MikaStars39/nano-eval huggingface dataset:
Find out the correct order of rate of reaction towards allylic substitution :(I) $CH_3 - CH = CH_2$(II) $CH_3 -CH_2 - CH = CH_2$(III) $CH_3 - \overset{CH_3}{\overset{\!\!\!\!\!\!|}{CH}} - CH = CH_2$

Options:
A. ${\rm{II}}\,\,{\rm{ > }}\,\,{\rm{III}}\,\,{\rm{ > }}\,\,{\rm{I}}$
B. ${\rm{I}}\,\,{\rm{ > }}\,\,{\rm{II}}\,\,{\rm{ > }}\,\,{\rm{III}}$
C. $${\rm{II}}\,\,{\rm{ > }}\,\,{\rm{I}}\,\,{\rm{ > }}\,\,{\rm{III}}$$
D. $${\rm{III}}\,\,{\rm{ > }}\,\,{\rm{I}}\,\,{\rm{ > }}\,\,{\rm{II}}$$
E. ${\rm{I}}\,\,{\rm{ > }}\,\,{\rm{III}}\,\,{\rm{ > }}\,\,{\rm{II}}$
F. ${\rm{III}}\,\,{\rm{ > }}\,\,{\rm{II}}\,\,{\rm{ > }}\,\,{\rm{I}}$
Answer: C
Which of the following is not correctly matched?

Options:
A. Energy of one mole of photons, $E=\frac{N_A hc}{3\lambda}$
B. Energy of one mole of photons, $E=\frac{N_A hc}{\lambda^2}$
C. Energy of one mole of photons, $E=\frac{N_A hc}{2\lambda}$
D. Energy of one mole of photons, $E=N_A\frac{hc}{\lambda}$
E. Energy of one mole of photons, $E=\frac{N_A hc}{4\lambda}$
F. Energy of one mole of photons, $E=\frac{N_A hc}{6\lambda}$
G. Energy associated with Bohr's orbit,$E=\frac{-2.18\times10^{-18}J\times Z^2}{n^2}$
H. Kinetic energy of the ejected electron, $hv= hv_0+\frac{1}{2} mv^2$
I. Energy gap between two orbits, $\triangle E = R_H(\frac{1}{n_1^2}-\frac{1}{n_2^2})$
J. Energy of one mole of photons, $E=\frac{N_A hc}{5\lambda}$
Answer: D
Two long conducting co-axial cylinders with have radii R and 2R. Their cross section is shown in the figure. The inner one carries a $ \lambda $ charge $ x $ per unit length. A particle of mass m and negative charge - q is given velocity $ v_0 $ perpendicular to axis of cyinders and perpendicular to radial direction. at a distance $ \frac {3R} {2} $ from the common axis of cylinders. The velocity v so that it moves in a circular path is

Options:
A. $${ v }_{ 0 }=\sqrt { \frac { qx }{ 9\pi \epsilon _{ 0 }m } }$$
B. $${ v }_{ 0 }=\sqrt { \frac { qx }{ 5\pi \epsilon _{ 0 }m } }$$
C. $$ { v }_{ 0 }=\sqrt { \frac { qx }{ 2\pi \epsilon _{ 0 }m } } $$
D. $${ v }_{ 0 }=\sqrt { \frac { qx }{ 6\pi \epsilon _{ 0 }m } }$$
E. $${ v }_{ 0 }=\sqrt { \frac { qx }{ 3\pi \epsilon _{ 0 }m } }$$
F. $$ { v }_{ 0 }=\sqrt { \frac { qx }{ 4\pi \epsilon _{ 0 }m } } $$
G. $${ v }_{ 0 }=\sqrt { \frac { 2qx }{ 3\pi \epsilon _{ 0 }m } }$$
H. $$ { v }_{ 0 }=\sqrt { \frac { qx }{ \pi \epsilon _{ 0 }m } } $$
I. $${ v }_{ 0 }=\sqrt { \frac { qx }{ 8\pi \epsilon _{ 0 }m } }$$
J. $$ { v }_{ 0 }=\sqrt { \frac { qx }{ \epsilon _{ 0 }m } } $$
Answer: H
20 mL of 0.1 ${ MH }_{ 2 }{ SO }_{ 4 }$ solution is added to 30 mL of 0.2 $M\quad { NH }_{ 4 }OH$ solution. The pH of the resulatant mixture is : $\left[ { pk }_{ b }\quad of\quad { NH }_{ 4 }OH=\quad 4.7 \right] .$

Options:
A. 8.9
B. 5.0
C. 5.2
D. 8.8
E. 9.3
F. 9.4
G. 8.7
H. 9.1
I. 9.2
J. 9.0
Answer: F
A source of constant voltage $V$ is connected to a resistance $R$ and two ideal inductors $L_{1}$ and $L_{2}$ through a switch $S$ as shown. There is no mutual inductance between the two inductors. The switch $S$ is initially open. At $t=0$, the switch is closed and current beings to flow. Which of the following options is/are correct?

Options:
A. The ratio of the currents through $L_{1}$ and $L_{2}$ is fixed at all times $(t\ >\ 0)$
B. After a long time, the current through $L_{1}$ will be $\dfrac {V\ \ L_{1}}{2R\ L_{1}+L_{2}}$
C. At $t=0$, the current through the resistance $R$ is $\dfrac{V}{3R}$
D. At $t=0$, the current through the resistance $R$ is $\dfrac{V}{2R}$
E. At $t=0$, the current through the resistance $R$ is $\dfrac{V}{4R}$
Answer: A
A small object , tied at the end of a string of length r is launched into a vertical cicle with a speed $ 2\sqrt{gr}$ at the lowest points . Its speed when the string is horizontal is

Options:
A. $$= \sqrt{3gr} $$
B. \sqrt{7gr}
C. \sqrt{6gr}
D. \sqrt{5gr}
E. $$ = \sqrt{2gr} $$
F. 0
G. \sqrt{9gr}
H. = \sqrt{4gr}
I. $$\sqrt{8gr}$$
J. $$ > \sqrt{3gr}$$
Answer: J
The velocity of a particle moving in straight line depends on its position $x$ w.r.t. to fixed origin according to the relation $16x^{2} + 4v^{2} = 9$. The acceleration of particle when it is at $x = 2\ m$ is:

Options:
A. $$-8\ m/s^{2}$$
B. $$-4\ m/s^{2}$$
C. $$-14\ m/s^{2}$$
D. Not defined
E. $$-6\ m/s^{2}$$
F. $$-12\ m/s^{2}$$
G. $$-16\ m/s^{2}$$
H. $$-10\ m/s^{2}$$
I. $$-2\ m/s^{2}$$
Answer: A
A body of mass $6\ kg$ is hanging from another body of mass $10\ kg$ as shown in figure. This combination is being pulled up by a string with an acceleration of $2\ ms^{-2}$, the tension $T_1$ is $(g = 10ms^{-2})$.

Options:
A. 230\ N
B. 180\ N
C. 250\ N
D. $$240\ N$$
E. 200\ N
F. 228\ N
G. $$150\ N$$
H. $$192\ N$$
I. 210\ N
J. $$220\ N$$
Answer: D
A dipole of dipole moment $\overline P  = 2\widehat i - 3\widehat j + 4\widehat k$ is placed at point $A\left( {2, - 3,1} \right)$. The electric potential due to this dipole at the point $B\left( {4, - 1,0} \right)$ is equal to ( All the parameters specified here are in S.I. units.)

