MikaStars39/nano-eval huggingface dataset:
Two wires PQ and QR, carry equal currents I as shown in figure. One end of both the wires extends to infinity. $\angle PQR = \theta$. The magnitude of the magnetic field at O on the bisector angle of the these two wires at a distance r from point Q is

Options:
A. $$\displaystyle \frac{\mu_0}{4 \pi } \frac{I}{r} \left ( \frac{(1 - cos \theta/2)}{(2 \cdot cos \theta/2)} \right )$$
B. $$\displaystyle \frac{\mu_0}{4 \pi } \frac{I}{r} \left ( \frac{(1 - cos \theta/2)}{(sin \theta/2)} \right )$$
C. $$\displaystyle \frac{\mu_0}{4 \pi } \frac{I}{r} \left ( \frac{(1 + sin \theta/2)}{(cos \theta/2)} \right )$$
D. $$\displaystyle \frac{\mu_0}{4 \pi }\frac{I}{r} \left ( \frac{(1 + cos \theta/2)}{(sin \theta/2)} \right )$$
E. $$\displaystyle \frac{\mu_0}{4 \pi } \frac{I}{r} \left ( \frac{(1 + sin \theta/2)}{(2 \cdot sin \theta/2)} \right )$$
F. $$\displaystyle \frac{\mu_0}{4 \pi }\frac{I}{r} cot (\theta/2)$$
G. $$\displaystyle \frac{\mu_0}{4 \pi } \frac{I}{r} \left ( \frac{(1 - cos \theta/2)}{(2 \cdot sin \theta/2)} \right )$$
H. $$\displaystyle \frac{\mu_0}{4 \pi }\frac{I}{r} tan \left ( \frac{\theta}{2} \right )$$
I. $$\displaystyle \frac{\mu_0}{4 \pi }\frac{I}{r} sin \left ( \frac{\theta}{2} \right )$$
J. $$\displaystyle \frac{\mu_0}{4 \pi } \frac{I}{r} \left ( \frac{(1 - sin \theta/2)}{(sin \theta/2)} \right )$$
Answer: D
A bucket filled with water is tied to a rope of length $0.5\ m$ and is rotated in a circular path in vertical plane. the least velocity it should have at the lowest point of circle so that water does not spill is $(g=10ms^{-2})$:

Options:
A. $$\sqrt{10}\ m/s$$
B. $$4\sqrt{5}\ m/s$$
C. $$2\ m/s$$
D. $$5\ m/ s$$
E. $$6\ m/s$$
F. $$4\ m/s$$
G. $$\sqrt{5}\ m/s$$
H. $$2\sqrt{5}\ m/s$$
I. $$3\ m/s$$
J. $$3\sqrt{5}\ m/s$$
Answer: D
A block of mass M = 5kg is moving on a horizontal plane. An object of mass m = 1 kg is dropped onto the block, hitting it with a vertical velocity of $v_1$ 10 m/s. The speed of the block at the same instant is $v_ 2$ = 2 m/s on the floor. The collision ois momentary and object stricks to the block upon collision. What will be the speed of the block just after the collision if the coefficient of friction between the block and the horizontal plane is $\mu$ = $0.4$

Options:
A. 1.3 m/s
B. 1.4 m/s
C. 1.6 m/s
D. 1.45 m/s
E. 1.67 m/s
F. 1.5 m/s
G. 0.8 m/s
H. 1.2 m/s
I. 1 m/s
J. 1.35 m/s
Answer: B
$0.1M$ KI and $0.2M \,\,AgNO_3$ are mixed in $3:1$ volume ratio.The depression of freezing point of the resulting solution will be $[K_f(H_2O)=1.86K\,kg mol^{-1}]$

Options:
A. $1.86$K
B. $0.45$K
C. 2.74K
D. 1.36K
E. 2.78K
F. 2.76K
G. $2.68$K
H. $0.93$K
I. 2.70K
J. $0.279$K
Answer: J
An organic compound, $C_3H_8O$ does not give a precipitate with $2,4$-dinitrophenyl hydrazine reagent and does not react with metallic sodium. It could be:

Options:
A. $$CH_3-CH=CH-CH_3$$
B. $$CH_3-CH=CH-OH$$
C. $$CH_3-O-CH_3$$
D. $$CH_2=CH-CH_3$$
E. $$CH_3-CH_2-O-CH_2$$
F. $$CH_3-CH_2-CH_2OH$$
G. $$CH_2=CH-CH_2OH$$
H. $$CH_2=CH-O-CH_3$$
I. $$CH_3-CH_2-CHO$$
J. $$CH_3-CH_2-O-CH_3$$
Answer: H
Accrording to Maxwell's distribution law, the fraction of molecules per unit change in speed [velocity distribution function, F(c)] travelling with the speed from C to C +dC is given as.$f\left( c \right) =\dfrac { 1 }{ dC } .\left( \dfrac { dN }{ N }  \right) =4\pi .{ \left( \dfrac { M }{ 2\pi RT }  \right)  }^{ 3/2 }.{ C }^{ 2 }.{ e }^{ -{ MC }^{ 2 }RT }$
At what speed of the molecules, the value of the Maxwell distribution function f(c) for the temperature T will be the same as that of the temperature n times higher?

Options:
A. $$\sqrt { \dfrac { 3RTn }{ M\left( n-1 \right) } }$$
B. $$\sqrt { \dfrac { 3RT\ln(n) }{ M(n-1) } }$$
C. $$\sqrt { \dfrac { 3RTn }{ M(n+1) } }$$
D. $$\sqrt { \frac { 3RTnInn }{ M\left( n-1 \right) } } $$
E. $$\sqrt { \dfrac { 3RTn }{ M(n-2) } }$$
F. $$\sqrt { \frac { 3RTInn }{ M\left( n-1 \right) } } $$
G. $$\sqrt { \dfrac { 3RTIn }{ M\left( n-1 \right) } }$$
H. $$\sqrt { \dfrac { 3RTn^2 }{ M\left( n-1 \right) } }$$
I. most probable speed
J. RMS speed
Answer: F
A body of mass $2$ kg slides down with an acceleration of $3\, m/s^2$ on a rough inclined plane having a slope of $30^\circ$. The external force required to take the same body up the plane with the same acceleration will be:
($g=10\, m/s^2$)

Options:
A. $$10\, N$$
B. $$15\, N$$
C. $$18\, N$$
D. $$16\, N$$
E. $$14\, N$$
F. $$20\, N$$
G. $$6\, N$$
H. $$17\, N$$
I. $$12\, N$$
J. $$4\, N$$
Answer: F
The composition of the equilibrium mixture $\left ( Cl_{2}\rightleftharpoons 2Cl \right )$ which is attained at $1200^{\circ}C$ is determined by measuring the rate of effusion through a pin hole. It is observed that the given mixture effuses 1.5 times as fast as an unknown gas 'X' effuses under the same conditions. Calculate the fraction of chlorine molecules dissociated into atoms.[Molecular Mass of unknown gas=90]

Options:
A. 0.58
B. 0.64
C. 0.14
D. 0.4
E. 0.54
F. 0.62
G. 0.68
H. 0.75
I. 0.52
J. 0.8
Answer: C
The electric field and the electric potential at a point are E and V respectively:

Options:
A. If E $=$ 0, V cannot be zero.
B. If E $=$ 0, V must be zero.
C. If V $=$ 0, E must be zero.
D. If E $\neq$ 0, V must be zero.
E. If E $\neq$ 0, V must be non-zero.
F. If V $\neq$ 0, E can be non-zero.
Answer: A
Half-life of radium is $1620$ years. How many radium nuclei decay in $5$ hours in $5gm$ radium? (atomic weight of radium $=223$)

Options:
A. $$9.3 \times 10^{12}$$
B. $$1.1 \times 10^{13}$$
C. $$3.23\times {10}^{15}$$
D. $$1.72\times {10}^{20}$$
E. $$3.15\times {10}^{15}$$
F. $$3.13 \times 10^{16}$$
G. $$9.0 \times 10^{12}$$
H. $$8.1 \times 10^{12}$$
I. $$3.3\times {10}^{17}$$
J. $$9.5 \times 10^{12}$$
Answer: E
For the reaction, $2NO(g)+O_2(g)\to 2NO_2(g); \Delta H=-35.0\ kcal $ at $300\ K$ If $6.0$ moles of $NO$ reacts with $3.0$ moles of $O_2$ at constant pressure of $1.0$ atm and temperature $300\ K$ to from $NO_2$, then the correct statement(s) is/are:

Options:
A. The internal energy of system decreased by $105.8\ kcal$
B. The internal energy of system decreased by $1.8\ kcal$
C. The internal energy of system decreased by $107.2\ kcal$
D. The amount of heat released by the reaction is $35\ kcal$
E. The internal energy of system decreased by $101.2\ kcal$
F. The internal energy of system decreased by $104.2\ kcal$
G. The magnitude of work done by the system is $1.8\ kcal$
H. The internal energy of system decreased by $103.2\ kcal$
I. The internal energy of system decreased by $108.8\ kcal$
J. The internal energy of system decreased by $105.2\ kcal$
Answer: B
A sample of air is saturated with benzene (vapour pressure $=100mm$ $Hg$ at $298K$), $750mm$ $Hg$ pressure. If it is isothermally compressed to one-third of its initial volume, the final pressure of the system is:

Options:
A. $$2150\ torr$$
B. $$2250\ torr$$
C. $$2450\ torr$$
D. $$2300\ torr$$
E. $$2000\ torr$$
F. $$1950\ torr$$
G. $$2050\ torr$$
H. $$2100\ torr$$
I. $$2350\ torr$$
Answer: B
$FeO$ crystallizes in the cubic system, in which there is four formula units in each unit cell. The density of the crystal is $4.0g/{cm}^{3}$. The side length of each unit cell is
$$\left( Fe=56,{ N }_{ A }=6\times { 10 }^{ 23 } \right) $$

Options:
A. $$4.292\mathring { A }$$
B. $$4.296\mathring { A }$$
C. $$2.974\mathring { A } $$
D. $$4.896\mathring { A }$$
E. $$4.227\mathring { A } $$
F. $$4.932\mathring { A } $$
G. $$4.848\mathring { A }$$
H. $$4.249\mathring { A }$$
I. $$2.424\mathring { A } $$
Answer: F
Mass of ${ H }_{ 2 }{ C }_{ 2 }{ O }_{ 4 }$ required to reduce 100 ml of 0.008M $KMn{ O }_{ 4 }$ in acidic medium is x g and to neutralize 100 ml of 0.01 M $Mg{ \left( OH \right)  }_{ 2 }$ is y g, then the relation between x and y is:

Options:
A. 3x = 2y
B. x = y
C. x = 2y
D. x = 5y
E. x = 6y
F. 2x = 3y
G. x = 3y
H. x = 4y
I. 2x = y
J. x = \frac{3}{2}y
Answer: B
Three identical square metal plates $M_1, M_2$ and $M_3$ of side 10 cm & 5 mm thick are arranged as shown in figure. The plates are separated by sheets of paper 0.5 mm thick & of dielectric constant 5. The outer plates are connected together & connected to lower potential while inner plate to higher potential terminal of a battery. Capacitance between the terminals of the battery is : $(\epsilon _{0} = 8.9 10^{-12} C^{2}/N-m^{2})$

Options:
A. 356 pF
B. $$1780pF$$
C. $$445 pF$$
D. 8900 pF
E. $$222.5 pF$$
F. 4450 pF
G. 2225 pF
H. 3560 pF
I. 890 pF
Answer: B
The number of grams/weight of $NH_{4}Cl$ required to be added to $3\ litres$ of $0.01\ M\ NH_{3}$ to prepare the buffer of $pH = 9.45$ at temperature $298\ K$ ($K_{b}$ for $NH_{3}$ is $1.85\times 10^{-5})$.

Options:
A. $$0.654\ gm$$
B. $$0.254\ gm$$
C. $$4.55\ gm$$
D. $$0.344\ gm$$
E. $$3.55\ gm$$
F. $$0.455\ gm$$
G. $$0.354\ gm$$
H. $$0.444\ gm$$
I. $$0.554\ gm$$
J. $$0.454\ gm$$
Answer: G
If moment of inertia of a point particle at a distance r from an axis would have defined as $L = mr$ instead of L=${ mr }^{ 2 }$, then moment of inertia of a uniform rod of mass M and length L about an axis passing through centre of mass and perpendicular to the rod will be (moment of inertia is still a scalar )

Options:
A. $$\dfrac { ML }{ 5 }$$
B. $$\dfrac { { ML }^{ 2 } }{ 12 } $$
C. $$\dfrac { ML }{ 3 }$$
D. $$\dfrac { ML }{ 4 } $$
E. $$\dfrac { ML }{ 6 }$$
F. $$\dfrac { ML }{ 7 }$$
G. $$\dfrac { ML }{ 12 } $$
H. 0
I. $$\dfrac { ML }{ 8 }$$
J. $$\dfrac { ML }{ 2 }$$
Answer: D
For the equilibrium reaction, $3Fe+4{H}_{2}O\Leftrightarrow {Fe}_{3}{O}_{4}+4{H}_{2}$ the relation between ${K}_{p}$ and ${K}_{c}$

Options:
A. ${K}_{P}={K}_{C} \times {(2TR)}^{2}$
B. ${K}_{P}<{K}_{C} \times {(2TR)}^{2}$
C. $${K}_{P}={K}_{C}\times{(TR)}^{2}$$
D. $${K}_{P}> {K}_{C}$$
E. $${K}_{P}< {K}_{C}$$
F. ${K}_{P}={K}_{C} \times {(0.5TR)}^{2}$
G. ${K}_{P}={K}_{C} \times {(1.5TR)}^{2}$
H. ${K}_{P}<{K}_{C} \times {(TR)}^{2}$
I. $${K}_{P}={K}_{C}$$
J. $${K}_{P} = {K}_{C} \times {(1.2TR)}^{2}$$
Answer: D
A solid sphere of mass $2 kg$ is resting inside a cube as shown in $Fig.$. The cube is moving with a velocity $\overrightarrow { v } =\left( 5t\hat { i } +2t\hat { j }  \right) m{ s }^{ -1 }$. Here t is the time in seconds. All surfaces are smooth. The sphere is at rest with respect to the cube. What is the total force exerted by the sphere on the cube?

Options:
A. $$29 \ N$$
B. $$\sqrt{65} \ N$$
C. $$\sqrt{74} \ N$$
D. $$28 \ N$$
E. $$\sqrt{29} N$$
F. $$26 \ N$$
G. $$\sqrt{89} N$$
H. $$27 \ N$$
I. $$25 \ N$$
J. $$\sqrt{53} \ N$$
Answer: F
A vessel contains a mixture of $7g$ of nitrogen and $8g$ of oxygen at temperature $T=300K$. If the pressure of the mixture is $1atm$, its density is 
$$\left[ R=\cfrac { 25 }{ 3 } J/mol\quad K \right] $$

Options:
A. $$1.4kg/{m}^{3}$$
B. $$1.0kg/{m}^{3}$$
C. $$1.5kg/{m}^{3}$$
D. $$0.8kg/{m}^{3}$$
E. $$2kg/{m}^{3}$$
F. $$1.2kg/{m}^{3}$$
G. $$1.6kg/{m}^{3}$$
H. $$1.3kg/{m}^{3}$$
I. $$0.6kg/{m}^{3}$$
J. $$1.8kg/{m}^{3}$$
Answer: I
A block of ice of mask $M=10kg$ is moved back and forth over the flat horizontal surface of a large block of ice. Both block are at $0^oC$ and the force that produces the back-and -forth motion acts only horizontally. The coefficienrt of friction between the two surfaces is $0.060$. If $m=15.2 g$ of water is produced the total distance travelled by the upper block relative to the lower is $(L_{ice}=3.34\times 10^5 J/kg)$

Options:
A. $$432 m$$
B. $$268 m$$
C. $$434 m$$
D. $$369 m$$
E. $$435 m$$
F. $$587 m$$
G. $$216 m$$
H. $$368 m$$
I. $$863 m$$
Answer: I
A thin rod of length $L$ is vertically straight on horizontal floor. This rod falls freely to one side without slipping at its bottom. The linear velocity of centre of rod when its top end touches floor is : (Given acceleration due to gravity $=g$)

Options:
A. $$\sqrt{\dfrac{3gL}{5}}$$
B. $$\sqrt{\dfrac{3gL}{8}}$$
C. $$\sqrt{\dfrac{3gL}{10}}$$
D. $$\sqrt{\dfrac{3gL}{6}}$$
E. $$\sqrt{\dfrac{3gL}{2}}$$
F. $$\sqrt{\dfrac{3gL}{7}}$$
G. $$\sqrt{3gL}$$
H. $$\sqrt{\dfrac{3gL}{4}}$$
I. $$\sqrt{2gL}$$
J. $$\sqrt{\dfrac{3gL}{9}}$$
Answer: H
A block of mass 10 kg is released on a fixed wedge inside a cart which is moving with constant velocity 10 $\displaystyle ms^{-1}$ towards right. There is no relative motion between block and cart. Then work done by normal reaction on block in two seconds from ground frame will be: $\displaystyle \left ( g= 10ms^{-2} \right )$

