sqrt{2gh\left(1-\dfrac{1}{n}\right)}$
H. If the block overshoots P, the angle of projectile is $\alpha$
I. If the block overshoots P, the angle of projectile is $\alpha - 10^\circ$
J. Block will reach the top of the wedge if $u=\sqrt{2gh\left(1+\dfrac{1}{n}\right)}$
Answer: B
An inductor of inductance 2.0 mH is connect across a charge capacitor of capacittance 5.0 uF , and the resulting LC circuit is set oscillating at natural frequency. Let Q denote the instantaneous charge on the capacitor, and I the current in the circuit. It is found the maximum value of Q is 200 uC .When Q=100uC the value of $\dfrac{dI}{dt} $ is found to be $10^{x}$ A/s. Find the value of x is then

Options:
A. x=-4
B. x=-2
C. x=-8
D. x=3
E. x=-6
F. x=-5
G. x=-7
H. x=5
I. x=4
J. x=-3
Answer: A
Assertion (A) : A large soap bubble expands while a small bubble shrinks, when they are connected to each other by a capillary tube.Reason (R) : The excess pressure (due to surface tension) inside a spherical bubble is more due to its small volume.

Options:
A. Both Assertion (A) and Reason (R) are true, but Reason (R) is not a plausible explanation of the Assertion(A).
B. Assertrion (A) is false, but the Reason (R) is true.
C. Both Assertion (A) and Reason (R) are true, but Reason (R) is not a complete explanation of the Assertion(A).
D. Assertion (A) is true, but the Reason (R) is false.
E. Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation of the Assertion(A) .
F. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion(A).
G. Both Assertion (A) and Reason (R) are false.
H. Both Assertion (A) and Reason (R) are true, but Reason (R) is the correct explanation of the Assertion(A) with a slight modification in the reason.
I. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion(A)
J. Both Assertion (A) and Reason (R) are true and the Reason (R) is correct explanation of the Assertion(A) .
Answer: J
An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii $r_{e}, r_{p}, r_{\alpha}$ respectively in a uniform magnetic field $B$. The relation between $r_{e}, r_{p}, r_{\alpha}$ is

Options:
A. r_{e} > r_{p} < r_{\alpha}
B. $$r_{e} < r_{p} < r_{\alpha}$$
C. $$r_{e} < r_{p} = r_{lpha}$$
D. $$r_{p} > r_{e} = r_{\alpha}$$
E. r_{p} < r_{e} = r_{\alpha}
F. $$r_{e} > r_{p} = r_{\alpha}$$
G. $$r_{e} < r_{lpha} < r_{p}$$
H. r_{e} > r_{p} > r_{\alpha}
I. $$r_{p} < r_{e} < r_{\alpha}$$
Answer: D
The equations of a traveling and stationary waves are ${y}_{1}=a\sin{(\omega t-kx)}$ and ${y}_{2}=a\sin{kx}\cos{\omega t}$. The phase difference between two points ${x}_{1}=\cfrac{\pi}{4k}$ and ${x}_{2}=\cfrac{4\pi}{3k}$ are ${\phi}_{1}$ and ${\phi}_{2}$ respectively for two waves, where $k$ is the wave number. The ratio of ${\phi}_{1}/{\phi}_{2}$ is

Options:
A. $$12/13$$
B. $$13/12$$
C. $$6/7$$
D. $$14/15$$
E. $$12/14$$
F. $$16/3$$
G. $$7/6$$
H. $$15/14$$
Answer: C
$CaCl_{2}.6H_{2}O(s)\rightleftharpoons CaCl_{2}(s) + 6H_{2}O(g)\ K_{p} = 6.4\times 10^{-17} atm^{6}$Excess solid $CaCl_{2}.6H_{2}O$ and $CaCl_{2}$ are taken in a container containing some water vapours at a pressure of $1.14\ torr$ at a particular temperature.