Options:
A. 6 \times {10^9}\;{\text{Volts}}
B. $$ - 2 \times {10^9}\;{\text{Volts}}$$
C. 1 \times {10^9}\;{\text{Volts}}
D. 5 \times {10^9}\;{\text{Volts}}
E. $$2 \times {10^9}\;{\text{Volts}}$$
F. - 4 \times {10^9}\;{\text{Volts}}
G. -1 \times {10^9}\;{\text{Volts}}
H. $$ - 3 \times {10^9}\;{\text{Volts}}$$
I. $$3 \times {10^9}\;{\text{Volts}}$$
J. 4 \times {10^9}\;{\text{Volts}}
Answer: E
A string of length 1$\mathrm { m }$ and linear mass density 0.01$\mathrm { kgm } ^ { - 1 }$ is stretched to a tension of $ 100 \mathrm { N }$ . When both ends of the string are fixed, the three lowest frequencies for standing wave are $f _ { 1 }$ f and $f _ { 3 }$ . When only one end of the string is fixed, the three lowest frequencies for standing wave are $n _ { 1 } , n _ { 2 }$ and $n _ { 3 }$ . Then

Options:
A. $$f _ { 3 } = 5 f _ { 1 } = n _ { 2 } = 125 \mathrm { Hz }$$
B. n _ { 3 } = 5 n _ { 1 } = f _ { 3 } = 130 \mathrm { Hz }
C. $$n _ { 3 } = 5 n _ { 1 } = f _ { 3 } = 125 \mathrm { Hz }$$
D. n _ { 3 } = 5 n _ { 1 } = f _ { 3 } = 150 \mathrm { Hz }
E. n _ { 3 } = 5 n _ { 1 } = f _ { 3 } = 160 \mathrm { Hz }
F. n _ { 3 } = 5 n _ { 1 } = f _ { 3 } = 175 \mathrm { Hz }
G. $$n _ { 2 } = \frac { f _ { 1 } + f _ { 2 } } { 2 } = 75 H z$$
H. $$f _ { 3 } = n _ { 2 } = 3 f _ { 1 } = 150 \mathrm { Hz }$$
I. n _ { 3 } = 5 n _ { 1 } = f _ { 3 } = 140 \mathrm { Hz }
J. n _ { 3 } = 5 n _ { 1 } = f _ { 3 } = 135 \mathrm { Hz }
Answer: C
An open capillary tube is lowered in a vessel with mercury. The difference between the levels of the mercury in the vessel and in the 
capillary tube $\triangle h=4.6\ mm$. What is the radius of curvature of the mercury meniscus in the capillary tube? Surface tension 
of mercury is $0.46\ N/m$, density of mercury is $13.6\ gm/cc$.

Options:
A. Information insufficient
B. \dfrac { 1 }{ 510 }\ m
C. \dfrac { 1 }{ 1360 } \ m
D. \dfrac { 1 }{ 920 }\ m
E. $$\dfrac { 1 }{ 850 }\ m$$
F. $$\dfrac { 1 }{ 340 }\ m$$
G. \dfrac { 1 }{ 710 }\ m
H. $$\dfrac { 1 }{ 1020 }\ m$$
I. $$\dfrac { 1 }{ 680 }\ m$$
J. \dfrac { 1 }{ 1700 }\ m
Answer: I
The effects are produced at a given point in space by two waves described by the equations, $\quad { y }_{ 1 }={ y }_{ m }\sin { \omega t } $ and ${ y }_{ 2 }={ y }_{ m }\sin { (\omega t+\theta ) } $ where ${ y }_{ m }$ is the same for both the waves and $\theta $ is a phase angle. Tick the incorrect statement among the following.

Options:
A. The maximum amplitude that can be achieved at a point is twice the amplitude of either wave and occurs at $\theta = \frac{\pi}{2}$
B. When the intensity is zero, the net amplitude is zero, and at this point $\theta =\pi$
C. The maximum amplitude that can be achieved at a point is twice the amplitude of either wave and occurs at $\theta = \frac{3\pi}{4}$
D. The maximum amplitude that can be achieved at a point is twice the amplitude of either wave and occurs at $\theta = \frac{\pi}{4}$
E. The maximum amplitude that can be achieved at a point is twice the amplitude of either wave and occurs at $\theta = \frac{3\pi}{2}$
F. The maximum intensity that can be achieved at a point is twice the intensity of either wave and occurs if $\theta =0$
G. The maximum amplitude that can be achieved at a point is half the amplitude of either wave and occurs at $\theta =0$
H. The maximum amplitude that can be achieved at a point is twice the amplitude of either wave and occurs at $\theta =2\pi$
I. The maximum intensity that can be achieved at a point is four times the intensity of either wave and occurs if $\theta =0$
J. The maximum amplitude that can be achieved at a pot its twice the amplitude of either wave and occurs at $\theta =0$
Answer: J
A block of mass $m$ is pressed against a vertical wall by applying a force $P$ at an angle $\theta$ to the horizontal as shown in the figure. As a result, if that block is prevented from falling down and $\mu$ is the coefficient of static friction between the block and the wall, the value of P $=$

Options:
A. $$\dfrac{mg}{\mu sin \theta + cos \theta}$$
B. \dfrac{mg}{\mu sin \theta - cos \theta}
C. \dfrac{mg}{\mu cos \theta + cos \theta}
D. $$\dfrac{mg}{\mu cos \theta - sin \theta}$$
E. \dfrac{mg}{\mu cos \theta + sin \theta}
F. \dfrac{mg}{\mu sin \theta + sin \theta}
G. \dfrac{mg}{\mu cos \theta - cos \theta}
Answer: D
A cylinderical rod has breaking stress ${ 10 }^{ 5 }N/{ m }^{ 2 }$ .The maximum possible length of the rod that can hang freely vertically downwards under gravity is 4m. If g = ${ 10 }m/{ s }^{ 2 }$, the density of the material is CGS units is :

Options:
A. 2
B. 1.3
C. 1.75
D. 1.1
E. 1
F. 1.2
G. 2.25
H. 1.5
I. 2.5
J. 1.25
Answer: E
There are $24$ cells, each of emf $2 V$ and internal resistance $0.4 \Omega$. What should be the number $m$ of rows and number $n$ of cells in each row so that the maximum current is delivered in a load of resistance $0.6 \Omega$

Options:
A. $$m = 5, n = 4.8$$
B. $$m = 3, n = 7$$
C. $$m = 4, n = 6$$
D. $$m = 6, n = 3$$
E. $$m = 5, n = 5$$
F. $$m = 8, n = 3$$
G. $$m = 6, n = 4$$
H. $$m = 3, n = 8$$
I. $$m = 4, n = 5$$
Answer: G
What is the solubility of $MnS$ in pure water, assuming hydrolysis of $S^{2-}$ ions? $K_{sp}$ of $MnS = 2.5 \times 10^{-10}$, $K_{a1} = 1 \times 10^{-7}$ and $K_{a2} = 1 \times 10^{-14}$ for $H_{2}S$.

Options:
A. $$1.58 \times 10^{-5}M$$
B. $$1.58 \times 10^{-6}M$$
C. $$6.3 \times 10^{-4}M$$
D. $$1.58 \times 10^{-4}M$$
E. $$1.26 \times 10^{-4}M$$
F. $$5.3 \times 10^{-4}M$$
G. $$2.5 \times 10^{-4}M$$
H. $$5.3 \times 10^{-5}M$$
I. $$6.3 \times 10^{-5}M$$
J. $$6.3 \times 10^{-3}M$$
Answer: C
A ray incident at an angle of incidence $  60^{\circ}  $ entres a glass sphere of refractive index $  \mu=\sqrt{3}  $ , this ray is reflected and refracted at the farther surface of the sphere, the angle between reflected and refracted rays
atthis surface is

Options:
A. 80^{\circ}
B. 75^{\circ}
C. 55^{\circ}
D. $$ 45^{\circ} $$
E. $$ 90^{\circ} $$
F. $$ 30^{\circ} $$
G. 50^{\circ}
H. 40^{\circ}
I. $$ 60^{\circ} $$
J. 35^{\circ}
Answer: F
The equation: $log\dfrac{k_{c}}{k_{P}}-log\dfrac{1}{RT}=0$, is applicable to which of the following equilibrium reactions?

Options:
A. $$CaCO_{3}(s) \rightleftharpoons CaO(s) + 3CO_{2}(g)$$
B. $$H_{2}(g)+I_{2}(g)\rightleftharpoons 2HI(g)$$
C. $$CaCO_{3}(s) \rightleftharpoons CaO(s) + CO_{2}(g) + H_{2}O(g)$$
D. $$CaCO_{3}(s) \rightleftharpoons CaO(s) + H_{2}O(g)$$
E. $$CaCO_{3}(s) \rightleftharpoons CaO(s) + CO_{2}(g) + \dfrac{1}{2}H_{2}O(g)$$
F. $$CaCO_{3}(s) \rightleftharpoons CaO(s) + 2CO_{2}(g)$$
G. $$2NO_{2}(g)\rightleftharpoons N_{2}O_{4}(g)$$
H. $$CaCO_{3}(s)\rightleftharpoons CaO_{3}(s)+CO_{2}(g)$$
I. $$NH_{3}(g)\rightleftharpoons \dfrac{1}{2}N_{2}(g)+\dfrac{3}{2}H_{2}(g)$$
J. $$CaCO_{3}(s) \rightleftharpoons CaO(s)+CO_{2}(g)$$
Answer: H
An electron and a proton are injected into a uniform magnetic field perpendicular to it with the same momentum. If the two particles are injected into a uniform transverse electric field with same kinetic energy, then

Options:
A. both trajectories are straight lines
B. electron trajectory is equally curved
C. proton trajectory is less curved
D. electron trajectory is less curved
E. electron trajectory is more curved
F. both trajectories are equally curved
G. both trajectories are parallel
H. proton trajectory is equally curved
I. proton trajectory is more curved
Answer: C
The conc. of ${Fe}^{3+}$ ions in a sample of water is found to be $50\times {10}^{-5}M$. Calculate the $pH$ at which $99$% of ${Fe}^{3+}$ will be precipitated. ${ K }_{ { sp }_{ Fe{ \left( OH \right)  }_{ 3 } }\quad  }={10}^{-36}$.

