MikaStars39/nano-eval huggingface dataset:
When a charge of amount Q is given to an isolated metal plate X of surface area A, its surface charge density becomes $\sigma _1$. When an isolated identical plate  Y  is brought close to X, the surface charge density on X becomes $\sigma_2$. When y is earthed, the surface charge density becomes $\sigma_3$.

Options:
A. $$\sigma_3=\dfrac{Q}{A}$$
B. $$\sigma_3=\dfrac{Q}{4A}$$
C. $$\sigma_1=\dfrac{Q}{A}$$
D. $$\sigma_3=\dfrac{Q}{3A}$$
E. $$\sigma_3=\dfrac{Q}{2A}$$
F. $$\sigma_3=\dfrac{Q}{6A}$$
G. $$\sigma_3=\dfrac{2Q}{A}$$
H. $$\sigma_1=\dfrac{Q}{2A}$$
I. $$\sigma_3=\dfrac{Q}{5A}$$
J. $$\sigma_1=\sigma_2$$
Answer: A
A massive stone pillar $20m$ high and of uniform cross section rests on a rigid base and supports a vertical load of $5.0\times { 10 }^{ 5 }N$ at its upper end. If the compressive stress in the pillar is not exceed $16\times { 10 }^{ 6 }N/m$, what is the minimum cross-sectional area of the pillar? (Density of the stone$=2.5\times { 10 }^{ 3 }kg/{ m }^{ 3 }$ take $g=10N/kg$

Options:
A. 0.55{ m }^{ 2 }
B. 0.30{ m }^{ 2 }
C. 0.50{ m }^{ 2 }
D. $$0.15{ m }^{ 2 }$$
E. 0.32{ m }^{ 2 }
F. 0.42{ m }^{ 2 }
G. $$0.45{ m }^{ 2 }$$
H. $$0.35{ m }^{ 2 }$$
I. 0.40{ m }^{ 2 }
J. $$0.25{ m }^{ 2 }$$
Answer: G
For a particular reaction at equilibrium $\Delta H^{o}=-38.3\ KJ$ and $\Delta S^{o}=-113\ JK^{-1}$. This reaction is

Options:
A. spontaneous at all temperature above $339\ K$
B. spontaneous at all temperature above 338 K
C. non-spontaneous at all temperatures
D. spontaneous at all temperature below $336\ K$
E. spontaneous at all temperature below $337\ K$
F. spontaneous at all temperature below $339\ K$
G. spontaneous at all temperature
H. spontaneous at all temperature above $340\ K$
I. spontaneous at all temperatures above 336 K
J. spontaneous at all temperature below $338\ K$
Answer: G
Two speakers are driven by the same oscillator of frequency $f$. They are located a distance $d$ from each other on a vertical pole. An observer far away walks slowly, straight towards the lower speaker and his ears are at the same level as the lower speaker. Let $v$ represent the speed of sound and assume that there is no reflection from the ground.Let $n_{max}$ be the number of times the observer hears the minimum in sound intensity and let $\lambda = v/f$ be wavelength of the sound emitted. Choose the correct alternative(s).

Options:
A. A minimum occurs at $\dfrac {4d^{2} - 2\lambda^{2}}{4\lambda}$
B. A minimum occurs at $\dfrac {4d^{2} - \lambda^{2}}{8\lambda}$
C. A minimum occurs at $\dfrac {4d^{2} + \lambda^{2}}{4\lambda}$
D. A minimum occurs at $\dfrac {4d^{2} - \lambda^{2}}{6\lambda}$
E. A minimum occurs at $\dfrac {4d^{2} - 3\lambda^{2}}{4\lambda}$
F. A minimum occurs at $\dfrac {4d^{2} + \lambda^{2}}{8\lambda}$
G. $$n_{max} \leq \dfrac {d}{\lambda} - 0.5$$
H. A minimum occurs at $\dfrac {4d^{2} + 9\lambda^{2}}{12\lambda}$
I. A minimum occurs at $\dfrac {4d^{2} - \lambda^{2}}{4\lambda}$
Answer: I
A man of mass 50 kg falls freely from a height of 40 m into a swimming pool and just come to rest at the bottom of the pool. Assume that the average upward force on the man due to water is 1000 N. If depth of water in the pool is $d$, then $d$ is $(g=10 \ m/s^2)$

Options:
A. 5 m
B. 30 m
C. 15 m
D. 40 m
E. 32 m
F. 35 m
G. 20 m
H. 45 m
I. 10 m
J. 25 m
Answer: G
Mark the correct relationship from the following.

Options:
A. Nernst equation is $E_{cell}=E^o_{cell}+\dfrac{0.0591}{n}log\dfrac{[Products]}{[Reactants]}$
B. Nernst equation is $E_{cell}=E^o_{cell}+\dfrac{0.0591}{n}log\dfrac{[Reactants]}{[Products]}$
C. EMF of a cell $Zn|Zn^{2+}_{(a_1)}||Cu^{2+}_{(a_2)}|Cu$ is $E=E^o-\dfrac{0.591}{n}log\dfrac{[a_2]}{[a_1]}$
D. Nernst equation is $E_{cell}=E^o_{cell}+\dfrac{0.591}{n}log\dfrac{[Products]}{[Reactants]}$
E. Equilibrium constant is related to emf as log K$=\dfrac{nFE}{2.303RT}$
F. Nernst equation is $E_{cell}=E^o_{cell}-\dfrac{0.0591}{n}log\dfrac{[Products]}{[Reactants]}$
G. Nernst equation is $E_{cell}=E^o_{cell}-\dfrac{0.0591}{n}log\dfrac{[Reactants]}{[Reactants]}$
H. Nernst equation is $E_{cell}=E^o_{cell}+\dfrac{0.591}{n}log\dfrac{[Reactants]}{[Products]}$
I. Nernst equation is $E_{cell}=E^o_{cell}-\dfrac{0.0591}{n}log\dfrac{[Reactants]}{[Products]}$
J. For the electrode $M^{n+}/M$ at $273K,$ $E=E^o+\dfrac{0.0591}{n}log[M^{n+}]$
Answer: F
Upon mixing equal volume of $0.02$$\mathrm { M }$ $\mathrm { AgNO } _ { 3 }$ & $0.16$$\mathrm { M }$ $\mathrm { KCN }$ solutions: (Assume no hydrolysis of any ions). Given $\mathrm { Ag } ^ { + } ( \mathrm { aq } . ) + 2 \mathrm { CN } ^ { - } ( \mathrm { aq } ) \rightleftharpoons \left[ \mathrm { Ag } ( \mathrm { CN } ) _ { 2 } \right] ^ { - } ( \mathrm { aq } )$ $K _ { C } = 10 ^ { 18 }$

Options:
A. \left[ \mathrm { NO } _ { 3 } ^ { - } \right] = 0.01 \mathrm { M } ; \left[ \mathrm { K } ^ { + } \right] = 0.10 \mathrm { M }
B. \left[ \mathrm { NO } _ { 3 } ^ { - } \right] = 0.01 \mathrm { M } ; \left[ \mathrm { K } ^ { + } \right] = 0.14 \mathrm { M }
C. $$\left[ \mathrm { Ag } ( \mathrm { CN } ) _ { 2 } ^ { - } \right] = 0.01 \mathrm { M }$$
D. \left[ \mathrm { NO } _ { 3 } ^ { - } \right] = 0.01 \mathrm { M } ; \left[ \mathrm { K } ^ { + } \right] = 0.12 \mathrm { M }
E. \left[ \mathrm { NO } _ { 3 } ^ { - } \right] = 0.02 \mathrm { M } ; \left[ \mathrm { K } ^ { + } \right] = 0.16 \mathrm { M }
F. $$[ \mathrm { CN } ] = 0.06 \mathrm { M }$$
G. All of these
H. \left[ \mathrm { NO } _ { 3 } ^ { - } \right] = 0.01 \mathrm { M } ; \left[ \mathrm { K } ^ { + } \right] = 0.16 \mathrm { M }
I. \left[ \mathrm { NO } _ { 3 } ^ { - } \right] = 0.02 \mathrm { M } ; \left[ \mathrm { K } ^ { + } \right] = 0.08 \mathrm { M }
J. $$\left[ \mathrm { NO } _ { 3 } ^ { - } \right] = 0.01 \mathrm { M } ; \left[ \mathrm { K } ^ { + } \right] = 0.08 \mathrm { M }$$
Answer: J
In the system shown, the wire connecting two masses has linear mass density of $\dfrac{1}{20}kg/m$. A tuning fork of $50 Hz$ is found to be in resonance with the horizontal part of wire between pulley and block