Options:
A. $$1320 J$$
B. $$1020 J$$
C. $$920 J$$
D. $$960 J$$
E. $$240 J$$
F. $$1040 J$$
G. $$880 J$$
H. $$1200 J$$
I. $$1080 J$$
J. $$840 J$$
Answer: D
Velocity versesus displacement of a particle moving in a  straight line is $v=\sqrt { 9+4x } $ (Here V is in m/s and x is in metre). Mass of the particle is 2 kg. Work done by all  the forces on the particle in a time interval from t=0 to t=2s is

Options:
A. 35 J
B. 12 J
C. 30 J
D. 20 J
E. 40 J
F. 14 J
G. 10 J
H. 15 J
I. 25 J
J. 18 J
Answer: G
In a closed loop, which has some inductance but negligible resistance, uniform but time varying magnetic field is applied directed into the plane of the loop. Variation of field with time is shown. Initially current in the loop was zero. Then

Options:
A. current in the loop is zero at $t = 1\ s$
B. emf induced in the loop is maximum at $t = 1\ s$
C. emf induced in the loop will be maximum at $t = 2\ s$
D. current in the loop will be maximum at $t = 2\ s$
E. emf induced in the loop will change direction at $t = 2\ s$
F. direction of emf in the loop will change at $t = 2\ s$
G. current in the loop will be zero at $t = 2\ s$
H. current in the loop will be minimum at $t = 2\ s$
I. None of the above
J. emf induced in the loop is zero at $t = 2\ s$
Answer: J
A sphere of radius $R$ has a charge density $\rho(r) =\rho_0(\dfrac rR)$ where $\rho_0$ is a constant and $r$ is the distance from the centre of the sphere. Find the electric field for $r>R$

Options:
A. $$E=\dfrac{4\rho_0 R^3}{3\epsilon_0r^2}$$
B. $$E=\dfrac{\rho_0R}{2\epsilon_0}$$
C. $$E=\dfrac{\rho_0 r^2}{4\epsilon_0 R}$$
D. $$E=\dfrac{\rho_0r^3}{4\epsilon_0R^2}$$
E. $$E=\dfrac{4\rho_0 R^2}{3\epsilon_0 r^2}$$
F. $$E=\dfrac{3\rho_0 R^2}{4\epsilon_0 r^2}$$
G. $$E=\dfrac{\rho_0 r^2}{3\epsilon_0 R^2}$$
H. $$E=\dfrac{\rho_0 R^3}{4\epsilon_0r^2}$$
I. $$E=\dfrac{3\rho_0 R^3}{8\epsilon_0 r^2}$$
J. $$E=\dfrac{2\rho_0 R^3}{4\epsilon_0r^2}$$
Answer: D
$1$ mole of an ideal gas undergoes an isothermal expansion as energy is added to it as heat Q. Graph shows the volume V versus Q. The gas temperature is nearly equal to : (use R= $8.31 $J/K mole).

Options:
A. $$313.6K$$
B. $$353.8K$$
C. $$313.1K$$
D. $$269.5K$$
E. $$312.8K$$
F. $$312.1K$$
G. $$312.6K$$
H. $$208.4K$$
I. $$268.2K$$
J. $$312.9K$$
Answer: G
The pH of a buffer solution made by mixing $25 ml$ of $0.02 M\quad NH_4OH$ and $25 ml$ of $0.2 M\quad NH_4Cl$ at $25^oC$ is : $(pK_b\quad of\quad NH_4OH = 4.8)$

Options:
A. $$8.2$$
B. $$5.8$$
C. $$6.0$$
D. $$6.2$$
E. $$7.8$$
F. $$5.3$$
G. $$3.8$$
H. $$4.8$$
I. $$6.8$$
Answer: A
A ring rolls without slipping on the ground. Its centre C moves with a constant speed $u$. P is any point on the ring. The speed of P with respect to the ground is $v$.

Options:
A. $v=u$, if CP is horizontal.
B. $$v= \sqrt{u}$$
C. $v= \sqrt{4u}$, if CP is horizontal.
D. $v= \sqrt{2u}$, if CP is horizontal.
E. $$v= \sqrt{3u}$$
F. $$v= \sqrt{3u}, if CP is horizontal.$$
G. v= 2u, if CP is horizontal.
Answer: D
A car of mass $M$ is moving on a horizontal circular path of radius $r$. At an instant, its speed is $v$ and is increasing at the rate $a$. Then

Options:
A. The friction coefficient between the ground and the car is not less than \(\frac{v^2 + 2a}{rg}\)
B. The friction coefficient between the ground and the car is not less than v^2/rg
C. The friction coefficient between the ground and the car is not less than \(\frac{v^2 + a}{rg}\)
D. The friction coefficient between the ground and the car is not less than \(\frac{a}{g}\)
E. The friction coefficient between the ground and the car is not less than \(\frac{v^2 + a^2}{rg}\)
F. The acceleration of the car is towards the centre of the path
G. The friction coefficient between the ground and the car is $\displaystyle \mu = \tan^{-1} \frac{v^2}{rg}$
H. The friction coefficient between the ground and the car is not less than \(\frac{a^2}{rg}\)
I. The friction coefficient between the ground and the car is not less than a/g
Answer: I
For the cell $Pt | H_2 (g)|$ solution $X||KCl(saturated)|Hg_2Cl_{2(s)}|Hg||Pt$ the observed EMF at $25^oC$ was 600 mV, when solution X was replaced by a standard phosphate buffer with $pH=7.00$, the EMF was 777 mV, The $pH$ of solution X is

Options:
A. 3
B. 8
C. 9
D. 10
E. 4
F. 7
G. 1
H. 2
I. 6
J. 5
Answer: A
The position $x$ of a particle moving along $x-$axis is given as $x=4\sin \left(\dfrac {4\pi}{3}t+\dfrac {\pi}{2}\right)$, where $x$ is in meter and $t$ is in second. The ratio of average speed and magnitude of average velocity of the particle for the time, interval, $0 \le t \le 2.25\ s$ is

Options:
A. $$5$$
B. $$4$$
C. $$3$$
D. $$6$$
E. $$7$$
F. $$1$$
G. $$2$$
Answer: C
As per VSEPER theory in which molecular species nitrogen has $NX_{3}L$ type arrangement , but lonepair is $NOT$ localised on nitrogen .(N= nitrogen , X = $\sigma $ bonds , L=lone pair )

Options:
A. $$N(CH_{3})_{4}$$
B. $$N(CH_{3})_{4}^{+}$$
C. $$N(CH_{3})_{4}^{2+}$$
D. $$N(CH_{3})_{3}$$
E. $$CH_{3}N(SiH_{3})_{2}$$
F. $$CH_{3}NH_{2}$$
G. $$CH_{3}N(CH_{3})_{2}$$
H. $$N(CH_{3})_{2}$$
Answer: F
A 30 kg box has to move up an inclined slope of $30^\circ$ to horizontal at a uniform velocity of 5 m/sec. If $\mu_k=\dfrac{5}{3\sqrt 3}$, the horizontal force required to move up is $(g=10 m/sec^{2})$

Options:
A. $$300 N$$
B. \dfrac{300 \sqrt{3}}{4} N
C. 300 \times 2 \sqrt{2} N
D. $$300 \sqrt{2} N$$
E. 300 \times 3 \sqrt{3} N
F. $$300 \times 2 \sqrt{3} N$$
G. 300 \times \sqrt{3} N
H. $$\dfrac{300 \sqrt{3}}{2} N $$
I. \dfrac{300 \sqrt{3}}{3} N
J. 300 \times \sqrt{2} N
Answer: F
A loop is formed by two parallel conductors conducted by a solenoid with inductance L and a conducting rod of mass m, which can freely (without friction )slide over the conductors. The conductors are located in a horizontal plane in a uniform vertical magnetic field with induction B. The distance between the conductors is equal to l At the moment t=0 a constant force F starts acting on the rod . Angular frequency of motion rod is . If F and m are doubled then :