Options:
A. 4.4
B. none of these
C. 5.9
D. 4.2
E. 4.1
F. 4.0
G. 4.3
H. 5.8
I. 3.7
J. 4.5
Answer: I
A radioactive sample has an initial activity of $28\ dpm$, half hour later the activity is $14\ dpm$. How many atoms of the radioactive nuclide were there originally?
$$(\ln\ 2=0.7)$$

Options:
A. $$200$$
B. 480
C. $$300$$
D. $$12$$
E. $$600$$
F. 150
G. 24
H. 18
I. 240
Answer: D
A voltmeter has range $0\rightarrow V$ with a series resistance R. With a series resistance 2R, the range is $0\rightarrow V$'. The correct relation between v and v' is

Options:
A. $$v^{1} = 4v$$
B. $$v^{1}> 2v$$
C. $$v^{1} = \frac{3}{2}v$$
D. $$v^{1}=2v$$
E. $$v^{1} >>2v$$
F. $$v^{1} = \frac{3}{4}v$$
G. $$v^{1}<2v$$
H. $$v^{1} = 3v$$
I. $$v^{1} = \frac{5}{2}v$$
J. $$v^{1} = \frac{4}{3}v$$
Answer: G
A body of mass $4.9 kg$ hangs from a spring and oscillates with a period $0.5 s$. On the removal of the body, the spring is shortened by (Take $g =10 ms^{-2}  ,\pi^2 =10)$

Options:
A. $$6.25cm$$
B. 0.64cm
C. 6.35cm
D. $$0.63cm$$
E. $$63cm$$
F. $$0.625cm$$
G. 0.635cm
H. $$0.615cm$$
I. 6.2cm
J. 0.62cm
Answer: A
A battery is connected between two pipes $A$ and $B$ on the circumference of a uniform conducting ring of radius $r$ and resistance $R$. One of the arcs $AB$ of the ring subtends an angle $\theta$ at the center. The value of the magnetic induction at the center due to the current in the ring is:

Options:
A. proportional to \(\theta - 180^\circ\)
B. inversely proportional to \(r^2\)
C. zero, only if $\theta={180}^{o}$
D. proportional to \(\theta - 45^\circ\)
E. inversely proportional to $r$
F. proportional to \(\theta - 90^\circ\)
G. proportional to $\theta$
H. proportional to $2\theta$
I. proportional to $2({180}^{o}-\theta)$
J. zero for all values of $\theta$
Answer: J
10 g of argon gas is compressed isothermally and reversibly at a temperature of $27^{o}C$ from 10 L to 5 L. Calculate q and $\Delta H$ for this process. (Atomic weight of $Ar=40 u$)

Options:
A. +103.99 cal, 0
B. -103.99 cal, 0
C. +103.99 cal, 3 J
D. -103.99 cal, +3 J
E. -103.99 cal, 3 J
F. -39.99 cal, 0
G. +39.99 cal, 3 J
H. +39.99 cal, 0
I. -103.99 cal, -3 J
J. -39.99 cal, 3J
Answer: A
A pendulum suspended from the ceiling of an elevator at rest has time period ${ T }_{ 1 }$. When the elevator moves up with an acceleration 'a' its time period of oscillation becomes ${ T }_{ 2 }$ when the elevator moves down with an acceleration 'a', its period of oscillation become ${ T }_{ 3 }$ then

Options:
A. $${ T }_{ 1 }=\dfrac { { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { {T_{ 2 }}^{ 2 }-4{T_{ 3 } }^{ 2 } } }$$
B. $${ T }_{ 1 }=\sqrt { T_{ 2 }{ ^{ 2 }T_{ 3 } }^{ 2 } } $$
C. $${ T }_{ 1 }=\dfrac { { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { 2{T_{ 2 }}^{ 2 }-{T_{ 3 } }^{ 2 } } }$$
D. $${ T }_{ 1 }=\dfrac { \sqrt { 2 } { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { {T_{ 2 }}^{ 2 }+{T_{ 3 } }^{ 2 } } } $$
E. $${ T }_{ 1 }=\dfrac { { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { {T_{ 2 }}^{ 2 }-2{T_{ 3 } }^{ 2 } } }$$
F. $${ T }_{ 1 }=\dfrac { { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { {T_{ 2 }}^{ 2 }+2{T_{ 3 } }^{ 2 } } }$$
G. $${ T }_{ 1 }=\dfrac { { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { {T_{ 2 }}^{ 2 }+{T_{ 3 } }^{ 2 } } } $$
H. $${ T }_{ 1 }=\sqrt { { T }_{ 2 }{ T }_{ 3 } } $$
I. $${ T }_{ 1 }=\dfrac { { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { {T_{ 2 }}^{ 2 }-{T_{ 3 } }^{ 2 } } }$$
J. $${ T }_{ 1 }=\dfrac { { T }_{ 2 }{ T }_{ 3 } }{ \sqrt { {T_{ 2 }}^{ 2 }-3{T_{ 3 } }^{ 2 } } }$$
Answer: G
At time $t=0$, a car moving along a straight line has a velocity of $16\ m { s }^{ -1 }$. It slow down with an acceleration of $0.5t\ m { s }^{ -2 }$, where $t$ is in seconds. Mark the correct statement(s).

Options:
A. The direction of velocity changes at $t=8\ s$.
B. The distance travelled by the particle in $10\ s$ is $97\ m$
C. The distance travelled in $4\ s$ is approximately $59\ m$
D. The distance travelled by the particle in $10\ s$ is $94\ m$
E. The distance travelled by the particle in $10\ s$ is $95\ m$
F. The distance travelled by the particle in $10\ s$ is $96\ m$
G. The distance travelled by the particle in $10\ s$ is $98\ m$
Answer: D
A rain drop of radius $1.5 mm$, experiences a drag force $F =$ $12\times 10^{-5}\times v \:N$, while falling through air from a height $2 km$, with a velocity $v$. The terminal velocity of the rain drop will be nearly (use $g=10 m/s^2)$ :

Options:
A. 120 m/s
B. 100 m/s
C. 160 m/s
D. $$7m/s$$
E. 250 m/s
F. $$150 m/s$$
G. $$3 m/s$$
H. $$200m/s$$
I. $$60 m/s$$
Answer: H
For the reaction $X_{2}O (l) \rightarrow 2XO_{2}(g); \triangle E = 2.1 kcal , \triangle S = 20 cal-K^{-1}$ at 300 K is

Options:
A. -2.1 kcal
B. 9.3 kcal
C. 2.7 kcal
D. -9.2 kcal
E. -9.3 kcal
F. -9.5 kcal
G. -9.4 kcal
H. -2.7 kcal
I. 2.1 kcal
J. -2.3 kcal
Answer: E
A stone is dropped in a well of depth $500$m and a sound is heard after $80/7$ seconds. Then speed of sound is: $[g=10 m/s^2]$

Options:
A. 360 m/s
B. 390 m/s
C. 340 m/s
D. $350$ m/s
E. 410 m/s
F. $330$ m/s
G. 385 m/s
H. $300$ m/s
I. 375 m/s
J. $400$ m/s
Answer: J
The correct stability order of the following resonating structures is:(I) $H_2C = \overset{+}{N} = \overset{-}{N}$                     (II) $H_2\overset{+}{C}-N=\overset{-}{N}$(III) $H_2\overset{-}{C} - \overset{+}{N} \equiv N$                  (IV) $H_2\overset{-}{C} - N = \overset{+}{N}$

Options:
A. (IV) > (III) > (II) > (I)
B. (III) > (IV) > (II) > (I)
C. (III) > (II) > (I) > (IV)
D. (I) > (IV) > (III) > (II)
E. (IV) > (III) > (I) > (II)
F. (III) > (II) > (IV) > (I)
G. (II) > (I) > (III) > (IV)
H. (I) > (III) > (II) > (IV)
I. (I) > (II) > (IV) > (III)
Answer: I
The equivalent weight of $CuSO_{4}$ (mol. wt. $=M$) in the following reaction is$5CuS + 8KMnO_4 +12 H_2SO_4 \rightarrow 5CuSO_4 +4 K_2SO_{4} + 8MnSO_4 +12H_2O $

Options:
A. \(\dfrac{M}{7} \)
B. $$\dfrac {M}{6}$$
C. $$\dfrac {M}{2}$$
D. \dfrac{M}{5}
E. \dfrac{M}{12}
F. $$\dfrac {M}{8}$$
G. \dfrac{M}{9}
H. \dfrac{M}{11}
I. \dfrac{M}{10}
J. $$\dfrac {M}{4}$$
Answer: F
A transparent slab of thickness d has a refractive index n(z) that increases with z. Here z is the vertical  distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices ${ n }_{ 1 }$ and ${ n }_{ 2 }$ $(>{ n }_{ 1 })$, as shown in the figure. A ray of light incident with angle ${ \theta  }_{ i }$ from medium 1 and emerges in medium 2 with refraction angle ${ \theta  }_{ f }$ with lateral displacement $l$.
Which of the following statement(s) is (are) true?