Options:
A. 1 : \sqrt{2} - \frac{1}{2}
B. $$2 : 1$$
C. 1 : \sqrt{2} - 3
D. 1 : \sqrt{3} - 1
E. 1 : \sqrt{2} + 1
F. $$1 :\sqrt{2}-1$$
G. 1 : \sqrt{2} - 2
H. $$1 : 2$$
I. 1 : \sqrt{2} + 2
J. $$1 :\sqrt{2}$$
Answer: F
A conducting rod of length $l$ and mass $m$ is moving down a smooth inclined plane of inclination $\theta $ with constant velocity $V$. A current $i$ is flowing in the conductor in a direction perpendicular to paper inwards. The magnitude of field $\vec{B}$ is :(Given acceleration due to gravity $=g$)

Options:
A. $$\dfrac{mg}{il}\cos\theta$$
B. $$\dfrac{mg\cos\theta }{il}$$
C. $$\dfrac{mg\sin\theta }{il}$$
D. \dfrac{mg\sin\theta }{il\cos\theta}
E. \dfrac{mg\cos\theta}{il\sin\theta}
F. $$\dfrac{mg}{il}\tan\theta $$
G. $$\dfrac{mg }{il\sin\theta }$$
H. $$\dfrac{mg\sin\theta }{il\sin\theta}$$
I. \dfrac{mg}{il\cos\theta}
J. $$\dfrac{mg}{il}\sin\theta $$
Answer: F
Two point charges (Q each) are placed at (0, y) and (0, -y). A point charge q of the same polarity can move along the x-axis. Then :

Options:
A. for any position of q other than origin, the force is directed along the x-axis
B. for any position of q other than origin, the force is directed towards the origin
C. for any position of q other than origin, the force is directed away from origin
D. for any position of q other than origin, the force is directed away from the y-axis
E. the charge q is in equilibrium at the origin
F. for any position of q other than origin, the force is directed towards the x-axis
G. the charge q performs an oscillatory motion about the origin
H. for any position of q other than origin, the force is directed along the y-axis
I. the charge q performs a linear motion away from the origin
J. the force on q is maximum at $x = \pm y / \sqrt{2}$
Answer: C
The system is released from rest with the spring initially stretched $75$ $mm$. Calculate the velocity $v$ of the block after it has dropped $12$ $mm$. The spring has a stiffness of $1050$ $N/m$. Neglect the mass of the small pulley.

Options:
A. $$\displaystyle v= 0.571ms^{-1}$$
B. v= 0.321ms^{-1}
C. v= 0.471ms^{-1}
D. v= 0.411ms^{-1}
E. v= 0.421ms^{-1}
F. $$\displaystyle v= 0.471ms^{-1}$$
G. v= 0.551ms^{-1}
H. $$\displaystyle v= 0.371ms^{-1}$$
I. v= 0.401ms^{-1}
J. $$\displaystyle v= 0.521ms^{-1}$$
Answer: H
A pendulum of mass  $1{ kg }$  and length  $l = 1  { m }$  is released from rest at angle  $\theta = 60 ^ { \circ }.$  The power delivered by all the forces acting on the bob at angle  $\theta = 60 ^ { \circ }$  will be  $\left(  { g } = 10  { m } / { s } ^ { 2 } \right).$

Options:
A. $$26.6 W$$
B. 14.4 W
C. 15.4 W
D. $$13.4 W$$
E. $$20.4 W$$
F. 12.4 W
G. 16.4 W
H. 19.4 W
I. zero
J. 18.4 W
Answer: D
A vessel X contains 1 mole of O_2 gas (molar mass 32) at a temperature T and pressure P. Another identical vessel Y contains one mole of He gas (molar mass 4) at temperature 2 T, then

Options:
A. pressure in the container Y is 8P
B. kinetic energy of O_2 molecules = kinetic energy of He mmolecules
C. pressure in the container Y is 2P
D. Pressure in the container Y is P/8
E. kinetic energy of He molecules = $\dfrac { 8 }{ 5 } $ (kinetic energy of O_2 molecules)
F. pressure in the container Y is 3P
G. pressure in the container Y is P/16
H. pressure in the container Y is P/4
I. pressure in the container Y is 4P
J. pressure in the container Y is P/2
Answer: C
Find the stress developed inside a tooth cavity filled with copper when hot tea at a temperature of $57^\circ \text{C}$ is drunk. (Take the temperature of the tooth to be $37^\circ \text{C}$, $\alpha = 1.7 \times 10^{-5} {^\circ \text{C}}^{-1}$, and bulk modulus for copper $= 140 \times 10^9 \, \text{N/m}^2$)

Options:
A. $$1.37 \times { 10 }^{ 8 }N{ m }^{ -2 }$$
B. $$1.53 \times { 10 }^{ 8 }N{ m }^{ -2 }$$
C. $$ 2.12\times { 10 }^{ 4 }N{ m }^{ -2 } $$
D. $$1.33 \times { 10 }^{ 8 }N{ m }^{ -2 }$$
E. $$1.52 \times { 10 }^{ 8 }N{ m }^{ -2 }$$
F. $$ 4.13\times { 10 }^{ 8 }N{ m }^{ -2 } $$
G. $$1.32 \times { 10 }^{ 8 }N{ m }^{ -2 }$$
H. $$1.24 \times { 10 }^{ 8 }N{ m }^{ -2 }$$
I. $$ 3.12\times { 10 }^{ 4 }N{ m }^{ -2 } $$
J. $$ 1.43\times { 10 }^{ 8 }N{ m }^{ -2 } $$
Answer: J
A fixed mass of gas is taken through a process $A\rightarrow B\rightarrow C\rightarrow A$. Here $A\rightarrow B$ is isobaric $B\rightarrow C$ is adiabatic and $C\rightarrow A$ is isothermal
The pressure at $C$ is given by $(\gamma=1.5)$

Options:
A. \frac{10^5}{48}\,N/m^2
B. \frac{10^5}{96}\,N/m^2
C. $${10}^5\,N/m^2$$
D. $$\frac{10^5}{64}\,N/m^2$$
E. \frac{10^5}{16}\,N/m^2
F. zero
G. \frac{10^5}{128}\,N/m^2
H. $$\frac{10^5}{32}\,N/m^2$$
I. \frac{10^5}{80}\,N/m^2
Answer: D
$E^{0}_{Zn^{2+}/Zn}=-0.76\ V$. The EMF of the cell 
$Zn/Zn_{(1\ M)}^{2+}||HCl(pH=2)|H_{2(1\ atm)}$, $Pt$ is

Options:
A. 0.920\ V
B. 0.760\ V
C. $$0.878\ V$$
D. 0.761\ V
E. $$-0.878\ V$$
F. 0.778\ V
G. 0.702\ V
H. $$0.701\ V$$
I. $$0.642\ V$$
J. 0.762\ V
Answer: C
A horizontal rod length $1$ m is rotated about a vertical axis passing through one of its ends. The number of rev/sec at which the rod breaks is ( Breaking stress of material of rod $3 \times 10^9 N/m^2$ and density of material of rod $= 6000 kg/m^3$)

Options:
A. $$880$$
B. 510
C. 360
D. $$390$$
E. $$420$$
F. 480
G. 630
H. 540
I. $$1000$$
J. $$159$$
Answer: E
A bead of mass $m$ moves with constant speed $v_0$ along a thin and smooth wire bent to form a curve defined as $y=(0.5)x^2$, as shown in the figure.The wire lies in a horizontal plane. The impulse imparted on the bead while it moves from point $A$ $(\sqrt{3m},1.5m)$ to point  $B$ $(-\sqrt{3m},1.5m)$is

Options:
A. \dfrac{2mv_o}{\sqrt3}
B. $$\sqrt3 mv_o$$
C. \dfrac{mv_0}{\sqrt{3}}
D. \dfrac{4mv_o}{\sqrt3}
E. $$\dfrac{mv_o}{\sqrt3}$$
F. \dfrac{mv_o}{\sqrt{2}}
G. $$mv_o$$
H. $$zero$$
I. \dfrac{3mv_o}{\sqrt3}
Answer: B
Calculate solubility of $Pb{I}_{2}$ (${K}_{sp}=1/4\times {10}^{-8}$) in water at ${25}^{o}$, which is $90$% dissociated.