Options:
A. $\omega $ become $\sqrt 2 $ times
B. $\omega $ become $\frac{1}{\sqrt{8}}$ times
C. $\omega $ become $2$ times
D. $\omega $ become $\frac{1}{\sqrt{3}}$ times
E. \omega \text{ become } \frac{1}{\sqrt{4}} \text{ times}
F. $\omega$ become $\frac{1}{4}$ times
G. $\omega $ become $\frac{1}{2}$ times
H. $\omega $ become $\frac{1}{{\sqrt 2 }}$ times
I. \omega \text{ become } \frac{1}{3} \text{ times}
J. $\omega $ become $roverscore{3}$ times
Answer: G
A solution containing 1 mol per liter of each $Cu(NO_3)_2,\, AgNO_3,\, and\, Hg_2(NO_3)_2$ is being electrolyzed by using inert electrodes. The value of standard electrode potentials in volts (reduction potential) are$Ag\, |\, Ag^{\bigoplus}\, =\, +0.80,\, 2Hg\, |\, Hg_2^{2+}\, =\, +0.79$$Cu\, |\, Cu^{2+}\, =\, +0.34,\, Mg\, |\, Mg^{2+}\, =\, -2.37$With increasing voltage, the sequence of deposition of metals at the cathode will be:

Options:
A. $$Ag, Hg, Mg, Cu$$
B. $$Ag, Cu, Hg$$
C. $$Mg, Ag, Hg, Cu$$
D. $$Cu, Hg, Ag$$
E. $$Ag, Cu, Hg, Mg$$
F. $$Ag, Mg, Cu, Hg$$
G. $$Ag, Cu, Mg, Hg$$
H. $$Ag, Hg, Cu$$
I. $$Mg, Cu, Hg, Ag$$
J. $$Ag, Hg, Cu, Mg$$
Answer: H
Two resistors of resistance ${R}_{1}=(100\pm 3)\Omega$ and ${R}_{2}=(200\pm 4)\Omega$ are connected in parallel. The effective resistances of the combinations is

Options:
A. $$(66.7\pm 0.9)\Omega$$
B. $$(66.7\pm 0.4)\Omega$$
C. $$(66.7\pm 1.0)\Omega$$
D. $$(66.7\pm 0.5)\Omega$$
E. $$(66.7\pm 1.4)\Omega$$
F. $$(66.7\pm 0.6)\Omega$$
G. $$(66.7\pm 1.8)\Omega$$
H. $$(66.7\pm 1.2)\Omega$$
I. $$(66.7\pm 1.5)\Omega$$
J. $$(66.7\pm 7)\Omega$$
Answer: G
A body of mass m = 1 kh is dropped from height h = 40 cm on a horizontal platform fixed to one end of an elestic spring the other being fixed to a base as shown in fig.As a result the spring is compressed by an amount X = 10 cm.The force constant of the spring is (g = 10 ms^{-2}) $

Options:
A. 850 N/m
B. 1000 N/m
C. 1200 N/m
D. 500 N/m
E. 600 N/m
F. 800 N/m
G. 750 N/m
H. 1100 N/m
I. 700 N/m
J. 900 N/m
Answer: E
The amount of $(NH_{4})_{2}SO_{4}$ having degree of dissocation 75% which should be dissolved in 1500 ml of 1 M $NH_{4}OH$ to decrease its degree of dissociation by 200 times, is [$k_{b}$ of $NH_{4}OH$ = $1.8$ $\times10^{-5}$]:

Options:
A. 112.1gm
B. 56.0gm
C. 28.0gm
D. 140.1gm
E. 65.4gm
F. 224.2 gm
G. 112.2gm
Answer: A
A number of blocks each of mass $0.4\ kg$ are placed one over the other. The minimum horizontal force required to pull the eighth block from the top without disturbing the remaining is $10\ N$. If $g = 10\ ms^{-2}$, the coefficient of static friction between any two blocks is:

Options:
A. $$1/5$$
B. $$1/11$$
C. $$1/10$$
D. $$1/6$$
E. $$1/7$$
F. $$1/12$$
G. $$1/4$$
H. $$1/3$$
I. $$1/8$$
J. $$1/9$$
Answer: D
A hot body, obeying Newton's law of cooling is cooling down from its peak value ${80}^{o}C$ to ${40}^{o}C$. How much time will it take to cool down from ${62}^{o}C$ to ${32}^{o}C$? (given $\ln { 2 } =0.693,\ln { 5 } =1.609$)

Options:
A. $8.6$ minutes
B. 7.6 minutes
C. 7.5 minutes
D. $3.75$ minutes
E. $6.5$ minutes
F. $9.6$ minutes
G. 8.2 minutes
Answer: A
Thermal decomposition of gaseous $X_2$ to X at 298 K takes place according to the following equation  $X_2(g) \rightleftharpoons 2X(g)$The standard reaction Gibbs energy, $\Delta_r G^0$ of this reaction is positive. At the start of the reaction, there is one mole of $X_2$ and no X. As the reaction proceeds the number of moles of X formed is given by $\beta$. Thus $\beta_{ equilibrium}$ is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2 bar. Consider the gases to behave ideally. ($ R = 0.083 \ L \ bar \ K^{-1} \ mol^{-1}$). The equilibrium constant $K_p$ for this reaction at 298 K in terms of $\beta_{eqb}$ is

Options:
A. $$\dfrac{4\beta_{eqb}^2}{4 - 2\beta_{eqb}}$$
B. $$\dfrac{8\beta_{eqb}^2}{4 - \beta_{eqb}^2}$$
C. $$\dfrac{4\beta_{eqb}^2}{2 - 2\beta_{eqb}}$$
D. $$\dfrac{2\beta_{eqb}^2}{4 - \beta_{eqb}^2}$$
E. $$\dfrac{8\beta_{eqb}^2}{2 - \beta_{eqb}}$$
F. $$(\dfrac{4\beta_{eqb}^2}{4 - 2\beta_{eqb}^2} )$$
G. $$\dfrac{4\beta_{eqb}^2}{4 + \beta_{eqb}^2}$$
H. $$\dfrac{4\beta_{eqb}^2}{2 - \beta_{eqb}}$$
I. $$\dfrac{8\beta_{eqb}^2}{4 - 2\beta_{eqb}}$$
J. $$\dfrac{4\beta_{eqb}^2}{4 - \beta_{eqb}^2}$$
Answer: B
A particle with charge q and , mass m starts its motion from origin with a velocity $\overrightarrow v  = a\widehat i$. A uniform magnetic field $\overrightarrow B  = \dfrac{b}{4}\widehat i + \dfrac{{\sqrt 3 b}}{4}\widehat j$ exists everywhere in space. The component of the velocity vector in y-direction when the value of z-coordination becomes maximum is

Options:
A. $${4\sqrt 3 a}$$
B. $$\dfrac{{\sqrt 3 }}{3}a$$
C. $$\dfrac{{\sqrt 3 }}{6}a$$
D. $$\dfrac{{3\sqrt{3}}}{8}a$$
E. $${2\sqrt 3 a}$$
F. $$\dfrac{{\sqrt 3 }}{8}a$$
G. $$\dfrac{{\sqrt 3 }}{4}a$$
H. $$\dfrac{3\sqrt{3}}{4}a$$
I. $$\dfrac{{\sqrt 3 }}{5}a$$
J. $$\dfrac{{\sqrt 3 }}{2}a$$
Answer: G
The surface of a black body is at a temperature $727^{\circ}C$ and its cross section in $1\ m^{2}$. Heat radiated from this surface in one minute in Joules is (Stefan's constant $= 5.7\times 10^{-8}W/m^{2} k^{4})$

Options:
A. $$34.2\times 10^{5}$$
B. $$24.2 \times 10^{5}$$
C. $$34.2 \times 10^{4}$$
D. $$2.5\times 10^{5}$$
E. $$3.42 \times 10^{6}$$
F. $$25.2 \times 10^{5}$$
G. $$34.2 \times 10^{3}$$
H. $$34.2 \times 10^{6}$$
I. $$2.5\times 10^{6}$$
J. $$3.42\times 10^{5}$$
Answer: A
A continous line of charge of length $3d$ lies along the x-axis, extending from $x=+d$ to $x=+4d$. The lies carries a uniform linear charge density $'\lambda$.In terms of $d$, $\lambda$ and necessary constant, find the magnitude of the electric field at the origin.

Options:
A. \dfrac{\lambda}{ 4 \pi \varepsilon _0 2d}
B. $$\dfrac{\lambda}{9 \pi \varepsilon_0 d}$$
C. $$ \dfrac{\lambda}{ 5 \pi arepsilon _0 d}$$
D. $$\dfrac{\lambda}{4 \pi \varepsilon_0 d}$$
E. $$\dfrac{3 \lambda}{8 \pi \varepsilon_0 d}$$
F. $$\dfrac{\lambda}{7 \pi \varepsilon_0 d}$$
G. $$\dfrac{\lambda}{ 6 \pi \varepsilon _0 d}$$
H. $$\dfrac{3 \lambda}{16 \pi \varepsilon_0 d}$$
I. $$\dfrac{\lambda}{ 2 \pi \varepsilon _0 d}$$
Answer: D
Statements that are true of d-and f-block elements are:
1. The colour of $K_{2}Cr_{2}O_{7}$ is due to d-d transition.
2.  Actinides have larger number of oxidation states than lanthanides.
3.  Tripositive ions of $Pr(At. No. =59)$ and $Tm(At. No. =69)$ have the same (green) colour.
4.  Actinides have paramagnetic moments less than theoretically predicted values.

