MikaStars39/nano-eval huggingface dataset:
A block of mass $m$ height $2 h$ and width $2b$ rests on a flat car which moves horizontally with constant acceleration 'a' as shown in the figure. The coefficient of friction is $\mu$. The following data are given $b=0.6 m$, $h= 0.9 m$ $\mu = 0.5$ and $g = 10 m s^{-2}$. Then,

Options:
A. the shortest distance in which it can be stopped from a speed of $20ms^{-1}$ with constant deceleration so that the block is not disturbed is $35m$.
B. the shortest distance in which it can be stopped from a speed of $20ms^{-1}$ with constant deceleration so that the block is not disturbed is $38m$.
C. the shortest distance in which it can be stopped from a speed of $20ms^{-1}$ with constant deceleration so that the block is not disturbed is $50m$.
D. the value of acceleration at which block topples about $P$, assuming sufficient friction to prevent slipping is $\displaystyle\ \frac{60}{9} ms^{-2}$.
E. None of these
F. the shortest distance in which it can be stopped from a speed of $20ms^{-1}$ with constant deceleration so that the block is not disturbed is $40m$.
G. the shortest distance in which it can be stopped from a speed of $20ms^{-1}$ with constant deceleration so that the block is not disturbed is $39m$.
H. the shortest distance in which it can be stopped from a speed of $20ms^{-1}$ with constant deceleration so that the block is not disturbed is $42m$.
I. the value of acceleration at which slipping of the block on the car starts is $a$ = $ 5ms^{-2}$.
J. the shortest distance in which it can be stopped from a speed of $20ms^{-1}$ with constant deceleration so that the block is not disturbed is $45m$.
Answer: F
The displacement of a particle varies according to the relation ${x=3 sin 100 t+ 8 cos^{2}50 t}$. Which of the following is/are correct about this motion.

Options:
A. the maximum displacement of the particle from the origin is 8 units
B. the amplitude of the resultant S.H.M is ${\sqrt73}$ units
C. the maximum displacement of the particle from the origin is 6 units
D. the maximum displacement of the particle from the origin is 9 units
E. the maximum displacement of the particle from the origin is 11 units
F. the motion of the particle is not S.H.M
G. the maximum displacement of the particle from the origin is 10 units
H. the amplitude of the resultant S.H.M is ${\sqrt{81}}$ units
I. the maximum displacement of the particle from the origin is 7 units
Answer: D
Three partieles, each of mass $m$ are situated at the vertices of an equilateral triangle of side $a$. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original mutual separation $a$. Find the initial velocity that should be given to each particle and also the time period of the circular motion. $\displaystyle \left ( F= \frac{Gm_{1}m_{2}}{r^{2}} \right )$

Options:
A. \displaystyle T= 2\pi \sqrt{\frac{a^{3}}{5Gm}}
B. $$\displaystyle  T= 2\pi \sqrt{\frac{a^{3}}{3Gm}}$$
C. \displaystyle T= 2\pi \sqrt{\frac{a^{3}}{6Gm}}
D. $$\displaystyle  T= 2\pi \sqrt{\frac{a^{3}}{Gm}}$$
E. $$\displaystyle v= \sqrt{\frac{Gm}{2a}}$$
F. \displaystyle T= 2\pi \sqrt{\frac{a^{3}}{2Gm}}
G. \displaystyle T= 2\pi \sqrt{\frac{a^{3}}{4Gm}}
Answer: B
A spider is hanging by means of its own silk thread directly above a transparent fixed sphere of radius R = 20 cm as shown in the figure. The refractive index of the material of the sphere is equal to $\sqrt{2}$ and the height of the spider from the centre of the sphere is 2R ( = 40 cm). An insect, initially sitting at the bottom, starts crawling on the sphere along a vertical circular path with the constant speed of $\pi$/4 cm/s. For ho long time the insect will be invisible for the spider, assume that it crawls once round the vertical circle.

Options:
A. $\dfrac{140}{3}$ sec
B. $\dfrac {320}{3}$ sec
C. $\dfrac{100}{3}$ sec
D. $\dfrac {240}{3}$ sec
E. $\dfrac {80}{3}$ sec
F. \dfrac{200}{3} sec
G. \dfrac{120}{3} sec
H. \dfrac{180}{3} sec
I. $\dfrac {160}{3}$ sec
J. 80 sec
Answer: J
If $'\lambda_1'$ and $'\lambda_2'$ are wavelengths of first lines is Lymas and Balnar series of H-atom respectively then wavelength of radiation needed to excite the electron from $n=1$ and $n=3$ is?