Options:
A. 2.3 \times 10^{-3} \ M
B. $$7.5 \times 10^{-4} \ M$$
C. 1.8 \times 10^{-3} \ M
D. 1.5 \times 10^{-3} \ M
E. $$1.6 \times 10^{-3} \ M$$
F. $$8.2 \times 10^{-3} \ M$$
G. 1.7 \times 10^{-3} \ M
H. $$3.4 \times 10^{-3} \ M$$
I. 1.4 \times 10^{-3} \ M
J. 1.9 \times 10^{-3} \ M
Answer: E
A buffer solution $0.04$M in $Na_2HPO_4$ and $0.02$M in $Na_3PO_4$ is prepared. The electrolytic oxidation of $1.0$ milli-mole of the organic compound RNHOH is carried out in $100$ mL of the buffer. The reaction is      $RNHOH+H_2O \rightarrow RNO_2+4H^{+}+4e^{-}$The approximate pH of solution after the oxidation is complete is:[Given: for $H_3PO_4$, $pK_{a1}=2.2$; $pK_{a2}=7.20$; $pK_{a3}=12$]

Options:
A. $$7.5$$
B. $$7.70$$
C. $$None of these$$
D. $$6.90$$
E. $$7.45$$
F. $$7.40$$
G. $$7.10$$
H. $$7.30$$
I. $$7.60$$
J. $$7.20$$
Answer: A
How many moles of sulphuric acid must be dissolved to produce 250 mL of an aqueous solution of $pH = 3.52$? Assume complete dissociation.

Options:
A. 1.5 \times 10^{-4}
B. $$3.75 \times 10^{-5}$$
C. $$3.75 \times 10^{-6}$$
D. $$3.75 \times 10^{-4}$$
E. $$7.5 \times 10^{-5}$$
F. 7.5 \times 10^{-6}
G. $$7.5 \times 10^{-4}$$
H. 7.5 \times 10^{-3}
Answer: E
Two identical positive point charges $Q$ each fixed at a distance $d = 2a$ apart in air. A point charge $q$ lies at mid-point on the joining the charges. Now $q$ executes simple harmonic motion, if:
(i) $q$ is positive and it is given a small displacement along the line joining the charges. 
(ii) $q$ is negative and it is given a small displacement. If frequencies of oscillations are $n_a$ and $n_b$, in case (i) and case (ii) then the ratio $\dfrac{\sqrt 2 n_a}{n_b}$ is:

Options:
A. $$3$$
B. $$6$$
C. $$5$$
D. $$2$$
E. $$4$$
F. 10
G. 12
H. $$8$$
Answer: B
A magnetic field $B={ B }_{ 0 }\hat { i } $ exists in a region . Four conducting wires , each in the shape of a quartes circle of radius a and centre at the origin , are placed such that they join points A (-a,0,0) B (0,a,0), C(0,0,a) D(a,0,0) and E(0,-a,0) successively . They carry current I from A to E throught B,C and D . The net force on this wire mesh is

Options:
A. $$-\left( 2la{ B }_{ o } \right) \hat { i } $$
B. $$\left( \sqrt { 2la{ B }_{ o } } \right) \hat { j } $$
C. $$-\left( la{ B }_{ o } \right) \hat { j } $$
D. $$-\left( 2la{ B }_{ o } \right) \hat { k }$$
E. $$-\left( \sqrt{la{ B }_{ o }} \right) \hat { k }$$
F. $$\left( la{ B }_{ o } \right) \hat { k } $$
G. $$-\left( \frac{3}{2}la{ B }_{ o } \right) \hat { j }$$
H. $$-\left( 2la{ B }_{ o } \right) \hat { j }$$
I. $$-\left( \sqrt{la{ B }_{ o }} \right) \hat { j }$$
J. $$-\left( \sqrt { 2la{ B }_{ o } } \right) \hat { k } $$
Answer: C
Two blocks A and B of masses m and 2m respectively placed on a smooth floor are connected by a spring. A third body C of mass m moves with velocity $v_0$ along the line joining A and B and collides elastically with

Options:
A. the common velocity of A and B at time t$_0$ is v/5.
B. the common velocity of A and B at time t$_0$ is v/7.
C. the common velocity of A and B at time t$_0$ is 3v/5.
D. the common velocity of A and B at time t$_0$ is v/3.
E. the common velocity of A and B at time t$_0$ is 2v/3.
F. the spring constant is k $= \displaystyle \frac{3mv_0^2}{2x_0^2}$.
G. the common velocity of A and B at time t$_0$ is v/4.
H. the common velocity of A and B at time t$_0$ is v/6.
I. none of these
J. the spring constant is k $= \displaystyle \frac{2mv_0^2}{3x_0^2}$.
Answer: D
A particle is moving with constant angular acceleration $=\alpha$ in a circular path of radius $\sqrt {3}m$. At $t = 0$, it was at rest and at $t = 1\sec$, the magnitude of its acceleration becomes $\sqrt {6} m/s^{2}$, then $\alpha$ is

Options:
A. $$2\ rad/s^{2}$$
B. $$\sqrt {2}\ rad/s^{2}$$
C. 3\ rad/s^{2}
D. \sqrt {3}\ rad/s^{2}
E. 1\ rad/s^{2}
F. 4\ rad/s^{2}
G. $$\sqrt {5}\ rad/s^{2}$$
Answer: B
A ,man of mass $m=60\ kg$ is standing on weighing machine fixed on a triangular wedge of angle $\theta ={ 60 }^{ \circ  }$ as shown in the figure. The wedge is moving up with an upward acceleration $a=2{ m/s }^{ 2 }.$ The weight registered by machine is 180x N, then the value of x is

Options:
A. 1200\ N
B. 1620\ N
C. 1320\ N
D. 1080\ N
E. $$360\ N$$
F. $$1440\ N$$
G. 1800\ N
H. 1560\ N
I. $$240\ N$$
J. $$600\ N$$
Answer: E
Calculate $\Delta H^0_{f}$ for $ UBr_4$ from $\Delta G^0$ of reaction and the $S^0$ values.$ U(s)+2Br_2(l)\rightarrow UBr_4(s) ; \, \Delta G^0=-788.6 KJ; \, S^0(J/K-mol) 50.3 , 152.3 , 242.6$

Options:
A. -798.1 KJ/mol
B. -755.1 KJ/mol
C. -841.2 KJ/mol
D. -804.3 KJ/mol
E. -768.4 KJ/mol
F. -787.4 KJ/mol
G. -781.2 KJ/mol
H. -822.1 KJ/mol
I. -801.4 KJ/mol
J. -814.2 KJ/mol
Answer: B
Two particles A and B are revolving with constant angular velocity on two concentric circles of radius $1  m$ and $2  m$ respectively. The positions of the particles at $t = 0$ are shown in figure. If $m_{A} = 2  kg$, $m_{B} = 1  kg$ and $P_{A}$ and $P_{B}$  are linear momentum of the particles then what is the maximum value of $\left |P_{A} + P_{B}  \right |$ in $kgm/s$ in subsequent motion of the two particles.