Options:
A. $CaCl_{2}.6H_{2}O(s)$ acts as hygroscopic substance
B. $CaCl_{2}.6H_{2}O(s)$ acts as effluoroscent substance
C. $CaCl_{2}(s)$ acts as a deliquescent substance
D. $CaCl_{2}.6H_{2}O(s)$ acts as a desiccant substance
E. $CaCl_{2}.6H_{2}O(s)$ acts as a deliquescent substance
F. $CaCl_{2}(s)$ acts as efflorescent substance
G. $CaCl_{2}(s)$ acts as drying agent under given condition
Answer: B
In S.H.M. potential energy of the particle at the mean position is $E_1$ and the kinetic energy is $E_2$, then :

Options:
A. E_1 = \dfrac{E_2}{3}
B. E_1 = \dfrac{3E_2}{2}
C. E_1 = \dfrac{3E_2}{4}
D. Total kinetic energy at x = $\dfrac{\sqrt 3A}{2}$ is $E_1 + \dfrac{3E_2}{4}$
E. Total potential energy at x = $\dfrac{\sqrt 3A}{2}$ is $E_1 + \dfrac{3E_2}{4}$
F. Total kinetic energy at x = $\dfrac{A}{\sqrt2}$ is $E_1 + \dfrac{E_2}{4}$
G. E_1 = \dfrac{E_2}{4}
H. $$E_1 = E_2$$
I. $$E_1 = \dfrac{E_2}{2}$$
J. E_1 = \dfrac{2E_2}{3}
Answer: E
The acceleration vector along x-axis of a particle having initial velocity ${v_0}$  changes with distance x as $a = \sqrt x $.The distance covered by the particles, When its speed becomes twice that of initial speed is:-

Options:
A. $${\left( {\frac{1}{4}{v_0}} \right)^{4/3}}$$
B. $${\left( {\frac{2}{3}{v_0}} \right)^{4/3}}$$
C. $${\left( {\frac{1}{3}{v_0}} \right)^{4/3}}$$
D. $${\left( {\frac{7}{6}{v_0}} \right)^{4/3}}$$
E. $${\left( {\frac{5}{3}{v_0}} \right)^{4/3}}$$
F. $${\left( {\frac{3}{2}{v_0}} \right)^{4/3}}$$
G. $$2{v_0}$$
H. $${\left( {\frac{4}{3}{v_0}} \right)^{4/3}}$$
I. $${\left( {\frac{1}{2}{v_0}} \right)^{4/3}}$$
J. $${\left( {\frac{9}{4}{v_0}} \right)^{4/3}}$$
Answer: B
The EMF of the cell $M\left | M^{n+}(0.2\:M) \right |\left | H^{+}(1\:M) \right |H_{2}(g)\:(1\:atm)$, Pt at $25^{\circ}C$ is 0.81 V. Calculate the valency of the metal if the standard oxidation potential of the metal is 0.76 V.

Options:
A. $$n=2$$
B. $$n=1$$
C. $$n=7$$
D. $$n=4$$
E. None of these
F. $$n=3$$
G. $$n=8$$
H. $$n=5$$
I. $$n=6$$
Answer: A
A small ball of density $\rho _{0}$ is released from rest from the surface of a liquid whose density varies with depth $h$ as $\rho =\dfrac{\rho _{0}}{2}(a+\beta h)$Mass of the ball is $m$.Select the most appropriate option:

Options:
A. The maximum speed of the ball is $\dfrac{2-a}{\sqrt{2\beta }} $.
B. Both (a) and (b) are correct.
C. Both (a) and (b) are wrong.
D. The maximum speed of the ball is $\dfrac{2a-1}{\sqrt{2\beta}}$
E. The maximum speed of the ball is \(\dfrac{2-a}{\sqrt{2\beta}}\).
F. The maximum speed of the ball is \(\dfrac{a+2}{\sqrt{2\beta}}\).
G. The particle will execute SHM.
H. The maximum speed of the ball is \(\dfrac{a-2}{\sqrt{2\beta}}\).
I. The particle will not execute SHM.
J. The maximum speed of the ball is $\dfrac{2+a}{\sqrt{2\beta}}$.
Answer: G
$$H_{2} (g)+CO_{2}(g) \rightleftharpoons  H_{2}O(g)+ CO(g); K_{p}=4$$
At $1660^{o}C\ 0.80$ mole $H_{2}$ and $0.80$ mole $CO_{2}$ are taken into a $5L$ container to attain the equilibrium. What is the equilibrium concentration of $CO_{2}(g)$.