Options:
A. $l$ is independent of $n(z)$
B. $${ n }_{ 1 }\sin { { \theta }_{ 1 } } = \left( { n }_{ 2 }-{ n }_{ 1 } \right) \sin { { \theta }_{ f } } $$
C. $l$ is independent of ${ n }_{ 1 }$
D. $l$ is dependent on ${ n }_{ 2 }$
E. $l$ is independent of ${ n }_{ 2 }$
Answer: E
A spring is of mass M=3 kg and spring constant $ k= 4 \pi^2 N/m $ . the spring lies on a frictionless surface and one end of spring is fixed to a wall as shown in the figure. it can be assumed at any time that velocity of any point on the spring is directly proportional to its distance from the wall. what is the time period of oscillation of the spring ( in second ) if the free end of the spring is slightly pulled from its natural length and released.

Options:
A. 2
B. 1.5
C. 2.7
D. 2.8
E. 1
F. 2.9
G. 3
H. 4
I. 2.5
J. 3.5
Answer: E
A coil carrying a current of $i=10mA$ is placed in uniform magnetic field so that its axis coincides with the field direction. The coil consists of only one turn and is made of copper. The diameter of the wire is $0.1\,mm$ the radius of coil is $R=3\,cm$, at what value of external field $B$ will the coil rupture.
Breaking stress = $3 \times {10^8}N/{m^2}$

Options:
A. $$3.9 \times 10^5T$$
B. $$7.9 \times {10^4}T$$
C. 7.9 \times 10^5T
D. $$3.9 \times {10^4}T$$
E. $$7.9 \times 10^2T$$
F. $$7.9 \times {10^3}T$$
G. $$3.9 \times 10^4T$$
H. $$3.9 \times 10^2T$$
I. $$3.9 \times {10^3}T$$
J. $$7.9 \times 10^3T$$
Answer: B
A particle executes a periodic motion according to the
relation $x = 4\,\,{\cos ^2}\left( {50t} \right)\sin \left( {500t} \right)$.Therefore, the motion can be considered to be the
superposition of $n$  independent simple harmonic motions, where$n$ is

Options:
A. 3
B. 9
C. 1
D. 4
E. 5
F. 2
G. 8
H. 10
I. 6
J. 7
Answer: F
On subjecting mesityl oxide to the iodoform reaction, one of the products is the sodium salt of an organic acid. Which acids is obtained?

Options:
A. $$(CH_{3})_{2}CH-COOH$$
B. $$(CH_{3})_{2}C=CH-COOH$$
C. $$(CH_{3})_{2}C=CH-CH_{2}CH_{2}COOH$$
D. $$(CH_{3})_{2}C=CH-CH-CO-COOH$$
E. $$(CH_{3})_{2}C=CH-CH_{2}COOH$$
F. $$(CH_{3})_{2}C=CH-CH_{2}COOCH_{3}$$
G. (CH_{3})_{2}C=CH-CH-COCH_{3}
Answer: E
In the Geiger - Marsden scattering experiment, find the distance of closest approach to the nucleus of a $7.7Mev$ $\alpha$ - particle before it comes momentaily to rest and reverses its direction. ($'Z'$ for gold nucleus $= 79$)

Options:
A. 22fm
B. 25fm
C. $$30fm$$
D. 17.5fm
E. 12.5fm
F. 50fm
G. $$10fm$$
H. $$40fm$$
I. 15fm
J. $$20fm$$
Answer: G
$_{83}Bi^{211}$($t=130\sec$) decays to $_{81}TI^{207}$ by $\alpha $- emission.In an experiment starting with $5$ moles of $_{83}Bi^{211}$,how much pressure would be developed in a $350L$ closed vessel at $25^oC$ after $760$ sec ?[Antilog $(1.759)=57.41$]

Options:
A. $$0.22atm$$
B. 0.74atm
C. 0.40atm
D. $$0.54atm$$
E. 0.58atm
F. 0.38atm
G. $$0.34atm$$
H. 0.48atm
I. $$0.68atm$$
J. 0.62atm
Answer: I
At what concentration of $[OH^-]$ does the following half reaction has a potential of $0V$ when other species are at $1M$?
$$NO_3^-+H_2O+2e^- \rightarrow NO_2^-+2OH^-;E^0cell=-0.06V$$

Options:
A. 1.2M
B. 0.2M
C. $$0.1M$$
D. 1.5M
E. $$1.0M$$
F. 3.0M
G. $$2.0M$$
H. 0.5M
I. $$0.01M$$
J. 2.5M
Answer: G
The weight of a man standing on a weighing machine in a lift is noted when
(i) the lift is moving with a uniform velocity and
(ii) the lift is moving down with an acceleration $a$. The ratio of the two weights is $\frac {4}{3}$. What is the value of '$a$'?

Options:
A. $$a= \frac {g}{4}$$
B. $$a = \frac{g}{8}$$
C. $$a= \frac {g}{2}$$
D. $$a= \frac{g}{6}$$
E. $$a = \frac{2g}{3}$$
F. $$a= \frac {g}{5}$$
G. \( a = \frac{3g}{5} \)
H. $$a= \frac {g}{3}$$
I. $$a = \frac{g}{7}$$
Answer: C
In a thermodynamic process helium gas obeys the law $\displaystyle \frac { T }{ { P }^{ { 2 }/{ 5 } } } $ = constant. The heat given to a gas when temperature of m moles of $\displaystyle He$ is raised from $\displaystyle T$ to $2T$ is:

Options:
A. 4RT
B. 8RT
C. 6RT
D. 2RT
E. 10RT
F. 12RT
G. 20RT
H. 16RT
I. 5RT
J. Zero
Answer: J
Two vertical parallel glass plates are partially submerged in water. The distance between the plates is $d$ and the length $l$. Assume that the water between the plates does not reach the upper edges of the plates and that the wetting is complete. If water is replaced  by another liquid of surface tension $2\sigma$, then the force $F$ of attraction between two plates would become($\rho=$density of water and $\sigma=$ surface tension water)

Options:
A. $$16F$$
B. $$4F$$
C. $$6F$$
D. 3F
E. 12F
F. $$8F$$
G. $$5F$$
H. $$2F$$
I. 10F
Answer: B
For the nuclear fusion reaction $ _{ 1 }^{ 2 }H+_{ 1 }^{ 3 }H\rightarrow _{ 2 }^{ 4 }He+_{ 0 }^{ 1 }{ n } $ temperature to which gases must be heated is $ 3.7 \times 10^9 K $ Potential energy between two nuclei is closest to (Boltzmann's constant k = $ 1.38 \times 10^{-23} J/K ) $

Options:
A. -10^{-8} J
B. -10^{-15} J
C. -10^{-11} J
D. -10^{-16} J
E. $$ -10^{-14} J $$
F. -10^{-12} J
G. $$ -10^{-13} J $$
H. -10^{-9} J
I. $$ -10^{-10} J $$
Answer: I
Number of electrons lost during electrolysis of $0.14g$ of $N^{-3}$ is-($N_0$=Avagadro number)

Options:
A. 0.0075\,N_0
B. 0.02\,N_0
C. $$0.015\,N_0$$
D. 0.005\,N_0
E. $$0.03\,N_0$$
F. 0.01\,N_0
G. $$\dfrac{0.01}{2\,N_0}$$
H. $$0.03$$
I. 0.025\,N_0
Answer: E
A particle of mass m is moving in a circle of radius r such that its centripetal acceleration a is given as a = K$t^2$, where K is constant. The work done on the particle in first t seconds is