Options:
A. $$2$$
B. $$7$$
C. $$5$$
D. 6
E. 9
F. 4
G. $$1$$
H. 3
I. $$8$$
Answer: G
Valence bond theory describes the bonding in complexes in terms of coordinate-covalent bonds resulting from the overlap of filled ligand orbitals with vacant metal hybrid orbitals. This theory explains the magnetic behavior and geometrical shape of coordination compounds. The magnetic moment of a complex compound can be determined experimentally and theoretically by using spin only formula.Magnetic moment = $\sqrt{n(n+2)}$ B.M. (where n = No. of unpaired electrons}.The value of spin only magnetic moment for an octahedral complex of the following configuration is 2.84 B.M. The correct statement is :

Options:
A. $d^4$ (in weak field and in strong field ligand)
B. $d^3$ (in weak field and in strong field ligand)
C. $d^2$ (in weak field and in strong field ligand)
D. $d^2$ (in weak field ligand)
E. $d^3$ (in strong field ligand)
F. $d^5$ (in weak field and in strong field ligand)
G. $d^4$ (in weak field ligand)
H. $d^5$ (in strong field ligand)
I. $d^2$ (in strong field ligand)
J. $d^4$ (in strong field ligand)
Answer: C
A man of mass M stands on the floor of box of mass m as shown in figure. he raises himself and the box with an acceleration a= g/ 3 by means of a rope going over a fixed frictionless pulley. if the mass of the rope is negligible compared to (M + m ) and if M= 2 m , the tension in the rope will be

Options:
A. $$ \frac {2 mg}{3} $$
B. $$\frac{3 mg}{4}$$
C. $$\frac{5 mg}{3}$$
D. $$\frac{3 mg}{2}$$
E. $$\frac{7 mg}{4}$$
F. $$ \frac {4 mg}{3} $$
G. $$\frac{5 mg}{2}$$
H. $$2 mg$$
I. $$\frac{4 mg}{2}$$
Answer: H
When the rms voltage ${V_{_L}}$,${V_{_C}}$and ${V_{_R}}$ are measured respectively across the inductor L, the capacitor C and the resistor R in a series LCR circuit connected to an AC source is $100$ V, the ${V_{_R}}$ is close to [Given$\sqrt {10}  = 3]$]

Options:
A. 80 V
B. $90$ V
C. 60 V
D. $50$ V
E. 100 V
F. 70 V
G. 110 V
Answer: B
For the equilibrium, $ N_{2}O_{4} \leftrightharpoons 2NO_{2}; (G^{0}_{N_{2}O_{4}})_{298}$ = 100 kJ/mole and $(G^{0}_{NO_{2}})_{298}$ = 50 kJ/mole. When 5 mole / lit is each taken, the value of $\Delta G $ for the reaction.

Options:
A. -43.9 $kJ.mol^{-1}$
B. -42.9 $kJ.mol^{-1}$
C. -50 $kJ.mol^{-1}$
D. -49 $kJ.mol^{-1}$
E. -39.9 $kJ.mol^{-1}$
F. 38.9 $kJ.mol^{-1}$
G. -41.9 $kJ.mol^{-1}$
H. 30.99 $kJ.mol^{-1}$
I. -45.9 $kJ.mol^{-1}$
J. -40.9 $kJ.mol^{-1}$
Answer: C
A horizontal platform vibrates with simple harmonic motion in the horizontal direction with a period of $2$ sec. A body of mass $0.5\;Kg$ is placed on the platform. The co-efficient of static friction between the body and the platform is $0.3$. What is the maximum frictional force on the body when platform oscillating with an amplitude $0.2\;m$ ? Assume $\pi \cong 10=g$

Options:
A. $$0.75\;N$$
B. $$1\;N$$
C. $$0.8\;N$$
D. $$1.5\;N$$
E. $$2\;N$$
F. $$0.9\;N$$
G. $$0.5\;N$$
H. $$1.25\;N$$
I. $$0.6\;N$$
J. $$1.1\;N$$
Answer: B
For the reaction ${ CH }_{ 3 }COOH(I)+{ C }_{ 2 }{ H }_{ 5 }OH(I)+{ H }_{ 2 }O(I)$, 1 mole each of acid and alcohol were taken initially at equilibrium. How many moles of ester will be formed (${ K }_{ c }=4$)

Options:
A. $$1 mol$$
B. $$\frac{4}{5} mol$$
C. $$2 mole$$
D. $$\frac{1}{4} mol$$
E. $$\frac{1}{2} mol$$
F. $$\frac{3}{5} mol$$
G. $$\frac{3}{4} mol$$
H. $$\frac { 1 }{ 3 } mol$$
I. $$\frac{2}{5} mol$$
J. $$\frac { 2 }{ 3 } mol$$
Answer: J
The earth's magnetic field lines resemble that of a dipole at the centre of the earth. if the magnetic moment of this dipole is close to $8\times 10^{22}\ Am^{2}$, the value of earth's magnetic field near the equator is close to (radius of the earth $=6.4\times 10^{6}\ m$ )

Options:
A. 0.76 Gauss
B. $1.2$ Gauss
C. 0.72
D. $1.8$ Gauss
E. $0.56$ Gauss
F. $0.6$ Gauss
G. 0.65
H. 0.52 Gauss
I. $0.32$ Gauss
J. 0.48
Answer: F
Three similar oscillators, A, B, C have the same small damping constant $r$, but different natural frequencies $\omega_0 = (k/m)^{\frac{1}{2}} : 1200 Hz, 1800 Hz, 2400 Hz$. If all three are driven by the same source at $1800 Hz$, which statement is correct for the phases of the velocities of the three?

Options:
A. $$\phi_A < \phi_B = \phi_c$$
B. $$\phi_A = \phi_B > \phi_c$$
C. $$\phi_A = \phi_B = \phi_c$$
D. $$\phi_A < \phi_B = 0 < \phi_c$$
E. $$\phi_A = \phi_B < \phi_c$$
F. $$\phi_A > \phi_B < \phi_c$$
G. $$\phi_A > \phi_B = 0 > \phi_c$$
H. $$\phi_A > \phi_B > 0 > \phi_c$$
I. $$\phi_A < \phi_B < \phi_c$$
Answer: C
If $\alpha $  moles of a monoatomic gas are mixed with  $\beta $  moles  of a polyatomic gas and  mixture behaves like  diatomic gas, then  [ neglect the vibrational mode of freedom )

Options:
A. $$\alpha = -\frac{5}{2}\beta$$
B. \alpha = -4\beta
C. $$\alpha = -5\beta$$
D. $$2\alpha =\beta $$
E. $$\alpha = -6\beta$$
F. $$3\alpha =-\beta $$
G. $$\alpha = -2\beta$$
H. $$\alpha =2\beta $$
I. $$3\alpha =\beta $$
J. $$\alpha = -\frac{7}{2}\beta$$
Answer: D
A uniform rod AB of mass m and length l at rest on a smooth horizontal surface. an impulse P is applied to the end B . the angular velocity of rod just after the impact is

Options:
A. \omega = \frac{8P}{ml}
B. $$\omega = \frac{3P}{ml}$$
C. $$ \omega = \frac {7P}{ml} $$
D. $$ \omega = \frac {6P}{ml} $$
E. $$\omega = \frac{4P}{ml}$$
F. $$ \omega = \frac {36P}{ml} $$
G. $$ \omega = \frac {2P}{ml} $$
H. $$\omega = \frac{9P}{ml}$$
I. $$\omega = \frac{12P}{ml}$$
J. $$\omega = \frac{5P}{ml}$$
Answer: D
The earth's magnetic field at some place on magnetic equator of the earth is $0.5\times { 10 }^{ -4 }T$. Consider the radius of the earth at that place as $6400 km$. Then, magnetic dipole moment of the earth is ___________ ${ Am }^{ 2 }$. $\left( { \mu  }_{ 0 }=4\pi \times { 10 }^{ -7 }Tm{ A }^{ -1 } \right) $

Options:
A. 1.56 \times { 10 }^{ 23 }
B. 1.48 \times { 10 }^{ 23 }
C. 1.64 \times { 10 }^{ 23 }
D. $$1.05\times { 10 }^{ 23 }$$
E. $$1.15\times { 10 }^{ 23 }$$
F. 1.65 \times { 10 }^{ 23 }
G. 1.58 \times { 10 }^{ 23 }
H. $$1.31\times { 10 }^{ 23 }$$
I. $$1.62\times { 10 }^{ 23 }$$
J. 1.45 \times { 10 }^{ 23 }
Answer: I
The kinetic energy k of a particle moving a circle of radius R depends on the distance covered s as K= $as^{2}$ where a is a constant.  The total force acting on the particle as a function of distance is -

Options:
A. 2a\frac{s}{R}
B. 2a s
C. 2a\frac{s^{2}}{6R}
D. $$2as\left ( 1+\frac{s^{2}}{R^{2}} \right )^{1/2}$$
E. 2a\frac{s^{2}}{3R}
F. None of these
G. 2a\frac{s^{2}}{2R}
H. $$2a\frac{s^{2}}{R}$$
I. 2a\frac{s^{2}}{4R}
J. 2a\frac{s^{2}}{5R}
Answer: H
When photons of energy $4.25$ eV strike the surface of a metal A, the ejected photoelectrons have maximum kinetic energy, $T_A$( expressed in eV) and de Broglie wavelength $1_A$. The maximum kinetic energy of photoelectrons liberated from another metal B by photons of energy $4.20 V$ is $T_B=T_A- 1.50eV$. If the de-Broglie wavelength of these photoelectons is $I_B= 2 I_A$, then which is not correct?