Options:
A. $$0.539 M$$
B. $$0.523\ M$$
C. $$0.533\ M$$
D. $$0.537\ M$$
E. $$0.535\ M$$
F. $$0.543\ M$$
G. $$None\ of\ these$$
H. $$0.531 M$$
I. $$0.0534\ M$$
J. $$0.532 M$$
Answer: C
The volume occupied by $2mol$ of $N_2$ at $200K$ and $10.1325 MPa$ pressure is (Given that $\dfrac{P_cV_c}{RT_c}={\dfrac{3}{8}}$and $\dfrac{P_rV_r}{T_r}=2.21)$, where $P_r,V_r$ and $T_r$ are reduced pressure, reduced volume and reduced temperature.

Options:
A. $$272.0cm^3$$
B. $$136cm^3$$
C. $$544 cm^3$$
D. $$20.3cm^3$$
E. $$20.7cm^3$$
F. $$21.4cm^3$$
G. $$41.4cm^3$$
H. $$22.7cm^3$$
I. $$21.7cm^3$$
Answer: E
A simple pendulum of mass m and charge +q is suspended vertically by a massless thread of length l . at the point of suspension, a point charge +q is also foxed. if the pendulum is displaced slightly from equilibrium position, its time period will be

Options:
A. $$T= 2 \pi \sqrt \frac{l}{g - \frac{kq^2}{ml^2}}$$
B. will be greater than $ 2 \pi \sqrt \frac {l}{g}.
C. $$ T= 2 \pi \sqrt \frac {l}{g} $$
D. $$T= 2 \pi \sqrt \frac{l}{g + \frac{2kq^2}{ml^2}}$$
E. $$ t= 2 \pi \sqrt \frac {l}{ g+ \frac{kq^2}{ml^2} } $$
F. $$T= 2 \pi \sqrt \frac{l}{g + \frac{3kq^2}{ml^2}}$$
G. $$ T= 2 \pi $$
H. $$T= 2 \pi \sqrt \frac{l}{g + \frac{4kq^2}{ml^2}}$$
I. $$T= 2 \pi \sqrt \frac{l}{g + \frac{kq^2}{ml^2}}$$
J. $$T= 2 \pi \sqrt \frac{l}{g - \frac{2kq^2}{ml^2}}$$
Answer: C
1.5 g of a monobasic acid, when dissolved in 150g of water, lowers the freezing point by $0.165^{\circ}$C. 0.5 g of the same acid when titrated, after dissolution in water, requires 37.5 ml of N/10 alkali. Calculate the degree of dissociation of the acid ?($K_{f} $for water =$1.86^{\circ}$C mol$^{-1}).$

Options:
A. $11.34$%
B. $12.34$%
C. $34.64$%
D. $89.64$%
E. $11.64$%
F. 10.34%
G. 11.84
H. $55.34$%
I. 45.34%
J. $65.36$%
Answer: A
A magnet $20 \ cm$ long with its poles concentrated at its ends is placed vertically with its north pole on the table. At a point due $20\  cm$ south (magnetic) of the pole, a neutral point is obtained. If $H = 0.3 \ G$, then the pole strength of the magnet is approximately :

Options:
A. $18.5 $ A-cm
B. 18.0 A-m
C. 1.85 A-m
D. 185 A-cm
E. $18.5 $ A-m
F. 185 A-m
G. 1.85 A-cm
Answer: E
In the bottom of a vessel with mercury of density $\rho$ there is a round hole of radius $r$. At what maximum height of the mercury layer will the liquid still not flow out through this hole. (Surface tension = $T$)

Options:
A. $$\dfrac {2T} {rpg}$$
B. $$\dfrac T {2rpg}$$
C. $$\dfrac{T}{2rpg}$$
D. $$\dfrac{5T} {2rpg}$$
E. $$\dfrac{T}{3rpg}$$
F. $$\dfrac{7T}{4rpg}$$
G. $$\dfrac{T}{4rpg}$$
H. $$\dfrac{4T} {rpg}$$
I. $$\dfrac T {rpg}$$
J. $$\dfrac{3T} {rpg}$$
Answer: I
An ideal $CO_2$ gas obey the law $PV^x = constant$. The value of x for which it has non positive molar specific heat at normal temperature, is

Options:
A. $$1.5$$
B. $$1.4$$
C. $$-0.5$$
D. $$1.55$$
E. $$1.42$$
F. $$-1.4$$
G. $$1.38$$
H. $$1.3$$
I. $$1.45$$
J. $$1.35$$
Answer: B