Options:
A. 1,2,3,4
B. 1,3,4
C. 2,4
D. 1,2,3,5
E. 1,2,4
F. 1,2,5
G. 1,2,4,5
H. 1,2,3
I. 1,3,4,5
J. 2,3,4
Answer: J
A satellite revolves in a circular orbit with speed $V=\cfrac{1}{\sqrt{3}}{V}_{e}$. If satellite is suddenly stopped and allowed to fall freely on to earth, the speed with which it hits earth's surface is

Options:
A. $$\sqrt{\cfrac{2}{3}gR}$$
B. $$\sqrt{\cfrac{7}{3}gR}$$
C. $$\sqrt{\cfrac{gR}{3}}$$
D. $$\sqrt{\cfrac{1}{3}gR}$$
E. \sqrt{\cfrac{5}{3}gR}
F. $$\sqrt{2gR}$$
G. $$\sqrt{gR}$$
H. $$\sqrt{\cfrac{4}{3}gR}$$
I. $$\sqrt{\cfrac{8}{3}gR}$$
J. $$\sqrt{\cfrac{3}{2}gR}$$
Answer: G
A uniform rod of mass $m = 2\space kg$ and length $l$ is kept on a smooth horizontal plane. If the ends $A$ and $B$ of the rod move with speeds $v$ and $2v$ respectively perpendicular to the rod, then

Options:
A. kinetic energy of the rod (in Joule) is $K = \displaystyle\frac{mv^2}{2}$
B. kinetic energy of the rod (in Joule) is $K = \displaystyleꙨrac{mv^2}{3}$
C. kinetic energy of the rod (in Joule) is \( K = \frac{3mv^2}{4} \)
D. kinetic energy of the rod (in Joule) is $K = \displaystyle\frac{mv^2}{4}$
E. $$K = \frac{3mv^2}{8}$$
F. kinetic energy of the rod (in Joule) is \( K = \frac{9mv^2}{8} \)
G. kinetic energy of the rod (in Joule) is \( K = \frac{5mv^2}{8} \)
H. kinetic energy of the rod (in Joule) is \( K = \frac{7mv^2}{8} \)
Answer: A
$aK_2Cr_2O_7+bKCl+cH_2SO_4\rightarrow xCrO_2Cl_2+yKHSO_4+zH_2O$Find the value of $a,b,c,x,y,\ and\ z$.

Options:
A. a=4,b=2,c=6,x=2,y=6,z=3
B. a=2,b=4,c=6,x=2,y=6,z=3
C. a=4,b=2,c=6,x=6,y=2,z=3
D. a=4,b=1,c=6,x=2,y=6,z=3
E. a=3,b=4,c=6,x=2,y=6,z=3
F. a=2,b=4,c=6,x=2,y=6,z=5
G. a=4,b=2,c=6,x=3,y=6,z=2
H. a=6,b=4,c=2,x=6,y=3,z=w
I. a=2,b=4,c=6,x=2,y=6,z=4
J. a=1,b=4,c=6,x=2,y=6,z=3
Answer: J
An object of density $\sigma (< \rho)$ is dropped from rest at a height h into a liquid of density $\rho$ kept in a tall vertical cylindrical container. Neglect all dissipative effects and assume that there is no spilling of the liquid. The time taken by the object to reach the starting point (from which it is dropped) is $(\rho=2\sigma)$

Options:
A. 5\sqrt {\dfrac {2h}{g}}
B. $$4\sqrt {\dfrac {2h}{g}}$$
C. $$9\sqrt {\dfrac {2h}{g}}$$
D. 7\sqrt{\dfrac{2h}{g}}
E. $$3\sqrt {\dfrac {2h}{g}}$$
F. it will never reach starting point
G. 6\sqrt {\dfrac {2h}{g}}
H. $$2\sqrt {\dfrac {h}{g}}$$
I. $$2\sqrt {\dfrac {2h}{g}}$$
J. 8\sqrt {\dfrac {2h}{g}}
Answer: B
When an ideal gas is taken from state a to b, along a path acb, $84\, KJ$ of heat flows into the gas and the gas does $32 \, KJ$ of work. The following conclusions are drawn. Mark the one which is not correct.

Options:
A. If $U_a= 0, U_d= 42\, KJ$, heat absorbed in the process db is $10 \,KJ$
B. If $U_a = 0, \, U_d = \,42 \,KJ$, and the work done along the path adb is $10.5\, KJ$ then the heat adsorbed in the process ad is $52.5 \, KJ$.
C. If the work done along the path adb is $10.5 \, KJ$, then heat that will flow into the gas is $62.5 \, KJ$.
D. When the gas is returned from b to a along the curved path, the work done on the gas is $21 \, KJ $, and the system absorbs $73 \,KJ$ of heat.
Answer: D
Two bodies of masses $m_{1}$ and $m_{2}$  moving on the same direction with velocities $u_{1}$ and $u_{2}$ collide. The velocities after collision are $v_{1}$ and $v_{2}$ . If each sphere loses the same amount of kinetic energy, then

Options:
A. $$u_{1}+u_{2}+v_{1}-4v_{2}= 0$$
B. $$u_{1}-u_{2}+v_{1}+v_{2}= 0$$
C. $$u_{1}+u_{2}+v_{1}+2v_{2}= 0$$
D. $$u_{1}+u_{2}+v_{1}-v_{2}= 0$$
E. $$u_{1}+u_{2}+v_{1}+v_{2}= 0$$
F. $$u_{1}+u_{2}-v_{1}+v_{2}= 0$$
G. $$u_{1}+u_{2}-v_{1}-v_{2}= 0$$
H. $$u_{1}+u_{2}+v_{1}-3v_{2}= 0$$
I. $$u_{1}+u_{2}+v_{1}-2v_{2}= 0$$
Answer: E
A body of mass 2 kg is lying on a rough inclined plane of inclination of angle 30° coefficient of static friction = 0.2, [Take f=10 $m/s^{2}$]. Then the magnitude of the force parallel to the incline needed to make the block move

Options:
A. down the incline is 11 N
B. down the incline is 7 N
C. down the incline is 12 N
D. down the incline is 5 N
E. down the incline is  zero
F. down the incline is 18 N
G. down the incline is 10 N
H. down the incline is 15 N
I. up the incline is 21 N
J. up the incline is 13 N
Answer: E
When $0.575 \times 10^{-2} kg$ of Glauber's salt ($Na_2SO_4\cdot 10 H_2O$) is dissolved in $3$ g water, we get $1$ $dm^3$ of a solution of density $1077.2 \:kg\: m^{-3}$. Calculate the molarity, molality and mole fraction of $Na_2SO_4$ in the solution:

Options:
A. $$Molarity (M) = 0.2502, Molality (m) = 0.2502, Mole fraction=4.304 \times 10^{-3}$$
B. $Molarity (M) = 0.2502$, $Molality (m)=0.2402$, $Mole\  fraction=4.304\times 10^{-1}$
C. $Molarity (M) = 0.2502$, $Molality (m) = 0.2402$, $Mole\  fraction=4.304\times 10^{-3}$
D. $$Molarity (M) = 0.2502, Molality (m) = 0.2402, Mole fraction=4.304 \times 10^{-4}$$
E. $$Molarity (M) = 0.2402, Molality (m) = 0.2502, Mole fraction=4.304 \times 10^{-2}$$
F. $$Molarity (M) = 0.2402, Molality (m) = 0.2502, Mole fraction=4.304 \times 10^{-4}$$
G. None of these
H. $$Molarity (M) = 0.2502, Molality (m) = 0.2402, Mole fraction=4.304 \times 10^{-2}$$
I. $$Molarity (M) = 0.2402, Molality (m) = 0.2402, Mole fraction=4.304 \times 10^{-3}$$
J. $Molarity (M) = 0.2402$, $Molality (m)=0.2502$, $Mole \ fraction=4.304\times 10^{-3}$
Answer: C
A boy of mass m climbs up a conveyor belt with a constant acceleration. The speed of the belt is v = $\sqrt{gh/6}$ and the coefficient of friction between the boy and conveyor belt is $\mu = \frac{5}{3\sqrt3}$. The boy starts from A and moves with the maximum possible acceleration till he reaches the highest point B.Work done by friction on the boy is