Options:
A. $$T_B= 2.75 eV$$
B. The work function of B is $3.70 $ eV
C. $$T_A= 2.000eV$$
D. The work function of A is $2.25$ eV
Answer: D
A parallel plate capacitor $C$ with plates of unit area and separation $d$ is filled with of dielectric constant $K=2$. The level of liquid decreases at a constant speed $V$, the time constant as a function of time $t$ is

Options:
A. \frac { 6\in_0R } { 5d+4Vt }
B. rac { 6\in _0R} { 5d-5Vt }
C. \frac { 6\in_0R } { 5d-6Vt }
D. \frac { 6\in _0R } { 5d-4Vt }
E. $$\frac { 6\in _0R} { 5d+3Vt } $$
F. $$\frac { 15d+9Vt)\ 6\in_0R }{2d^2-3dVt-9V^2t^2 }$$
G. \frac { 6\in _0R } { 5d-2Vt }
H. $$\frac { 6\in _0R} { 5d-3Vt } $$
I. $$\frac {(15d-9Vt)\ 6\in_ 0R }{2d^2+3dVt-9V^2t^2 }$$
Answer: E
A coil has an inductance $3\ H$ and a fuse wire of negligible resistance is connected in series with a cell of emf $6\ V$ with no internal resistance. The fuse wire will blow when the current through it reaches $8\ A$. If the switch is closed at $t=0$, the fuse will blow :

Options:
A. at $t=7\ s$
B. at $t=2\ s$
C. at $t=\infty$
D. at $t=12\ s$
E. at $t=4\ s$
F. at $t=6\ s$
G. at $t=8\ s$
H. at $t=10\ s$
I. just after the switch is closed
J. at $t=9\ s$
Answer: G
Equal torques act on the discs $A$ and $B$ of the previous problem, initially both being at rest. At a later instant, the linear speeds of a point on the rim of $A$ and another point on the rim of $B$ are ${v}_{A}$ and ${v}_{B}$ respectively. We have

Options:
A. $${ v }_{ A }=\frac{1}{4}{ v }_{ B }$$
B. $${ v }_{ A }=\frac{1}{3}{ v }_{ B }$$
C. $${ v }_{ A }=\frac{3}{4}{ v }_{ B }$$
D. $${ v }_{ A }<{ v }_{ B }$$
E. The relation depends on the actual magnitude of the torques
F. $${ v }_{ A }=\frac{1}{2}{ v }_{ B }$$
G. $${ v }_{ A }>{ v }_{ B }$$
H. $${ v }_{ A }=\frac{2}{3}{ v }_{ B }$$
I. $${ v }_{ A }={ v }_{ B }$$
Answer: G
An object is tied to a string of length $l$ and is revolved in a vertical circle at the minimum velocity. When the object reaches the upper most point, the string breaks and it describes a parabolic path as shown in the figure under the gravitational force. The horizontal range $AC$ in the plane of $A$ would be :-

Options:
A. $$x = l$$
B. $$x = \sqrt{4l}$$
C. x = 3l
D. $$x = \sqrt{5l}$$
E. $$x = \sqrt{l}$$
F. $$x = 2\sqrt{2l}$$
G. $$x = 2l$$
H. $$x = \sqrt{3l}$$
I. $$x = \sqrt{2l}$$
Answer: A
An initially parallel cylindrical beam travels in a medium of refractive index $\mu(I) =\mu_0 + \mu_2 I$, where $\mu_0$ and $\mu_2$ are positive constants and $I$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The initial shape of the wave front of the beam is

Options:
A. Convex
B. Concave
C. Concave near the axis and planar near the periphery
D. Convex near the axis and concave near the periphery
E. Planar near the axis and convex near the periphery
F. Convex near the periphery and planar near the axis
G. Planar near the axis and concave near the periphery
H. Convex near the periphery and concave near the axis
I. Concave near the axis and convex near the periphery
J. Planar
Answer: J
Electrostatic potential V has been measured everywhere outside a spherical body of radius R made of unknown material. It was found that the potential is spherically symmetric, i.e. depends only on the distance r from the centre of the sphere as given by the expression V=A/r, where A is a constant. No measurement of the potential inside the sphere has been done. What can you conclude for charge distribution of the body?

Options:
A. It may be uniform or nonuniform but must have some symmetry.
B. To predict charge distribution precisely we need to know electric field outside the sphere.
C. It may be nonuniform but must have spherical symmetry.
D. It must be uniform.
E. It may be uniform or nonuniform without any spherical symmetry.
F. It may be uniform or nonuniform but must have spherical symmetry.
G. It may be uniform but cannot have spherical symmetry.
H. It must be nonuniform but can have spherical symmetry.
I. It may be nonuniform but cannot have spherical symmetry.
J. It may be uniform but must have spherical symmetry.
Answer: E
One mole of anhydrous $MgCl_2$ dissolve in water and liberates $25$ cal/mol of heat. $\Delta H_{hydration}$ of $MgCl_2=-30$ cal/mol. Heat of dissolution of $MgCl_2\cdot H_2O$ is:

Options:
A. $-55$ cal/mol
B. -65 cal/mol
C. -35 cal/mol
D. $-5$ cal/mol
E. +55 cal/mol
F. -15 cal/mol
G. $+5$ cal/mol
H. +15 cal/mol
I. $55$ cal/mol
J. -45 cal/mol
Answer: D
Consider a circular coil of wire carrying constant current I, forming a magnetic dipole. The magnetic flux through an infinite plane contains the circular coil and excluding the circular coil area is given by $\Phi _\text{i}$. The magnetic flux through the area of the circular coil area is given by $\Phi_0$ Which of the following options is correct?

Options:
A. $$\Phi _\text{i}= - \Phi _0$$
B. $$\Phi _\text{i} > \Phi _0$$
C. $$\Phi_{\text{i}} = 0.75 \Phi_0$$
D. $$\Phi _\text{i} < \Phi _0$$
E. $$\Phi _\text{i} = \Phi _0$$
F. $$\Phi_{\text{i}} = 0.5 \Phi_0$$
G. $$\Phi _{\text{i}} = 1.5 \Phi _0$$
H. $$\Phi _{\text{i}} = 2 \Phi _0$$
I. $$\Phi _{\text{i}} = 3 \Phi _0$$
J. $$\Phi _{\text{i}} = 4 \Phi _0$$
Answer: A
Which of the following complexes exhibits the highest paramagnetic behaviour? (Where gly = glycine,en= ethylenediamine and bpy = bipyridyl moities). 
(At number Ti = 22, V = 23, Fe = 26, Co = 27 )

Options:
A. [Co(OX)_3(H_2O)]^-
B. [Co(OX)_2(OH)_3]^-
C. $$ [V(gly)_2(OH)_2(NH_3)_2]^* $$
D. [Fe(OX)_2(OH)_2]^-
E. [Co(OX)_3(OH)]^-
F. $$ [Fe(en)(bpy)(NH_3)_2]^{2+} $$
G. [Co(OX)_2(OH)]^-
H. [Co(OX)_3(OH)]^2-
I. $$ [Co(OX)_2(OH)_2]^- $$
J. $$ [Ti(NH_3)_6]^{3+} $$
Answer: I
Two particles, masses M and 2M, go round a circle of radius R under the action of their mutual gravitational attraction. The angular velocity of each particle is

Options:
A. $$\omega =\sqrt{\frac{4GM}{R^3}}$$
B. $$\omega = \sqrt{\frac{8GM}{R^3}}$$
C. $$\omega = \sqrt{\frac{GM}{R^3}}$$
D. $$\omega =\sqrt{\frac{3GM}{R^3}}$$
E. $$\omega = \sqrt{\frac{5GM}{R^3}}$$
F. $$\omega = \sqrt{\frac{GM}{2R^3}}$$
G. $$\omega = \sqrt{\frac{7GM}{R^3}}$$
H. $$\omega =\frac{1}{2R} \sqrt{\frac{1}{GM}}$$
I. $$\omega =\sqrt{\frac{6GM}{R^3}}$$
J. $$\omega = \sqrt{\frac{2GM}{R^3}}$$
Answer: D
${K_2}$ for a monobasic acid whose 0.10M solution has pH of $4.52.$ (log3=0.48)