Options:
A. $$\frac{mKrt}{2}$$
B. $$\dfrac {mkr {t}^{2} } {2}$$
C. $$\dfrac {3mk {t}^{2} } {2r}$$
D. $$\dfrac {mk {t}^{2} } {2r}$$
E. mKr$t^2$
F. $$\dfrac {3mKr {t}^{2} } {4}$$
G. 4 mKr$t$
H. 3 mKr$t$
I. $$\dfrac {2mk {t}^{2} } {r}$$
J. 3 mKr$t^2$
Answer: A
A rod of length $1m$ stands vertically on the ground, with one end hinged to the ground. When it falls on the ground such that it rotates without slipping the velocity of the upper tip on striking the ground is 
($g=10m/{s}^{2}$)

Options:
A. $$15\sqrt{2}m/s$$
B. \cfrac{15\sqrt{2}}{6}m/s
C. \cfrac{15\sqrt{2}}{2}m/s
D. \cfrac{15\sqrt{2}}{3}m/s
E. \cfrac{15\sqrt{2}}{7}m/s
F. \cfrac{15\sqrt{2}}{5}m/s
G. $$\cfrac{15}{\sqrt{2}}m/s$$
H. Can be not predicted.
I. $$15m/s$$
J. \cfrac{15\sqrt{2}}{4}m/s
Answer: H
Three point like equal masses $m_1, m_2$ and $m_3$ are connected to the ends of a mass-less rod of length $L$ which lies at rest on a smooth horizontal plane. At $t = 0$, an explosion occurs between $m_2$ and $m_3$, and as a result, mass $m_3$ is detached from the rod, and moves with a known velocity $v$ at an angle of $30^o$ with the y-axis. Assume that the masses $m_2$ and $m_3$ are unchanged during the explosion.
What is the velocity of the centre of mass of the system consisting of three masses after the expulsion?

Options:
A. $$\dfrac{v}{4} (\widehat i + \sqrt{3} \widehat k)$$
B. $$\displaystyle \dfrac{v}{4} (- \widehat i + \sqrt{3} \widehat j)$$
C. $$\(\dfrac{v}{4} (-\widehat i - \sqrt{3} \widehat j) \)$$
D. $$\dfrac{v}{4} (\widehat i + 3 \widehat j)$$
E. $$-v$$
F. $$\dfrac{v}{4} (\widehat i + \sqrt{3} \widehat j)$$
G. $$\dfrac{v}{4} (\widehat i - \sqrt{3} \widehat j)$$
H. none of the above
I. $$\displaystyle \dfrac{v}{4} (\widehat i - 3 \widehat j)$$
J. \(\dfrac{v}{4} (3\widehat i - \widehat j)\)
Answer: H
Given ${ k }_{ 1 }=1500N/m,{ k }_{ 2 }=500N/m,{ m }_{ 1 }=2kg,{ m }_{ 2 }=1kg$. Find:(a) Potential energy stored in the springs in equilibrium, and(b) work done in slowly pulling down ${m}_{2}$ by $8cm$

Options:
A. (a) $0.3J$  (b) $1.5J$
B. (a) $0.3J$  (b) $0.5J$
C. (a) 0.5J (b) 1.2J
D. (a) 0.5J (b) 0.8J
E. (a) $0.7J$  (b) $0.9J$
F. (a) $0.2J$  (b) $4J$
G. (a) $0.8J$  (b) $1J$
H. (a) $0.6J$  (b) $2J$
I. (a) $0.6J$  (b) $1.2J$
J. (a) $0.4J$  (b) $1J$
Answer: J
Two miscible liquids A and B having vapour pressure in pure state ${P}_{A}^{o}$ and ${P}_{B}^{o}$ are mixed in mole fraction ${\chi}_{A}$ and ${\chi}_{B}$ to get a mixture having total vapour pressure of mixture ${P}_{M}$.Which of the following reactions are correct?

Options:
A. All of these
B. $$\displaystyle\frac{{\chi}_{A(l)}}{{\chi}_{A(V)}^{'}}=\frac{{P}_{M}}{{P}_{B}^{o}+{P}_{A}^{o}}$$
C. $$\displaystyle \frac{{\chi}_{A(l)}}{{\chi}_{A(V)}^{'}}= \frac{{P}_{M}}{{P}_{B}^{o} + {P}_{A}^{o} - 1}$$
D. $$\displaystyle\frac{{\chi}_{A(l)}}{{\chi}_{A(V)}^{'}}=\frac{{P}_{M}}{{P}_{B}^{o}}$$
E. $$\displaystyle \frac{{\chi}_{A(l)}}{{\chi}_{A(V)}^{'}}= \frac{{P}_{M}}{{P}_{B}^{o} - {P}_{A}^{o} + 1}$$
F. $$\displaystyle \frac{{\chi}_{A(l)}}{{\chi}_{A(V)}^{'}}= \frac{{P}_{M}}{{P}_{B}^{o} + {P}_{A}^{o} + 1}$$
G. $$\displaystyle\frac{{\chi}_{A(l)}}{{\chi}_{A(V)}^{'}}=\frac{{P}_{M}}{{P}_{A}^{o}}$$
H. $${\chi}_{A}=\displaystyle\frac{{P}_{M}-{P}_{B}^{o}}{{P}_{A}^{o}-{P}_{B}^{o}}$$
I. $$\displaystyle \frac{{\chi}_{A(l)}}{{\chi}_{A(V)}^{'}}= \frac{{P}_{M}}{{P}_{B}^{o} - {P}_{A}^{o}}$$
Answer: G
What is the relationship between the standard cell potentials $E^o$ , for the following two galvanic cell reactions?
(I)$2 Ag^+(aq) +Sn^{2+}(aq) \rightarrow Sn^{4+}(aq)+2Ag(s)$
(II) $Ag(s)+\frac { 1 }{ 2 } Sn^{ 4+} (aq) \rightarrow \frac { 1 }{ 2 } Sn^ { 2+} (aq)+Ag^ +(aq)$

Options:
A. E^O(I) = \frac{3}{2}E^o (II)
B. E^O(I) = \frac{1}{3}E^o (II)
C. $$E^O(I) = -2E^o (II)$$
D. $$E^O(I) = \frac{1}{2}E^o (II)$$
E. E^O(I) = \frac{1}{4}E^o (II)
F. E^O(I) = -\frac{1}{2}E^o (II)
G. $$E^O(I) = 2E^o (II)$$
H. $$E^O(I) = -E^o (II)$$
I. E^O(I) = 3E^o (II)
J. $$E^O(I) = E^o (II)$$
Answer: G
A disc of radius R rolls on a horizontal surface with linear velocity $ \overrightarrow {v} = v \hat {i} $ and angular velocity $ \overrightarrow {\omega} = - \omega \hat k $ there is a particle P on the circumference of the disc which has velocity in vertical direction. the height of that particle from the ground will be

Options:
A. $$R - \dfrac {v}{ 3 \omega}$$
B. $$ R + \dfrac {R}{ 2} $$
C. $$ R - \dfrac {R}{ 2} $$
D. $$R + \dfrac {v}{ 2 \omega}$$
E. $$R + \dfrac {v}{ 3 \omega}$$
F. $$R + \dfrac {v}{ 4 \omega}$$
G. $$R + \dfrac {v}{ \omega}$$
H. $$R - \dfrac {v}{ 2 \omega}$$
I. $$ R - \dfrac {v}{ \omega} $$
Answer: I
Which expression is correct for the work done in adiabatic reversible expansion of an ideal gas

Options:
A. W=nC_v \quad \ln \frac{T_1}{T_2}
B. W=nC_v \quad \ln \frac{V_2}{V_1}
C. $$W=-\int _{ 1 }^{ 2 }{ PdV } $$
D. W=nC_v \quad \ln \frac{T_2}{T_1}
E. $$W=P\triangle V$$
F. W=nC_v \quad \ln \frac{P_2}{P_1}
G. W=nC_v \quad \ln \frac{P_1}{P_2}
H. $$W=n_{ e }\quad In\frac { { T }_{ 2 } }{ { T }_{ 1 } } $$
I. $$W=nRT\quad In\frac { { V }_{ 2 } }{ { V }_{ 1 } } $$
J. W=nC_v \quad \ln \frac{V_1}{V_2}
Answer: E
The equilibrium between gaseous isomers A, B and C can be represented as:$ A(g)  \leftrightharpoons B(g) ; K_1 = ? $$ B(g) \leftrightharpoons C(g) ; K_2 = 0.4 $$ C(g) \leftrightharpoons A(g) ; K_3 = 0.6 $ If one mole of A is taken in closed vessel of volume 1 L, then