Options:
A. None of the above
B. Equal to the work done by gravity on the conveyor belt
C. Equal to the work done by the boy on the motor
D. Equal to the work done by the boy
E. Equal to the work done by friction on the motor
F. Equal to the work done by the boy on the conveyor belt
G. Zero
H. Equal to the work done by gravity on the boy
I. Equal to work done by the motor in running the conveyor belt.
J. Equal to work done by boy
Answer: I
If $K_1$ and $K_2$ are the equilibrium constants at temperature $T_1$ and $T_2$, where $T_2 > T_1$, then :

Options:
A. $K_2 = K_1$ when $\Delta H$ is -ve
B. $K_2 < K_1$ when $\Delta H$ is 0
C. $K_2 > K_1$ when $\Delta H$ is -ve
D. $K_2 > K_1$ when $\Delta H$ is 0
E. K_2 = K_1 when \Delta H is +ve
F. $K_2 > K_1$ when $\Delta H$ is +ve
G. $K_2=K$ when $\Delta H$ is 0
H. $K_2 < K_1$ when $\Delta H$ is -ve
I. $K_2 < K_1$ when $\Delta H$ is +ve
Answer: H
A vessel of volume $  V=5.0  $ liters contains 1.4 $ \mathrm{g}  $ of nitrogen at temperature $  T=1800 \mathrm{K}  $ . Find the pressure of the gas if 30$  \%  $ of its molecules are dissociated into atoms at this temperature.

Options:
A. $$1.94 \times 10^{4} N / m^{2}$$
B. $$2.84 \times 10^{3} N / m^{2}$$
C. $$ 1.94 \times 10^{2} N / m^{2} $$
D. None of these
E. $$2.94 \times 10^{2} N / m^{2}$$
F. $$ 2.84 \times 10^{2} N / m^{2} $$
G. $$1.94 \times 10^{1} N / m^{2}$$
H. $$2.94 \times 10^{3} N / m^{2}$$
I. $$ 1.94 \times 10^{3} N / m^{2} $$
Answer: C
For the adiabatic expansion of an ideal gas in vacuum, which of the following is true

Options:
A. $$\Delta U=W=\Delta H \neq 0, \Delta S = 0$$
B. $$\Delta U=W=\Delta H=0, \Delta S < 0$$
C. $$\Delta U=W=\Delta H=0, \Delta S=0$$
D. $$\Delta U=W=\Delta H=\Delta S=0$$
E. $$\Delta U=W=\Delta H=0, \Delta S > 0$$
F. $$\Delta U = W=\Delta S=0, \Delta S < 0$$
G. $$\Delta U=W=\Delta H=0, \Delta S \neq 0$$
H. $$\Delta U>\Delta H ;W=\Delta S=0$$
I. $$\Delta U=\Delta H=\Delta S=0, W \neq 0$$
J. $$\Delta U=W=\Delta H=0. \Delta S > 0$$
Answer: D
In a game of Carrom board, the Queen ( a wooden disc of radius $2cm$ and mass $50gm$) is placed at the exact center of the horizontal board. The striker is a smooth plastic disc of radius $3cm$ and mass $100gm$. The board is frictionless. The striker is given an initial velocity '$u$' parallel to the sides $BC$ and $AD$ so that it hits the Queen inelastically with coefficient of restitution$=2/3$. The impact parameter for the collision is '$d$' (as shown in the figure). The Queen rebounds from the edge $AB$ of the board inelastically with the same coefficient of restitution$=2/3$ and enters the hole $D$ following the dotted path as shown. The side of the board is $L$. Find the value of impact parameter '$d$' and the time which the Queen takes to enter hole $D$ after collision with the striker :

Options:
A. $$\dfrac { 8 }{ \sqrt { 17 } } cm,\quad 153L/30u$$
B. $$\dfrac { 5 }{ \sqrt { 17 } } cm,\quad 53L/100u$$
C. $$\dfrac { 8 }{ \sqrt { 17 } } cm,\quad 153L/100u$$
D. $$\dfrac { 5 }{ \sqrt { 17 } } cm,\quad 53L/80u$$
E. $$\dfrac { 5 }{ \sqrt { 17 } } cm,\quad 153L/90u$$
F. $$\dfrac { 5 }{ \sqrt { 17 } } cm,\quad 153L/80u$$
G. $$\dfrac { 5 }{ \sqrt { 17 } } cm,\quad 153L/30u$$
H. $$\dfrac { 8 }{ \sqrt { 17 } } cm,\quad 53L/30u$$
I. $$\dfrac { 8 }{ \sqrt { 17 } } cm,\quad 153L/80u$$
Answer: G
Which of the following does not hold when two particles of masses $m_1$ and $m_2$ undergo elastic collision?

Options:
A. When the collision is oblique and $m_1=m_2$ with $m_2$ stationary, after the collision the particle move in opposite directions.
B. when $m_1 >> m_2$ and $m_2$ is stationary, after head on collision $m_2$ moves with twice the velocity of $m_1$
C. When $m_1 =m_2$ and $m_2$ is stationary, there is maximum transfer of kinetic energy in head an collision
D. When $m_1=m_2$ and $m_2$ is stationary , there is maximum transfer of momentum in head on collision
Answer: C
An electric dipole of moment $\overrightarrow{p}$ is placed at the origin along the x-axis. The angle made by electric field with x-axis at a point P, whose position vector makes an angle $\theta$ with x-axis, is ( where $\tan\, lpha=\dfrac{1}{2} \tan\theta$)

Options:
A. $$\theta + 3\alpha$$
B. $$\theta - \alpha$$
C. $$\theta+\alpha$$
D. $$\theta + 4\alpha$$
E. $$\theta+ 2\alpha$$
F. $$\theta - 3\alpha$$
G. $$\theta$$
H. $$\theta + 2\alpha$$
I. $$\alpha$$
J. $$\theta - 2\alpha$$
Answer: C
A solid sphere starts from rest at the top of an incline of height $h$ and length $l$ and moves down. The force of friction between the sphere and the incline is $F$. The friction is insufficient to prevent slipping. The kinetic energy of the sphere at the bottom of the incline is $K$. Then

Options:
A. $$K = mgh - \frac{1}{2}Fl$$
B. $$K=mgh-Fl$$
C. The work done by the frictional force is $Fl$
D. $$K < mgh - Fl$$
E. $$K=mgh+Fl$$
F. $$K = mgh - \frac{3}{4}Fl$$
G. $$K> mgh-Fl$$
H. $$K = mgh - 2Fl$$
I. The heat produced is $Fl$
J. $$K = mgh - \frac{3}{2}Fl$$
Answer: G
The following equations of state are occasionally used for approximate calculation on gases : ${\text{Gas}}\left( X \right):RT\left( {1 + \frac{b}{{{V_m}}}} \right)$
$$Gas\left( Y \right){\text{ : }}P\left( {{V_m} - b} \right) = RT$$
Assuming that the gas $\left( X \right){\text{ and }}\left( Y \right)$ actually obeyed above equation of state respectively then choose the correct statement:

Options:
A. Neither $Gas\left( X \right)$ nor $Gas\left( Y \right)$ is more liquefiable than the other
B. $Gas\left( Y \right)$ is less liquefiable than $Gas\left( X \right)$
C. $Gas\left( X \right)$ is more liquefiable than $Gas\left( Y \right)$
D. $Gas\left( Y \right)$ is equally liquefiable as $Gas\left( X \right)$
E. $Gas\left( Y \right)$ is more liquefiable than $Gas\left( X \right)$
F. Neither $Gas\left( X \right)$ nor $Gas\left( Y \right)$ is liquefiable
G. $Gas\left( X \right)$ is less liquefiable than $Gas\left( Y \right)$
H. $Gas\left( X \right)$ and $Gas\left( Y \right)$ both are liquefiable
Answer: C
pH of $0.01M (NH_4)_2SO_4$ and $0.02M$ $NH_4OH$ buffer $(pK_{a\ (NH_4^+)}=9.26)$ is

Options:
A. $$9.26 + log(0.25)$$
B. $$9.26 +log2$$
C. $$9.26+ log1$$
D. $$9.26 + log0.5$$
E. $$9.26 + log(1.75)$$
F. $$4.74 +log2$$
G. $$9.26 + log(2.5)$$
H. $$4.74$$
I. $$9.26 + log(1.25)$$
J. $$9.26 + log(1.5)$$
Answer: B
A transverse wave $y=0.05\sin (20\pi x-50 \pi t)$ in meters is propagating along $+ve\ X-$ axis on a string. A light insect starts crawling on the string with the velocity of $5\ cm/s$ at $t=0$ along the $+ve\ X-$ axis from a point where $x=5\ cm$. After $5\ s$ the difference in the phase of its position is equal to:

Options:
A. $$240\pi$$
B. $$150\pi$$
C. $$265\pi$$
D. $$245\pi$$
E. $$255\pi$$
F. $$-5\pi$$
G. $$250\pi$$
H. $$-245\pi$$
I. $$251\pi$$
J. $$260\pi$$
Answer: H
Pressure P varies as $P=\cfrac { lpha  }{ \beta  } exp\left( \cfrac { -lpha  }{ { K }_{ B }\theta  } Z \right) $ where Z denotes distance. ${ K }_{ B }$ Boltzman's constant.$\theta $ temperature and $lpha ,\beta$ are constants.Derive the dimensions of $\beta $

Options:
A. $$\left[ { M }^{ -1 }{ LT }^{ -2 } \right] $$
B. $$\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right] $$
C. $$\left[ { M }^{ 0 }{ L }^{ -1 }{ T }^{ -2 } \right] $$
D. $$\left[ { M }^{ 0 }{ L }^{ 1 }{ T }^{ -1 } \right]$$
E. $$\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 2 } \right]$$
F. $$\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 1 } \right]$$
G. $$\left[ { M }^{ 0 }{ L }^{ -1 }{ T }^{ -1 } \right]$$
H. $$\left[ { M }^{ 0 }{ L }^{ 2 }{ T }^{ 0 } \right] $$
I. $$\left[ { M }^{ 1 }{ L }^{ 0 }{ T }^{ 0 } \right]$$
J. $$\left[ { M }^{ 0 }{ L }^{ 1 }{ T }^{ 0 } \right]$$
Answer: B
For the following $3A \rightleftharpoons 2B+2C$
What is the relation between $K_p$ total pressure at equilibrium (P) and degree of dissociation?

Options:
A. $K_p$=$\dfrac{4P \alpha^2}{(1-\alpha)^3}$
B. $K_p$=$\dfrac{P\alpha^4}{(1-\alpha)^3}$
C. $K_p$=$\dfrac{4P \alpha^4}{(1-\alpha)^4}$
D. $K_p=\dfrac{4P \alpha^3}{9(1-\alpha)^3}$
E. $K_p$=$\dfrac{P \alpha^2}{(1-\alpha)^3}$
F. $K_p$=$\dfrac{2P \alpha^4}{(1-\alpha)^3}$
G. $K_p$=$\dfrac{P \alpha^3}{(1-\alpha)^3}$
H. $K_p$=$\dfrac{16P \alpha^4}{27(1-\alpha)^3(3+\alpha)}$
I. $K_p$=$\dfrac{4P \alpha^4}{9(1-\alpha)^3}$
J. $K_p$=$\dfrac{P \alpha^4}{2(1-\alpha)^3}$
Answer: H
At constant temperature, the equilibrium constant $ \left( \mathrm { K } _ { \mathrm { P } } \right) $ for the decomposition reaction $\mathrm { N } _ { 2 } \mathrm { O } _ { 4 } \rightleftharpoons 2 \mathrm { NO } _ { 2 }$ is expressed by $ \mathrm { K } _ { \mathrm { p } } = \left( 4 \mathrm { x } ^ { 2 } \mathrm { P } \right) / \left( 1 - \mathrm { x } ^ { 2 } \right) $, where P = final pressure at equilibrium and x = extent of decomposition till equilibrium starting with reactant only. Which one of the following statements is false?

Options:
A. Kp increases with the increase of P
B. Kp decreases with the increase of x
C. Kp remains constant with changes in x and P
D. Kp increases with the decrease of P
E. Kp decreases with the decrease of P
F. Kp remains constant with changes in P
G. Kp increases with decrease of x
H. Kp decreases with the increase of P
I. Kp increases with increases of x
J. Kp remains constant with change in P and x
Answer: A
For the cell,$Ag| AgCl| KCl(1 M)|| KBr(1M)|AgBr|Ag$. Select the correct option.

Options:
A. $$\Delta G^0_{cell}=0$$
B. $E^0_{cell} \neq 0$ & $\Delta G^0_{cell} \neq 0$
C. $E^0_{cell}=0$ & $\Delta G^0_{cell}=0$
D. $$E^0_{cell}=0$$
E. $$E^0_{cell} \neq 0$$ & $$\Delta G^0_{cell} = 0$$
F. $$E^0_{cell} \neq 0$$
G. $E^0_{cell}=0$ & $\Delta G^0_{cell} \neq 0$
Answer: C
In one Joule's experiments water was churned with a paddle wheel driven by two loads,each of mass $1kg$,each falling through an verticle height of $2.1$m. There was $7kg$ of water (including  water equivalent of calorimeter,paddle etc) to be heated. After each during churning,weight are hauled up and allowed to fall. What will be the rise in temperature after $50$ falls? Given $J=4.2J\,cal^{-1}$ and $g=9.8 ms^{-2}$

Options:
A. $$0.39^oC$$
B. $$0.5^oC$$
C. $$0.07^oC$$
D. $$0.37^oC$$
E. $$0.32^oC$$
F. $$0.65^oC$$
G. $$0.45^oC$$
H. $$0.47^oC$$
I. $$0.57^oC$$
J. $$0.43^oC$$
Answer: J
For the reaction  : $3N_{2} + H_{2} \rightarrow 2NH_{3}$,   $\triangle H = - 24 kcal$ at $427^{0}C$ and $200 atm$Calculate magnitude of internal enerrgy change in $Kcal$ ($\triangle U$), if 168 g $N_{2}$ gas and 30 g $H_{2}$ gas are allowed to react completely (100% reaction yield) to form $NH_{3}$ gas at $427^{0}C$ and 200 atm.

Options:
A. 108 kcal
B. - 102 kcal
C. - 24 kcal
D. - 110 kcal
E. 102 kcal
F. 104 kcal
G. - 21.2 kcal
H. - 120 kcal
I. 110 kcal
J. - 106 kcal
Answer: J
If the magnetic field associated with a radiation of frequency 10 MHz is $B=6 \times 10^{-8}sin (Kx-\omega t)T$ then its energy density is ..........J $m^{-3}$

Options:
A. $$1.43 \times 10^{-13}$$
B. $$1.43 \times 10^{-7}$$
C. $$1.43 \times 10^{-12}$$
D. $$1.43 \times 10^{-9}$$
E. $$1.43 \times 10^{-14}$$
F. $$1.43 \times 10^{-10}$$
G. $$1.43 \times 10^{-11}$$
H. $$1.43 \times 10^{-15}$$
I. $$1.43 \times 10^{-8}$$
J. $$1.43 \times 10^{-16}$$
Answer: D
One ole of an ideal monatomic gas undergoes a process described by the equation $P{V^n}=$ constant$.$ the heat capacity of the gas during this process is

Options:
A. $$\frac{4R}{3}$$
B. $$\frac{2R}{3}$$
C. $$3R$$
D. $$2R$$
E. $$\frac{3R}{2}$$
F. $$none\ of\ these$$
G. $$\frac{5R}{3}$$
H. \(\frac{7R}{2} \)
I. $$R$$
J. $$\frac{5R}{2}$$
Answer: I
The potential energy a particle executing SHM along x-axis is given by U = U0 (1- cos ax ) the time period of oscillation is ------

Options:
A. $$\frac {2 \pi }{ a } \sqrt { \frac { 4U0 }{ m } }$$
B. $$\frac {2 \pi }{ a } \sqrt { \frac { m }{ 3U0 } }$$
C. $$\frac {2 \pi }{ a } \sqrt { \frac { U0 }{ 2m } }$$
D. $$\frac {2 \pi }{ a } \sqrt { \frac { m }{ 2U0 } }$$
E. $$ \frac {2 \pi }{ m } \sqrt { \frac { U0 }{ a} } $$
F. $$ \frac {2 \pi }{ a } \sqrt { \frac { U0 }{ m} } $$
G. $$\frac {2 \pi }{ a } \sqrt { \frac { 2U0 }{ m } }$$
H. $$ \frac {2 \pi }{ a } \sqrt { \frac { m }{ U0} } $$
I. $$\frac {2 \pi }{ a } \sqrt { \frac { U0 }{ 3m } }$$
J. $$ \frac {2 \pi }{ a } \sqrt { \frac { 3m }{ U0 } } $$
Answer: H
A string $60\ cm$ in length rigidly held at two ends sustains a standing wave $y = A\cos kx \sin \omega t$. Find the overtone to which those oscillations corresponds, if such magnitude of displacement amplitude that its value is $\dfrac {\sqrt {3}}{2}$ of maximum, occur at a regular interval of $10\ cm$.