Options:
A. $$9 \times {10^{ - 6}}$$
B. 3 \times {10^{ - 7}}
C. 3 \times 10^{-9}
D. 9 \times 10^{-7}
E. $${10^{ - 12}}$$
F. 9 \times {10^{ - 8}}
G. 3 \times 10^{-6}
H. $$9 \times {10^{ - 9}}$$
I. 3 \times 10^{-8}
J. $$3 \times {10^{ - 9}}$$
Answer: A
There exists a uniform electric field E = $4 \times 10^5 \space Vm^-1$ directed along negative x-axis  such that electric potential at origin is zero . A charge of $-200\mu C$ is placed at origin,and a charge of $+200\mu C$ is placed at (3 m, o). The electrostatic potential energy of the system is

Options:
A. 120 J
B. 240 J
C. -240 J
D. 150 J
E. Zero
F. 180 J
G. -150 J
H. -180 J
I. -120 J
J. 60 J
Answer: A
Two parallel plate air capacitors each of capacitance $C$ were connected in series to a battery with e.m.f. $E$. Then one of the capacitors was filled up with a uniform dielectric with relative permittivity $k$. What amount of charge flows through the battery?

Options:
A. \(\Delta q=\dfrac{1}{2}CE\dfrac{k-1}{k+2}\)
B. \(\Delta q=\dfrac{1}{2}CE\dfrac{2k+1}{k-1} \)
C. $$None of these$$
D. \(\Delta q=\dfrac{1}{2}CE\dfrac{k+1}{k+2} \)
E. $$\Delta q=\dfrac{1}{2}CE\dfrac{2k-1}{k+1}$$
F. $$\Delta q=\dfrac{1}{2}CE\dfrac{1-k}{1+k}$$
G. \(\Delta q=\dfrac{1}{2}CE\dfrac{k-1}{2k+1}\)
H. $$\Delta q=\dfrac{1}{2}CE\dfrac{k+1}{k-1}$$
I. $$\Delta q=\dfrac{1}{2}CE\dfrac{k+1}{k+3}$$
J. \(\Delta q=\dfrac{1}{2}CE\dfrac{k+1}{2k-1}\)
Answer: F
A balloon starts ascending in a vertical upward direction at a constant rate of $5m/{s^2}$. After $2s$ of its flight begins, a ball has been dropped from it, find the maximum height attained by ball. Also, calculate the position of the balloon relative to ground when ball strikes the ground. (Take g = $10m/{s^2}$)

Options:
A. 10m,137.1m
B. 10m,120m
C. 10m,130m
D. 5m,56m
E. $$5m,137.1m$$
F. $$15m,25m$$
G. 15m,137.1m
H. $$15m,56m$$
I. $$10m,56m$$
J. 15m,120m
Answer: I
If $x$ moles of $Mn{ O }_{ 4 }^{ - }$ required to oxidise $1$ mole of $Fe{ \left( H{ C }_{ 2 }{ O }_{ 4 } \right)  }_{ 2 }$ in acidic medium and $y$ moles of $Ca{ \left( OH \right)  }_{ 2 }$ are required to react with $1$ mole of $Fe{ \left( H{ C }_{ 2 }{ O }_{ 4 } \right)  }_{ 2 }$, then determine the value of $x$ and $y$.

Options:
A. x=1, y=3
B. $$x=2,\ y=3$$
C. None of the above
D. $$x=1,\ y=1$$
E. x=1,\ y=2
F. x=3, y=2
G. x=1, y=4
H. x=2, y=1
I. \( x=2,\ y=2 \)
Answer: D
A person is seeing two trains one of there is coming with speed of 4$\mathrm { m } / \mathrm { s }$ and another is going with same speed. If two trains blowing a whistle with frequency 243$\mathrm { Hz }$ . The beat frequency heard by stationary person will be (speed of sound in air $= 320 \mathrm { m } / \mathrm { s } )$

Options:
A. 1.01 Hz
B. 7.5 Hz
C. 9Hz
D. 4.5 Hz
E. 2.5 Hz
F. $$3Hz$$
G. $$0.01 Hz$$
H. $$12Hz$$
I. $$6Hz$$
J. 1.5 Hz
Answer: G
What is the solubility of solid zinc hydroxide at a $pH$ of $13$? Given that$Zn(OH)_{2}(g) \leftrightharpoons Zn(OH)_{2}(aq); K_{1} = 10^{-6}M$$Zn(OH)_{2}(aq) \leftrightharpoons Zn(OH)^{+} + OH^{-}; K_{2} = 10^{-7}M$$Zn(OH)^{+} \leftrightharpoons Zn^{2+} + OH^{-}; K_{3} = 10^{-4}M$$Zn(OH)_{2}(aq) + OH^{-} \leftrightharpoons Zn(OH)_{3}^{-}; K_{4} = 10^{3}M^{-1}$$Zn(OH)_{3}^{-} + OH^{-} \leftrightharpoons Zn(OH)_{4}^{2-}; K_{5} = 10M^{-1}$

Options:
A. $$10^{-17}$$
B. $$4 \times 10^{-4}$$
C. $$5 \times 10^{-4}$$
D. $$2 \times 10^{-4}$$
E. $$3 \times 10^{-4}$$
F. $$10^{-6}$$
G. $$4 \times 10^{-5}$$
H. 3 \times 10^{-5}
I. $$10^{-4}$$
J. $$1.5 \times 10^{-4}$$
Answer: D
An oil drop has a charge $-9.6\times {10}^{-19}C$ and mass $1.6\times {10}^{-15}gm$. When allowed to fall, due to air resistance force it attains a constant velocity. Then if a uniform electric field is to be applied vertically to make the oil drop ascend up with the same constant speed, which of the following are correct. $(g=10\ ms^{2})$ (Assume that the magnitude of resistance force is same in both the cases)

Options:
A. The intensity of electric field is $\dfrac {1}{3} \times{10}^{2}{NC}^{-1}$
B. The intensity of electric field is $\dfrac{1}{10} \times {10}^{5}{NC}^{-1}$
C. The intensity of electric field is $\dfrac {1}{6} \times {10}^{5}{NC}^{-1}$
D. The intensity of electric field is $\dfrac {1}{4} \times {10}^{5}{NC}^{-1}$
E. The intensity of electric field is $\dfrac {1}{7} \times {10}^{5}{NC}^{-1}$
F. The intensity of electric field is $\dfrac{1}{5} \times {10}^{5}{NC}^{-1}$
G. The intensity of electric field is $\dfrac {1}{9} \times {10}^{5}{NC}^{-1}$
H. The electric field is directed downward
I. The electric field is directed upward
J. The intensity of electric field is $\dfrac{1}{8} \times {10}^{5}{NC}^{-1}$
Answer: C
A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance $d$ of $1.2 m$ from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the lists.List IList IIP.Lift is accelerating vertically up.1.$d= 1.2 m$Q.Lift is accelerating vertically down with an acceleration less than the gravitational acceleration.2.$d > 1.2 m$R.Lift is moving vertically up with constant speed.3.$d < 1.2 m$S.Lift is falling freely.4.No water leaks out of the jar

Options:
A. P-1, Q-3, R-1, S-4
B. P-2, Q-3, R-2, S-4
C. P-2, Q-3, R-1, S-4
D. P-3, Q-2, R-1, S-4
E. P-1, Q-1, R-1, S-4
F. P-2, Q-3, R-1, S-1
G. P-1, Q-3, R-2, S-4
H. P-1, Q-1, R-2, S-4
I. P-2, Q-1, R-1, S-4
J. P-1, Q-3, R-1, S-1
Answer: E
If the half life of a first order reaction is 60 min, the approximate time in min, required to complete 90% of the reaction is $(log2=0.3)$