Options:
A. The value of \( K_1 \) is 1/0.36
B. The value of \( K_1 \) is 1/0.60
C. The value of \( K_1 \) is 1/0.48
D. Isomer C is thermodynamically most stable
E. The value of \( K_1 \) is 1/0.52
F. The value of \( K_1 \) is 1/0.30
G. Isomer A is thermodynamically least stable
H. Concentration of C is 1.2 M at equilibrium
Answer: G
A car is travelling in the x-direction and maintains constant horizontal speed $v$. The car goes over a bump whose shape is described by $\mathrm{y}_{0}=$ A $[1-\displaystyle \cos(\dfrac{\pi \mathrm{x}}{\ell})]$ for $ 0\leq \mathrm{x}\leq 2\ell$.In given Figure, the car is represented as mass $m$ attached to a mass-less spring (of relaxed length $\ell _{o}$ and spring constant $k$). If the motion of center of mass of car while passing over the bump is described by y = B cos $\left ( \dfrac{\pi vt}{\ell} \right )$The expression for B will be (ignore friction and spring is vertical always)

Options:
A. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}}$$
B. $$(k\ell ^{2}A)$$
C. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}-k\ell ^{2}A^2}$$
D. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}+k\ell ^{2}}$$
E. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}+k\ell ^{2}A}$$
F. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}+k\ell}$$
G. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}-k\ell ^{2}}$$
H. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}-k\ell}$$
I. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}-k\ell ^{2}A^3}$$
J. $$\dfrac{k\ell ^{2}A}{m\pi ^{2}v^{2}-k\ell ^{2}A}$$
Answer: G
Given the following cell at $25^{\circ}C$. What will be the potential of the cell ? Given $pK_{a}$ of $CH_{3}COOH\, =\, 4.74$ :$Pt \begin{vmatrix} H_2 \(1 atm) \end{vmatrix}\left.\begin{matrix} CH_3COOH \(10^{-3} M) \end{matrix}\right|\begin{vmatrix}NaOH \ (10^{-3}M)\end{vmatrix}\left.\begin{matrix} H_2 \ (1 atm)\end{matrix}\right|Pt$

Options:
A. -0.47 V
B. $$- 0.19 V$$
C. $$- 0.42 V$$
D. $$0.42 V$$
E. -0.43 V
F. -0.41 V
G. $$0.19 V$$
H. -0.44 V
I. -0.45 V
J. -0.39 V
Answer: C
At the moment $t=0$, a connector starts sliding from the apex of a parabola with a constant acceleration $\omega$. If the equation of the parabola is $y=ax^2$ and it is located in a uniform, perpendicular magnetic field B, the emf induced in the loop as a function of y is?

Options:
A. $$e=By \sqrt{\dfrac{9\omega}{2a}}$$
B. $$e=By \sqrt{\dfrac{5\omega}{2a}}$$
C. $$e=By \sqrt{\dfrac{7\omega}{2a}}$$
D. $$e=By \sqrt{\dfrac{2\omega}{a}}$$
E. None of these
F. $$e=By \sqrt{\dfrac{6\omega}{a}}$$
G. $$e=By \sqrt{\dfrac{\omega}{2a}}$$
H. $$e=By \sqrt{\dfrac{4\omega}{a}}$$
I. $$e=By \sqrt{\dfrac{3\omega}{2a}}$$
J. $$e=By \sqrt{\dfrac{8\omega}{a}}$$
Answer: J
75.2 g of ${ C }_{ 6 }{ H }_{ 5 }OH$ is dissolved in 960 g of solvent of ${ K }_{ f }=15\ K\ kg\ mol^{ -1 }$. If the depression in freezing point is 7 K, then find the percentage of phenol that dimerizes.

Options:
A. 60
B. 50
C. 80
D. 30
E. 45
F. 55
G. 65
H. 25
I. 35
J. 40
Answer: J
A charged particle having some mass is resting in equilibrium at a height H above the centre of a unifomly charged non-conducting horizontal ring of radius R. The force of gravity acts downwards. The equilibrium of the particle will be stable:

Options:
A. only if \( \mathrm{H} < \frac{\mathrm{R}}{3} \)
B. only if \( \mathrm{H} < \frac{\mathrm{R}}{\sqrt{4}} \)
C. only if $ \mathrm{H} < \frac{\mathrm{R}}{\sqrt{6}} $
D. only if $ \mathrm{H} < \frac{\mathrm{R}}{\sqrt{3}} $
E. only if $ \mathrm{H} < \frac{\mathrm{R}}{2} $
F. for all values of H
G. only if $ \mathrm{H}>\frac{\mathrm{R}}{\sqrt{2}} $
H. only if $ \mathrm{H}=\frac{\mathrm{R}}{\sqrt{2}} $
I. only if \( \mathrm{H} < \frac{\mathrm{R}}{\sqrt{5}} \)
J. only if $ \mathrm{H}<\frac{\mathrm{R}}{\sqrt{2}} $
Answer: G
Arrange the following resonating structure according to their contribution towards resonance hybrid?
$$(a)\ { CH }_{ 2 }=\overset { \oplus  }{ N } =\overset { \circleddash  }{ N }$$
$$(b)\ \overset { \circleddash  }{ C } { H }_{ 2 }-N=\overset { \oplus  }{ N }$$
$$(c)\ \overset { \oplus  }{ C } { H }_{ 2 }-N=\overset { \circleddash  }{ N }$$
$$(d)\ \overset { \circleddash  }{ C } { H }_{ 2 }-\overset { \oplus  }{ N } \equiv \ddot { N }$$

Options:
A. b>d>a>c
B. $$a>d>c>b$$
C. $$a>c>b>d$$
D. c>d>b>a
E. b>d>c>a
F. $$b>a>c>d$$
G. b>a>d>c
H. c>b>d>a
I. b>c>d>a
J. $$d>a>b>c$$
Answer: B
Consider the reaction of extraction of gold from its ore:
$$ Au(s)+2CN^{-}(aq)+\dfrac{1}{4}O_{2}(g)+\dfrac{1}{2}$$
$$ H_{2} O(1)\rightarrow Au(CN)_{2}^{-}(aq)+OH^{-}(aq)$$
Use the following data to calculate $ \triangle G^{0}$ for the above reaction.
$$ K_{f}[Au(CN)_{2}^{-}] = X $$
$$ O_{2}+2H_{2}O+4e^{-}\rightarrow 4oH^{-}; E^{0} = + 0.41 V$$
$$ Au^{3+}+ 3e^{-} \rightarrow Au; E^{0} = +1.50 V $$
$$ Au^{3+}+ 2e^{-} \rightarrow  Au^{+}; E^{0} = +1.40 V$$

Options:
A. -RT ln X + 1.31 F
B. $$ +RT ln X + 2.11 F $$
C. -RT ln X + 2.11 F
D. $$ -RT ln X-1.29 F $$
E. -RT ln X - 1.41 F
F. -RT ln X + 1.39 F
G. -RT ln X - 1.39 F
H. $$ -RT ln X+ 1.29 F $$
I. -RT ln X + 1.41 F
J. $$ -RT ln X - 2.11 F$$
Answer: H
In the options given below, let $E$ denote the rest mass energy of a nucleus and $'n\ '$ a neutron. The correct option is

Options:
A. $$\mathrm{E}(_{92}^{236}\mathrm{U})<\mathrm{E}(_{53}^{137}\mathrm{I})+\mathrm{E}(_{39}^{97}\mathrm{Y})+2\mathrm{E}(\mathrm{n})$$
B. $$\mathrm{E}(_{92}^{236}\mathrm{U})>\mathrm{E}(_{55}^{138}\mathrm{Cs})+\mathrm{E}(_{37}^{97}\mathrm{Rb})+2\mathrm{E}(\mathrm{n})$$
C. $$\mathrm{E}(_{92}^{236}\mathrm{U})>\mathrm{E}(_{53}^{137}\mathrm{I})+\mathrm{E}(_{38}^{96}\mathrm{Sr})+2\mathrm{E}(\mathrm{n})$$
D. $$\mathrm{E}(_{92}^{236}\mathrm{U})>\mathrm{E}(_{53}^{137}\mathrm{I})+\mathrm{E}(_{39}^{97}\mathrm{Y})+2\mathrm{E}(\mathrm{n})$$
E. $$\mathrm{E}(_{92}^{236}\mathrm{U})=\mathrm{E}(_{56}^{140} \mathrm{B}\mathrm{a} )+\mathrm{E}(_{36}^{94} \mathrm{K}\mathrm{r} )+2\mathrm{E}(\mathrm{n})$$
F. $$\mathrm{E}(_{92}^{236}\mathrm{U})>\mathrm{E}(_{55}^{138}\mathrm{Cs})+\mathrm{E}(_{36}^{94}\mathrm{Kr})+2\mathrm{E}(\mathrm{n})$$
G. $$\mathrm{E}(_{92}^{236}\mathrm{U})>\mathrm{E}(_{54}^{136}\mathrm{Xe})+\mathrm{E}(_{38}^{99}\mathrm{Sr})+2\mathrm{E}(\mathrm{n})$$
H. $$\mathrm{E}(_{92}^{236}\mathrm{U})>\mathrm{E}(_{53}^{138}\mathrm{I})+\mathrm{E}(_{39}^{96}\mathrm{Y})+2\mathrm{E}(\mathrm{n})$$
I. $$\mathrm{E}(_{92}^{236}\mathrm{U})>\mathrm{E}(_{56}^{140} \mathrm{B}\mathrm{a} )+\mathrm{E}(_{36}^{94} \mathrm{K}\mathrm{r} )+2\mathrm{E}(\mathrm{n})$$
J. $$\mathrm{E}(_{92}^{236}\mathrm{U})<\mathrm{E}(_{56}^{140} \mathrm{B}\mathrm{a} )+\mathrm{E}(_{36}^{94} \mathrm{K}\mathrm{r} )+2\mathrm{E}(\mathrm{n})$$
Answer: D
Two identical blocks of mass $m$ each are connected by a spring as shown in the figure. At any instant of time $t=0$ one block is given a velocity ${v}_{1}$ and other is given a velocity ${v}_{2}({v}_{1}> {v}_{2})$ in the same direction simultaneously as shown in the figure. The maximum energy stored in the spring is