Options:
A. I overtone
B. III overtone
C. VI overtone
D. XII overtone
E. VII overtone
F. II overtone
G. VIII overtone
H. IV overtone
I. IX overtone
J. V overtone
Answer: H
A uniform chain of length $1m$ and mass $'m'\ kg$ is held on a smooth horizontal surface of a table with half of its length hanging over one of the edges. It is released from rest. With what velocity its end would leave the surface? (Take $g=10\ ms^{-1}$)

Options:
A. $$1.37\ ms^{-1}$$
B. $$5.44\ ms^{-1}$$
C. $$3.13\ ms^{-1}$$
D. $$3.46\ ms^{-1}$$
E. $$4.48\ ms^{-1}$$
F. $$4.96\ ms^{-1}$$
G. $$1.93\ ms^{-1}$$
H. $$3.92\ ms^{-1}$$
I. $$2.74\ ms^{-1}$$
J. $$3.98\ ms^{-1}$$
Answer: E
1 mole of an ideal diatomic gas undergoes a reversible polytropic process $  \left(P V^{2}=   constant \right  )  $ . The gas expand from initial volume of $1$ litre and temp $300  \mathrm{K}  $ to find volume $3$ lit.Calculate change in internal energy (approx)

Options:
A. $$-4.3 \mathrm{kJ}$$
B. $$-4.2 \mathrm{kJ}$$
C. $$ -4.0 \mathrm{kJ} $$
D. $$-4.9 \mathrm{kJ}$$
E. $$ -4.6 \mathrm{kJ} $$
F. $$-4.1 \mathrm{kJ}$$
G. $$-4.7 \mathrm{kJ}$$
H. $$ -4.2 k J $$
I. $$-4.5 \mathrm{kJ}$$
J. $$ -4.4 \mathrm{kJ} $$
Answer: C
A solution of 0.2 mole Kl $(lpha$ =  100%) in 1000  g water freezes at $T_1\, ^{\circ}$C. Now to this solution 0.1 mole $HgI_2$ is added and the resulting solution freezes at $T_2\, ^{\circ}$C. Which of the following is correct?

Options:
A. $$T_1 > T_2$$
B. $$T_1 < T_2$$
C. $$T_1 \leq T_2$$
D. Can't be prediced
E. $$T_1 \neq T_2$$
F. $$T_1 \geq T_2$$
G. $$T_1 = T_2$$
Answer: B
The densities of a liquid at 0$^{0}$C and 100$^{0}$C are respectively 1.0127 and 1. A specific gravity bottle is filled with 300gm of the liquid at 0$^{0}$C up to the brim and it is heated to 100$^{0}$C. Then the mass of the liquid expelled in grams is [Coefficient of linear expansion of glass =$9\times 10^{-6/0}C$]

Options:
A. $$\dfrac{3.81}{1.1127}$$
B. $$\dfrac{3.81}{1.012}$$
C. $$\dfrac{3.81}{1.2127}$$
D. $$\dfrac{3}{10.1}$$
E. $$\dfrac{3}{1.01}$$
F. $$\dfrac{3}{1.11}$$
G. $$\dfrac{3.81}{1.127}$$
H. $$\dfrac{3.81}{1.0127}$$
I. $$\dfrac{3.81}{0.0127}$$
J. $$\dfrac{3.81}{1.01}$$
Answer: E
$K_{1}$ and $K_{2}$ are the first and second ionization constant of $H_{2}CrO_{4}$, and $K_{3}$ is the dissociation constant. Match the following.

Options:
A. $$0.1\ M\ H_{2}CrO_{4} \rightarrow pH = 7 - \dfrac{1}{2}pK_{2} - \dfrac{1}{2}pK_{1}$$
B. $$0.1\ M\ H_{2}CrO_{4} \rightarrow pH = 7 - \dfrac{1}{2}pK_{2} - \dfrac{1}{2}pK_{3}$$
C. $$0.1\ M\ KHCrO_{4} \rightarrow pH = 7 + \dfrac{1}{2}pK_{2} + \dfrac{1}{2}pK_{1}$$
D. $$0.1\ M\ H_{2}CrO_{4} \rightarrow pH = 7 - \dfrac{1}{2}pK_{1} - \dfrac{1}{2}pK_{2}$$
E. $$0.1\ M\ K_{2}CrO_{4} \rightarrow pH = \dfrac{1}{2}(pK_{1} + pK_{2})$$
F. $$0.1\ M\ (NH_{4})_{2}CrO_{4} \rightarrow pH = \dfrac{1}{2}pK_{3} - \dfrac{1}{2}\log C$$
G. $$0.1\ M\ H_{2}CrO_{4} \rightarrow pH = 7 - \dfrac{1}{2}pK_{2} + \dfrac{1}{2}pK_{3}$$
H. $$0.1\ M\ H_{2}CrO_{4} \rightarrow pH = 7 + \dfrac{1}{2}pK_{2} - \dfrac{1}{2}pK_{1}$$
I. $$0.1\ M\ H_{2}CrO_{4} \rightarrow pH = 7 + \dfrac{1}{2}pK_{1} - \dfrac{1}{2}pK_{2}$$
J. $$0.1\ M\ H_{2}CrO_{4} \rightarrow pH = 7 - \dfrac{1}{2}pK_{2} + \dfrac{1}{2}pK_{1}$$
Answer: H
A gas is enclosed in a cylinder with a movable friction less piston. Its initial thermodynamic state at pressure ${ { P }_{ i }={ 10 }^{ 5 } }\ Pa$ and volume ${ V }_{ i }={ 10 }^{ -3 }{ m }^{ 3 } $ changes to a final state as ${ P }_{ j }=\left( { 1 }/{ 32 } \right) { \times 10 }^{ 5 }\ pa$ and ${ V }_{ j }=8\times { 10 }^{ -3 }{ m }^{ 3 }$ in an adiabatic quasi-static process, such that ${ P }^{ 3 }{ V }^{ 5 }=constant$. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps : an isobaric expansion at ${ P }_{ i }$ followed by an isochoric(isovolumetric) process at volume ${ V }_{ j }$. The amount of heat supplied to the system in the two-step process is approximately.

Options:
A. $$588\ J$$
B. $$635\ J$$
C. $$294\ J$$
D. $$397\ J$$
E. $$712\ J$$
F. $$112\ J$$
G. $$456\ J$$
H. $$813\ J$$
I. $$742\ J$$
Answer: H
For which of the following compound, the formula $\alpha=\lambda_v/\lambda_{\infty}$ does not hold good?

Options:
A. \(H_2O\)
B. $$HCl$$
C. $$CH_3NH_2$$
D. \(H_2SO_4\)
E. \(HCl\)
F. H_2CO_3
G. $$CH_3COOH$$
H. $$HClO_4$$
I. CH_3COONa
J. \(HNO_3\)
Answer: H
A point mass oscillates along the x-axis according to the law $x = {x_0}cos\left( {\omega t - \frac{\pi }{4}} \right)$ acceleration of the particle is written as $a = Acos\left( {\omega t + \delta } \right)$, then

Options:
A. $$A = {X_0}{\omega},\delta = \frac{\pi}{4}$$
B. $$A = {X_0},\delta = \frac{3\pi}{4}$$
C. $$A = {X_0}{\omega},\delta = 0$$
D. $$A = {X_0},\delta = \frac{\pi }{4}$$
E. $$A = {X_0}{\omega ^2},\delta = \frac{{3\pi }}{4}$$
F. $$A = {X_0},\delta = \frac{\pi}{2}$$
G. $$A = {X_0}{\omega ^2},\delta = - \frac{\pi }{4}$$
H. $$A = {X_0}{\omega},\delta = \frac{\pi}{2}$$
I. $$A = {X_0}{\omega ^2},\delta = 0$$
J. $$A = {X_0}{\omega ^2},\delta = \frac{\pi }{4}$$
Answer: D
The slopes of the windscreen of two cars are $lpha_1=30^o$ and $lpha_2=15^o$ respectively. At what ratio $\dfrac {v_1}{v_2}$ of the velocities of the cars will their drivers see the hailstones bounced back by the windscreen on their cars in the vertical direction? Assume hailstones fall vertically downwards and collisions to be elastic.

Options:
A. $$4$$
B. $$\dfrac{1}{3}$$
C. $$\dfrac{5}{3}$$
D. $$2$$
E. $$\dfrac{1}{2}$$
F. $$\dfrac{2}{3}$$
G. $$1$$
H. $$3$$
I. \dfrac{3}{2}
J. $$\dfrac{4}{3}$$
Answer: H