Options:
A. $$100$$
B. 120
C. 160
D. 220
E. $$200$$
F. 180
G. 280
H. $$240$$
I. 260
J. $$50$$
Answer: H
A non-conducting insulating ring of mass $m$ and radius $R$, having charge $Q$ uniformly distributed over its circumference is hanging by a insulated thread with the help of a small smooth ring (not rigidly fixed with bigger ring). A time varying magnetic field $B = B_0 \sin\omega t$ is switched on at t $= 0$ and the ring is released at the same time.The average magnetic moment of Ring in time interval $0$ to $\dfrac{\pi}{\omega}$ is

Options:
A. \dfrac{B_0q^2R^2}{6\pi m}
B. zero
C. \dfrac{B_0q^2R^2}{3\pi m}
D. \dfrac{B_0q^2R^2}{5\pi m}
E. $$\dfrac{B_0q^2R^2}{2\pi m}$$
F. \dfrac{B_0q^2R^2}{8\pi m}
G. \dfrac{B_0q^2R^2}{9\pi m}
H. $$\dfrac{B_0q^2R^2}{4\pi m}$$
I. $$\dfrac{B_0q^2R^2}{\pi m}$$
J. \dfrac{B_0q^2R^2}{7\pi m}
Answer: I
A particle of mass M and positive charge Q, moving with a constant velocity $u_1=4\hat i ms^{-1}$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from $x=0$ to $x=L$ for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity $\vec u_2=2(\sqrt 3\hat i+\hat j)ms^{-1}$. The correct statement(s) is (are)

Options:
A. the direction of the magnetic field is +z direction
B. the magnitude of the magnetic field $\dfrac {75\pi M}{3Q}$ units
C. the magnitude of the magnetic field $\dfrac {50\pi M}{6Q}$ units
D. the magnitude of the magnetic field $\dfrac {150\pi M}{3Q}$ units
E. the direction of the magnetic field is -z direction
F. the magnitude of the magnetic field $\dfrac {25\pi M}{3Q}$ units
G. the magnitude of the magnetic field \dfrac {40\pi M}{3Q} units
H. the magnitude of the magnetic field $\dfrac {100\pi M}{6Q}$ units
I. the magnitude of the magnetic field $\dfrac {50\pi M}{3Q}$ units
J. the magnitude of the magnetic field $\dfrac {100\pi M}{3Q}$ units
Answer: I
$\alpha -$particles are projected towards the nuclei of the following metals, with the same kinetic energy. Towards which metal, the distance of closest approach is minimum ?

Options:
A. $$Cu (Z=29)$$
B. $$Al (Z=13)$$
C. Al (Z=14)
D. $$Au (Z=79)$$
E. $$Ag (Z=48)$$
F. $$Pd (Z=46)$$
G. Ag (Z=47)
Answer: A
$2 mole $ ideal monoatomic gas undergoes process, where pressure is directly proportion volume of the gas. If its temperature rises from $27^0C$ to $227^0C$, then heat absorbed by gas is (R=universal gas constant)

Options:
A. 650R
B. 720R
C. 750R
D. $$600R$$
E. $$400R$$
F. 500R
G. $$800R$$
H. 700R
I. $$200R$$
J. $$550R$$
Answer: I
For the reaction ${ N }_{ 2 }+3{ X }_{ 2 }\longrightarrow 2{ NX }_{ 3 }$ where $X=F, Cl$ (the average bond energies are $F-F=155kJ\ { mol }^{ -1 },N-F=272kJ\ { mol }^{ -1 },Cl-Cl=242kJ\ { mol }^{ -1 }$, $N-Cl=200kJ\ { mol }^{ -1 },NaN=941kkJ\ { mol }^{ -1 }$, the heats of formation of ${ NF }_{ 3 }$ and $N{ Cl }_{ 3 }$, respectively are closest to :

Options:
A. -151 and +467
B. $-151$ and $+311$
C. -226 and -311
D. -226 and +311
E. $+226$ and $-467$
F. -226 and -467
G. $+151$ and $-311$
H. $-226$ and $+467$
I. -151 and -467
Answer: H
A block of mass m is attached with a massless spring of force constant k. The block is placed over a rough inclined surface for which the coefficient of friction is $\mu = 3/4$. The minimum value of M required to move the block up the plane is (Neglect mass of string and pulley and friction in pulley)

Options:
A. \dfrac{2}{5}m
B. \dfrac{2}{3}m
C. \dfrac{1}{3}m
D. $$2m$$
E. \dfrac{4}{3}m
F. $$\dfrac{3}{5}m$$
G. $$\dfrac{3}{2}m$$
H. \dfrac{1}{2}m
I. \dfrac{3}{4}m
J. $$\dfrac{4}{5}m$$
Answer: F
Two identical coins connected by a massless string kept on the rough horizontal disc rotating with constant angular velocity $ \omega (>0) $ about the vertical axis passing through the center $ C_1 $ do not slip on disc. If $ f_1 and f_2 $ represent the friction on coin 'A' and friction on coin B respectively, which of the following possibility is  ruled out for $ r_2> r_1 $ ?

Options:
A. $$ f_1 = \omega r_2, f_2 = 0 $$
B. $$ f_1 = f_2 = 0 $$
C. $$f_1 = 0, f_2 = \omega r_1$$
D. $$f_1 = \omega r_1, f_2 = 0$$
E. $ f_ 1=0,f_ 2
eq 0 $
F. $$ f_1 = 0, f_2 = \omega r_2 $$
G. $ f_ 1 = 0,f_ 2\ =0 $
H. $$ f_ 1 = 0,f_ 2 = 0 $$
Answer: B
A particle of mass $'m'$ moves in a circular path of radius $'R'$ such that tangential acceleration is equal to normal acceleration. At $t=0$, the velocity of particle is $V_{0}$

Options:
A. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-5\pi/2})
B. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-\pi/2})$
C. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-3\pi/2})
D. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-2\pi})$
E. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-3\pi})$
F. Angular velocity of particle at time $'t'$ is $w=\dfrac {w_{0}}{1+w_{0}t}$
G. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-4\pi/3})
H. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-2\pi/3})
I. Angular velocity of particle at time $'t'$ is $w=\dfrac {w_{0}}{1-w_{0}t}$
J. time taken $t-$complete one revolution is $\dfrac {1}{V_{0}} (1-e^{-\pi})$
Answer: J
$A$ wire $AB$ of length 1m has a sliding support at one end $A$ and fixed support at end $B$. This wire is vibrated by a tuning fork of frequency $100 Hz$ due to which the wire $AB$ vibrates in its fundamental tone. 
The particle at point A is vibrating with an amplitude of 10mm and at t = 0, the photograph of the wire is as shown in the figure below. (all the particles are at mean position at t = 0). The transverse displacement of the particles as a function of x and t will be:

Options:
A. $$y (x, t) = (10 mm) \cos \left ( \frac{\pi} {2} x \right ) \cos \left ( 150 \pi t \right )$$
B. $$y (x, t) = (10 mm) \cos \left ( \frac{\pi} {2} x \right ) \cos \left ( 200 \pi t \right )$$
C. $$y (x, t) = (10 mm) \cos \left ( \frac{\pi} {2} x \right ) \sin \left ( 150 \pi t \right )$$
D. $$y (x, t) = (10 mm) \sin \left ( \frac{\pi} {2} x \right ) \sin \left ( 100 \pi t \right )$$
E. $$y (x, t) = (10 mm) \cos \left ( \frac{\pi} {2} x \right ) \sin \left ( 100 \pi t \right )$$
F. $$y (x, t) = (10 mm) \cos \left ( \frac{\pi} {2} x \right ) \cos \left ( 100 \pi t \right )$$
G. $$y (x, t) = (10 mm) \cos \left ( \frac{\pi} {2} x \right ) \sin \left ( 200 \pi t \right )$$
H. $$y (x, t) = (10 mm) \sin \left ( \frac{\pi} {2} x \right ) \sin \left ( 200 \pi t \right )$$
I. $$y (x, t) = (10 mm) \sin \left ( \frac{\pi} {2} x \right ) \cos \left ( 150 \pi t \right )$$
J. $$y (x, t) = (10 mm) \sin \left ( \frac{\pi} {2} x \right ) \cos \left ( 200 \pi t \right )$$
Answer: G
In a region at a distance r from z-axis magnetic field $B=B_0 rtk$ is present $B_0$ is constant t is time. Then the magnitude of induced electric field at a distance r from z-axis is given by