Options:
A. $$\cfrac{1}{5}m{({v}_{1}-{v}_{2})}^{2}$$
B. $$\cfrac{1}{2}m{({v}_{1}+{v}_{2})}^{2}$$
C. $$\cfrac{3}{4}m{({v}_{1}-{v}_{2})}^{2}$$
D. $$\cfrac{1}{4}m{({v}_{1}+{v}_{2})}^{2}$$
E. $$m{({v}_{1}+{v}_{2})}^{2}$$
F. $$\cfrac{1}{4}m{({v}_{1}-{v}_{2})}^{2}$$
G. $$m{({v}_{1}-{v}_{2})}^{2}$$
H. $$\cfrac{1}{2}m{({v}_{1}-{v}_{2})}^{2}$$
I. $$\cfrac{1}{3}m{({v}_{1}-{v}_{2})}^{2}$$
Answer: G
The standard reduction potential at $25^{\circ} C$ of the reaction $2H_2O + 2e^{-} \rightleftharpoons H_2 + 2\overset{\ominus}{O}H$ is -0.8277 V. The equilibrium constant for the reaction $2H_2O \rightleftharpoons H_3O^{\oplus} +\, \overset{\ominus}{O}$ at $25^{\circ}C$ is:

Options:
A. $$K_w = 9.35\, \times\, 10^{-15}$$
B. $$K_w = 9.40\, \times\, 10^{-15}$$
C. K_w = 9.45\, \times\, 10^{-15}
D. $$K_w = 9.24\, \times\, 10^{-15}$$
E. $$K_w = 9.14\, \times\, 10^{-15}$$
F. K_w = 9.55\, \times\, 10^{-15}
G. $$None \:\:of \:\:these $$
H. $$K_w = 9.83\, \times\, 10^{-15}$$
I. K_w = 9.64\, \times\, 10^{-15}
J. K_w = 9.38\, \times\, 10^{-15}
Answer: A
A man is raising himself and the crate on which he stands with an acceleration of $5m{ s }^{ -2 }$ by a massless rope-and-pulley arrangement. Mass of the man is $100 kg$ and that of a crate is 50kg. IF $g=10m{ s }^{ -2 }$, then the tension in the rope is:

Options:
A. $$1250N$$
B. $$1350N$$
C. $$1125N$$
D. $$2250N$$
E. $$2625N$$
F. $$1625N$$
G. $$750N$$
H. $$1500N$$
I. $$375N$$
J. $$1875N$$
Answer: C
The decomposition of $\displaystyle H_{2}O_{2}$ has a strong thermodynamic driving force $\displaystyle 2H_{2}O_{2}\rightarrow 2H_{2}O+O_{2}(g)\, \, \Delta H=-99kj/mole,\Delta s=+69JK^{-1}\, mole^{-1}$Addition of solution of KI causes $\displaystyle H_{2}O_{2}$ to decompose very rapidly as well Step -1 (slow) $\displaystyle H_{2}O_{2}+l\rightarrow lO+H_{2}O$Step -2 (fast)  $\displaystyle {H_{2}O_{2}+lO\rightarrow H_{2}O+r+O_{2}(g)}$Overall reaction: $\displaystyle {{2H_{2}O_{2}\rightarrow {H_2}O+O_{2}(g)}}$The above experiment causes a big fluff of foam to spiral out of the beaker ( Elephant toothpaste reaction)Identify the wrong statement.

Options:
A. LO (aq) is the reaction substrate
B. LO (aq) is the reaction inducer
C. the elephant tooth paste is hot to toch
D. LO (aq) is the reaction stabilizer
E. $\displaystyle H_{2}O_{2}$ is thermodynamically and kinetically unstable
F. LO (aq) is the reaction catalyst
G. a glowing splint if plunged into the elephant toothpaste will catch fire
H. LO (aq) is the reaction promoter
I. LO (aq) is the reaction intemediate
J. LO (aq) is the reaction product
Answer: G
A reversible adiabatic path on a P- V diagram foran ideal gas passes through state A where P = 0.7$\times $ ${ 10 }^{ 2  }$ N/${ m }^{ -2 }$ and v=0.0049 $ { m }^{ 3  }$, The ratio of specific heat of the gas is 1.4 , The slop of patch at A is:

Options:
A. $$-2.0\times{ 10 }^{ 7\quad }{ Nm }^{ -3 }$$
B. $$-1.0\times{ 10 }^{ 3\quad }{ Nm }^{ -3 }$$
C. $$-1.4 \times{ 10 }^{ 7 }{ Nm }^{ -3}$$
D. $$1.0 \times{ 10 }^{ 3\quad }{ Nm }^{ -3}$$
E. $$-1.4 \times{ 10 }^{ 5 }{ Nm }^{ -3 }$$
F. $$-1.4 \times{ 10 }^{ 6 }{ Nm }^{ -3}$$
G. $$2.0 \times{ 10 }^{ 3\quad }{ Nm }^{ -3 }$$
H. $$-1.4 \times 10^{5} \, {Nm}^{-3}$$
I. $$-2.0 \times{ 10 }^{ 6 }{ Nm }^{ -3 }$$
Answer: A
A cylinder $P$ of radius $r_P$ is being rotated at a constant angular velocity $w_P\hat{j}$ with the help of a motor about its axis that is fixed. Another cylinder $Q$ of radius $r_Q$ free to rotate about its axis that is also fixed is touched with and pressed on $P$ making an angle $\theta$ between their axes. Soon after the cylinders are pressed against each other, a steady state is reached and the cylinder $Q$ acquires a constant angular velocity. What can you conclude when the steady state is reached?

Options:
A. Angular velocity of cylinder $Q$ is $\bar{\omega_Q}=\dfrac {\omega_Pr_P}{T_Q \cos \theta } (\sin \theta \hat{i} +\cos\theta \hat{j})$
B. $(\sin \theta \hat{i} - \cos \theta \hat{j})$
C. Unit vector of frictional force on cylinder $Q$ is $(\sin \theta \hat{i} - \cos \theta \hat{j})$
D. Frictional force on each cylinder becomes vanishingly small.
E. Unit vector of frictional force on cylinder $Q$ is $(\cos \theta \hat{i} - \sin \theta \hat{j})$
F. Angular velocity of cylinder $Q$ is $\bar{\omega_Q}=\dfrac {\omega_Pr_P}{T_Q } (\sin \theta \hat{i} +\cos\theta \hat{j})$
G. Unit vector of frictional force on cylinder $Q$ is $(\sin \theta \hat{i} +\cos\theta \hat{j})$.
H. Frictional force on cylinder $Q$ is $\dfrac{\omega_P r_P}{T_Q} (\sin \theta \hat{i} + \cos \theta \hat{j})$
Answer: G
Two particles, each of mass $'m'$ are attached to the two ends of a light string of length $L$, which passes through a hole at the centre of a smooth table. One particle describes a circular path on the table with angular velocity $\omega _{1}$, and the other describes a conical pendulum with angular velocity $\omega _{2}$ below the table. If $l_{1}$ and $l_{2}$ are the lengths of portions of the string above and below the table, then:

Options:
A. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{l \tan \theta}{g}$$
B. $$ \dfrac{l_1}{l_2} = \dfrac{\omega_2^2}{\omega_1^2}$$
C. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{l \sin \theta}{g}$$
D. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{l \cos \theta}{g}$$
E. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{ml \cos \theta}{g}$$
F. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{l \sin \theta}{mg}$$
G. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{m l \sin \theta}{g}$$
H. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{l \sin 2\theta}{g}$$
I. $$\dfrac{1}{\omega_1^2} + \dfrac{1}{\omega_2^2} = \dfrac{ml}{g}$$
J. $$ \dfrac{l_1}{l_2} = \dfrac{\omega_2}{\omega_1}$$
Answer: D
10 g of ice cubes at $0^o C$ are released in a tumbler (water equivalent 55 g) at $40^o C$. If negligible heat is taken from surroundings, temperature of water in the tumbler becomes nearly ($L = 80 cal g^{-1}$)

Options:
A. $$31^oC$$
B. $$23^oC$$
C. $$21.5^oC$$
D. $$21^oC$$
E. $$17^oC$$
F. $$22^oC$$
G. $$15^oC$$
H. $$18^oC$$
I. $$20^oC$$
J. $$24^oC$$
Answer: C
A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure $P_0$, volume $V_0$ and temperature $T_0$. If the gas mixture is adiabatically compressed to a volume $V_0/4$, then the correct statement(s) is/are,
(Given $2^{1.2} = 2.3 ; \, 2^{3.2} = 9.2 ; \, R$ is gas constant)

Options:
A. The work |W| done during the process is $14 RT_0$
B. The work |W| done during the process is $17 RT_0$
C. The work |W| done during the process is $13 RT_0$
D. The average kinetic energy of the gas mixture after compression is in between $18 RT_0$ and $19 RT_0$
E. The work |W| done during the process is $16 RT_0$
F. The work |W| done during the process is $15 RT_0$
G. The work |W| done during the process is $11 RT_0$
H. The work |W| done during the process is $12 RT_0$
Answer: C
A solution of 1.25 g of a certain non-volatile substance in 20 g of water freeze at 271.94 K. Then the molecular mass of solute is $({ K }_{ f }=1.86K\ kg{ mol }^{- 1 })$

Options:
A. 112.66
B. 105.66
C. 150.66
D. 111.66
E. 180
F. 108.66
G. 119.66
H. 110.66
I. 100.66
Answer: I
Which one of the following is incorrect for ideal solution?

Options:
A. $$\Delta P={ P }_{ obs }-{ P }_{ calculated\ by\ Raoult's\ law }=0$$
B. \Delta { H }_{ mix } \neq 0
C. \Delta { G }_{ mix } \neq 0
D. \Delta { H }_{ mix } = -10 \text{ J}
E. $$\Delta { U }_{ mix }=0$$
F. \Delta { H }_{ mix } \neq 10 \text{ J}
G. \Delta { H }_{ mix } = 10 \text{ J}
H. $$\Delta { H }_{ mix }=0$$
I. \Delta { H }_{ mix } = 5 \text{ J}
J. $$\Delta { G }_{ mix }=0$$
Answer: H
Two litre solution of buffer mixture containing $1.0$ M $NaH_2PO_4$ and $1.0$ M $Na_2HPO_4$ is placed in two compartments (1 L in each) of an electrolytic cell. The platinum electrodes are inserted in each compartment and 1.25 A current is passed for 965 min. Assuming electrolysis of only water at each compartment, what will be the pH in the compartment after the passage of the above charge? (Given pKa for $H_2PO_4^-$  = $2.15$, log 7 = $0.85$)

Options:
A. Cathode: 2.85
B. Cathode: 2.90
C. Cathode: 3.10
D. Anode : 3:00
E. Cathode: 2.95
F. Cathode: 3.00
G. Anode : 4.30
H. Cathode: 2.80
I. Cathode: 3.15
J. Cathode: 1.30
Answer: F
A varying horizontal force $F= at$ acts on a block of mass m kept on a smooth horizontal surface. An identical block is kept on the first block. The coefficient of friction between the blocks is  $\displaystyle \mu.$ The time after which the relative sliding between the blocks prevails is

Options:
A. $$\displaystyle 2\mu mga$$
B. $$\displaystyle \frac{\mu mg}{a}$$
C. $$\frac{2\mu mg}{5a}$$
D. $$\displaystyle \frac{4\mu mg}{3a}$$
E. $$\displaystyle \frac{2 mg}{a}$$
F. $$\frac{2\mu mg}{4a}$$
G. $$\frac{2\mu mg}{3a}$$
H. $$\frac{\mu mg}{2a}$$
I. $$\displaystyle \frac{2\mu mg}{a}$$
J. $$\displaystyle \frac{3\mu mg}{a}$$
Answer: I
In what pH range will a $1\times10^{-4}$ M solution of an indicator will $K_{b}$ (indicator)$=1\times10^{-11}$ change colour?

Options:
A. $$9.0 \pm 1$$
B. $$11.0\pm 1$$
C. $$8.5 \pm 1$$
D. $$6.5\pm 1$$
E. $$5.5\pm 1$$
F. $$7.5 \pm 1$$
G. $$6.0\pm 1$$
H. $$3.0\pm 1$$
I. $$8.0\pm 1$$
Answer: H
A block of mass $4\ kg$ is placed on a horizontal surface is connected to another mass of $3\ kg$ by a mass less string going over a pulley as shown. At the instant $\theta = 60 ^ { \circ }$ and the blocks are at momentary rest, the tension in the string is $\left(g = 10 m s ^ { 2 } \right)$

Options:
A. 51.06 N
B. $43.06$ N
C. $43.45$ N
D. $35.72$ N
E. $140$ N
F. 54.06 N
G. 52.06 N
H. 52.06
I. $41.05$ N
Answer: C
A solid sphere has linear velocity $v_0=4m/s$ and angular velocity ${ \omega  }_{ 0 }=9rad/s$ as shown. Ground on which it is moving is smooth. It collides elastically with rough wall of coefficient of friction $\mu .$ Radius of sphere $1 m$ and mass $2 kg.$ Then what is the coefficient of friction if the sphere after colliding rolls without slipping.

Options:
A. $$\dfrac{7}{8}$$
B. $$\dfrac{1}{4}$$
C. $$\dfrac{3}{8}$$
D. $$\dfrac{2}{3}$$
E. $$\dfrac{5}{7}$$
F. $$\dfrac{3}{7}$$
G. $$\dfrac{1}{3}$$
H. $$\dfrac{1}{2}$$
I. $$\dfrac{5}{9}$$
J. $$\dfrac{3}{5}$$
Answer: B
A proton is fired from origin with velocity $\overrightarrow{v}=v_0\hat {j}+v_0\hat{k}$ in a uniform magnetic field $\overrightarrow{B}=B_0\hat{j}$. In the subsequent motion of the proton

Options:
A. its $x-$ and $z-$coordinate cannot be zero at the same time
B. its $x-$coordinate can never be positive
C. its $z-$coordinate can never be negative
D. its $x-$coordinate can never be zero
E. its $z-$coordinate can never be zero
F. None of the above
G. its $x-$coordinate can never be negative
H. its $y-$coordinate can never be positive
Answer: B
A particle moves along the parabolic path $y={ ax }^{ 2 }$ in such a way that the y-component of the velocity remains constant, say c. The x and y coordinates are in meters. Then acceleration of the particle at x=1 m is

Options:
A. $$-\dfrac { { c }^{ 2 } }{ { 4 a }^{ 2 } } \hat { i } $$
B. $$-\dfrac { { c } }{ a } \hat { i }$$
C. $$-\dfrac { { c }^{ 2 } }{ { 3 a }^{ 2 } } \hat { i } $$
D. $${ 2 ac }^{ 2 }\hat { j } $$
E. $$-\dfrac { { c } }{ 3a } \hat { i }$$
F. $$-\dfrac { { c } }{ 4a } \hat { i }$$
G. $$-\dfrac { { c }^{ 2 } }{ { 2a }^{ 2 } } \hat { i }$$
H. $$-\dfrac { { c } }{ 5a } \hat { i }$$
I. $$ac\hat { k } $$
J. $$-\dfrac { { c }^{ 2 } }{ 4a } \hat { i }$$
Answer: C
Calculate the molar solubility of $ Mg(OH{)  }_{2  }   $ in 1M $  N{ H }_{ 4 }Cl$
$ {  K}_{ sp }Mg(OH{ ) }_{  2}=1.8\times1 { 0}^{ -11}  $   $     {  K}_{ b }(N{  H}_{ 3 }{ ) }=1.8\times1 { 0}^{ -5}  $

Options:
A. 0.48
B. 0.24
C. 1.14
D. 1.24
E. 0.88
F. 0.98
G. 0.96
H. 0.94
I. 1.04
J. 1.48
Answer: G