Options:
A. $$-\dfrac{r^2}{3}B_0$$
B. $$-\dfrac{r^2}{8}B_0$$
C. $$-\dfrac{r^2}{5}B_0$$
D. $$-\dfrac{r^2}{6}B_0$$
E. $$-\dfrac{r^2}{9}B_0$$
F. $$-\dfrac{r^2}{2}B_0$$
G. $$-\dfrac{r^2}{4}B_0$$
H. $$-\dfrac{r^2}{7}B_0$$
I. $$-\dfrac{r^2}{12}B_0$$
J. $$-\dfrac{r}{2}B_0$$
Answer: A
$Ca{CO}_{3(s)}\rightarrow Ca{O}_{(s)}+{CO}_{2(g)}$ $\Delta H=+180KJ$If entropies of limestone, lime and carbon dioxide are $93,39$ and $213J/mol/K$. Then $\Delta {S}_{total}$ in ${JK}^{-1}$ at $300K$ is:

Options:
A. $$-441$$
B. $$169$$
C. $$139$$
D. $$179$$
E. $$159$$
F. $$149$$
G. $$-650$$
H. $$700$$
Answer: E
In the given figure, a smooth parabolic track lies in the vertical plane $(x-y)$ the shape of track is defined by equation $y=\cfrac { { x }^{ 2 } }{ a } $ where $a$ is a constant. A ring of mass $m$ which can slide freely on wire track, is placed at the position $(a,a)$. The track is rotated with  constant angular speed $\omega$ such that there is no relative slipping between the ring and the track then $\omega$ is equal to

Options:
A. $$\sqrt { \cfrac { 4g }{ a } }$$
B. $$\sqrt { \cfrac { g }{ a } } $$
C. $$\sqrt { \cfrac { 3g }{ 2a } }$$
D. $$\sqrt { \cfrac { 3g }{ 4a } }$$
E. $$\sqrt { \cfrac { g }{ 3a } }$$
F. $$\sqrt { \cfrac { g }{ 4a } }$$
G. $$None of these$$
H. $$\sqrt { \cfrac { g }{ 2a } } $$
I. $$\sqrt { \cfrac { 5g }{ 2a } }$$
J. $$\sqrt { \cfrac { 2g }{ a } } $$
Answer: J
When inner two concentric spheres of radius $r_1$ and $r_2$ ($r_1<r_2$) carries an electric charge ,the differential equation for the potential $V$ at the distance $r$ from the common centre is $\dfrac{d^2V}{dr^2} + \dfrac{2}{r} = 0$. The value of $V$ in terms of $r$ is?

Options:
A. $$Vr=c_1r^2+c_2$$
B. Vr=c_1r^{3}+c_2
C. $$Vr^2=c_1r-c_2$$
D. $$Vr=c_1r^{-1}+c_2$$
E. $$Vr=c_1r-c_2$$
F. $$Vr=c_1r^{-5}+c_2$$
G. $$Vr=c_1r^{-2}+c_2$$
H. $$Vr=c_1r^2-c_2$$
I. $$Vr=c_1r^{-3}+c_2$$
Answer: E
Which of the following complex compounds will exhibit highest paramagnetic behaviour? (At. No. : $Ti = 22,  Cr = 24,  Co = 27,  Zn = 30$)

Options:
A. $${ \left[ Zn{ \left( N{ H }_{ 3 } \right) }_{ 6 } \right] }^{ 2+ }$$
B. $${ \left[ Co{ \left( N{ H }_{ 3 } \right) }_{ 6 } \right] }^{ 3+ }$$
C. $${ \left[ Ti{ \left( N{ H }_{ 3 } \right) }_{ 6 } \right] }^{ 3+ }$$
D. $${ \left[ Cr{ \left( N{ H }_{ 3 } \right) }_{ 6 } \right] }^{ 3+ }$$
Answer: D
In Young's double-slit experiment, two wavelengths of light are used simultaneously where $\lambda_{2}=2\lambda_{1}$. In the fringe pattern observed o the screen,

Options:
A. $n$th order minima of $\lambda_{2}$ will coincide with $2n$th order minima of $\lambda_{1}$
B. maxima of wavelength $\lambda_{2}$ can coincide with minima of wavelength $\lambda_{1}$
C. fringe width of $\lambda_{2}$ will be double that of fringe width of $\lambda_{1}$ and $n$th order maxima of $\lambda_{2}$ will coincide with second order maxima of $\lambda_{1}$
D. none of the above
Answer: A
Two uniform solid spheres of equal radii R, but mass M and 4M have a centre to centre separation 6 R, the two spheres are held fixed on a horizontal floor. A projectile of mass m is projected from the surface of the sphere of mass M directly towards the centre of the second sphere. Obtain an expression for the minimum speed v of the projectile so that it reaches the surface of the second sphere

Options:
A. $$v= \left ( \dfrac{3R}{5GM} \right )^{\dfrac{1}{2}}$$
B. $$v= \left ( \dfrac{8R}{5GM} \right )^{\dfrac{1}{2}}$$
C. $$v= \left ( \dfrac{6GM}{5R} \right )^{\dfrac{1}{2}}$$
D. $$v= \left ( \dfrac{4GM}{3R} \right )^{\dfrac{1}{2}}$$
E. $$v= \left ( \dfrac{4GM}{5R} \right )^{\dfrac{1}{2}}$$
F. $$v= \left ( \dfrac{3GM}{5R} \right )^{\dfrac{1}{2}}$$
G. $$v= \left ( \dfrac{2GM}{5R} \right )^{\dfrac{1}{2}}$$
H. $$v= \left ( \dfrac{3GM}{4R} \right )^{\dfrac{1}{2}}$$
I. $$v= \left ( \dfrac{3GM}{6R} \right )^{\dfrac{1}{2}}$$
J. $$v= \left ( \dfrac{5GM}{3R} \right )^{\dfrac{1}{2}}$$
Answer: F
A $0.1 M$ solution of $[Cu(NH_{3})_{4}]^{+}$ is stirred with an excess of potassium cyanide sufficient to convert all the ammonium complex to the corresponding cuprocyanide complex $[Cu(CN)_{4}]^{-3}$ and in addition to provide the solution with an excess of $CN^{-}$ equal to $0.2 M$. Calculate the maximum $pH$ of the solution when the final solution is treated with hydrogen sulphide to maintain $[H_{2}S] = 0.1 M$ and the precipitation of cuprous sulphide is prevented. The instability constant for $[Cu(CN)_{4}]^{-3}$ is $5 \times 10^{-28}$, $K_{a.overall}$ of $H_{2}S = 1.6 \times 10^{-21}$

Options:
A. $$9.8$$
B. $$4.0$$
C. $$10.2$$
D. $$3.2$$
E. $$11.0$$
F. $$10.6$$
G. $$10.8$$
H. $$9.2$$
I. $$10.4$$
Answer: B
Select incorrect statement(s) among the following.

Options:
A. In the titration of a monoacidic weak base with a strong acid, the pH at the equivalent point is always calculated by $pH=\frac {1}{2}[pK_w-pH_b-logC]$
B. An acid-base indicator in a buffer solution of $pH=pK_{ln}+1$ is ionized to the extent of $\frac {1000}{11}$%
C. When $Na_3PO_4(aq)$ is titrated with HCl(aq), the pH of solution at second equivalent point is calculated by $\frac {1}{2}[pK_{a_1}+pK_{a_2}]$
D. Phenolphthalein is suitable indicator for the titration of HCl (aq) with $NH_4OH(aq)$
Answer: A
In a series $LCR$ circuit, $C={10}^{-11}\ Farad$, $L={10}^{-5}$ Henry and $R=100\ ohm$, when a constant $D.C$ voltage $E$ is applied to the circuit, the capacitor acquires a charge ${10}^{-9}C$. The $D.C$ source is replaced by a sinusoidal voltage source in which the peak voltage $E$. At resonance the peak value of the charge acquired by the capacitor will be:

Options:
A. $${10}^{-15}\ C$$
B. $${10}^{-6}\ C$$
C. $${10}^{-7}\ C$$
D. $${10}^{-8}\ C$$
E. $${10}^{-12}\ C$$
F. $${10}^{-10}\ C$$
G. $${10}^{-14}\ C$$
H. $${10}^{-13}\ C$$
I. $${10}^{-9}\ C$$
J. $${10}^{-11}\ C$$
Answer: D