Options:
A. \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}
B. \frac{1}{\lambda_1 \lambda_2}
C. \frac{\lambda_1 - \lambda_2}{\lambda_1 \lambda_2}
D. $$\displaystyle\frac{\lambda_1+\lambda_2}{\lambda_1\lambda_2}$$
E. $$\displaystyle\lambda_1+\lambda_2$$
F. $$\displaystyle\frac{1}{\lambda_1 +\lambda_2}$$
G. $$\displaystyle\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}$$
H. \frac{\lambda_1 + \lambda_2}{\lambda_1 + \lambda_2}
I. \frac{1}{\lambda_1 - \lambda_2}
J. \frac{\lambda_1 + \lambda_2}{\lambda_1 - \lambda_2}
Answer: F
A wire of mass 100 g is carrying a current of 2A towards increasing x in the form of $y={x}^{2}(-2m\le x\le +2m)$.This wire is placed in a magnetic field $ec{B}=-0.02\hat{k}$ tesla.The acceleration of the wire (in $m/{s}^{2}$) is

Options:
A. 3.2\hat{j}
B. $$-1.6\hat{j}$$
C. -0.8\hat{j}
D. $$-3.2\hat{j}$$
E. -1.6\hat{i}
F. zero
G. -0.8\hat{i}
H. 3.2\hat{i}
I. $$1.6\hat{j}$$
Answer: I
A car is going at a speed of $6m/s$, when it encounters a $15m$ slope of angle ${30}^{o}$. The friction coefficient between the road and tyre is $0.5$. The driver applies the brakes. The minimum speed of the car with which it can reach the bottom is ($g=10m/{s}^{2}$)

Options:
A. 7.29 m/s
B. $$7.49 m/s$$
C. $$8.45 m/s$$
D. 7.98 m/s
E. 6.98 m/s
F. 7.73 m/s
G. 7.25 m/s
H. $$3 m/s$$
I. $$4 m/s$$
J. 8.12 m/s
Answer: B
The potential energy of a $1kg$ particle free move along the x-axis is given by
$$V(x)=\left( \cfrac { { x }^{ 4 } }{ 4 } -\cfrac { { x }^{ 2 } }{ 2 }  \right) J$$
The total mechanical energy of the particle $2J$. Then, the maximum speed (in m/s) is

Options:
A. $$1/\sqrt {2}$$
B. $$2$$
C. 4/\sqrt{3}
D. 3/\sqrt{2}
E. 4/\sqrt{2}
F. $$\sqrt {2}$$
G. 3
H. 3/\sqrt{3}
I. \sqrt{3}
Answer: B
The plate current i in a triode valve is given $i _ { p } = K \left( V _ { p }\right.$ $+ \mu V _ { g } ^ { 3 / 2 }$ where $i _ { p }$ is in milliampare and $V _ { p }$ and $V _ { g }$ are in volt. $if r _ { p } = 10 ^ { 4 } \text { ohm, and } g _ { m } = 5 \times 10 ^ { - 3 }$ mho, then for $i _ { p } = 8 \mathrm { mA }$ and $V _ { p } = 300$ volt, what is the value of K and grid cut off voltage?

Options:
A. - 6 \mathrm { V } , ( 32 ) ^ { 3 / 2 }
B. $$- 6 \mathrm { V } , ( 1 / 30 ) ^ { 3 / 2 }$$
C. - 6 \mathrm { V } , ( 40 ) ^ { 3 / 2 }
D. - 6 \mathrm { V } , ( 20 ) ^ { 3 / 2 }
E. - 6 \mathrm { V } , ( 25 ) ^ { 3 / 2 }
F. $$- 6 \mathrm { V } , ( 30 ) ^ { 3 / 2 }$$
G. - 6 \mathrm { V } , ( 28 ) ^ { 3 / 2 }
H. - 6 \mathrm { V } , ( 22 ) ^ { 3 / 2 }
I. $$+ 6 \mathrm { V } , ( 30 ) ^ { 3 / 2 }$$
J. $$+ 6 \mathrm { V } , ( 1 / 30 ) ^ { 3 / 2 }$$
Answer: F
A variable external resistor has resistance given by $R=R_0T^2+R_1$. It is connected to a battery having internal resistance 'r' and EMF '$arepsilon$'. At what time is the maximum power delivered to the external resistance if $R_1 < r$?

Options:
A. $$\sqrt {\cfrac {R_{1}-r}{R_{0}}}$$
B. \sqrt {\cfrac {R_{0}-r}{R_{1}}}
C. \sqrt {\cfrac {R_{0}+r}{R_{1}}}
D. \sqrt {\cfrac {R_{1}+r}{2R_{0}}}
E. \sqrt {\cfrac {R_{1}+r}{R_{0}}}
F. $$\cfrac {2R_0}{R_1}$$
G. \sqrt {\cfrac {R_{1}+2r}{R_{0}}}
H. $$\sqrt {\cfrac {r-R_{1}}{R_{0}}}$$
I. Zero
J. \sqrt {\cfrac {R_{0}-R_{1}}{r}}
Answer: A
When a perfectly transparent glass slab $\left( \mu=1.5 \right )$ is introduced in front of upper slit of a usual double slit experiment, the intensity at $O$ reduces to $1/2$ times of its earlier value. Minimum thickness of slab would be

Options:
A. 3\lambda/8
B. $$2/lambda$$
C. 3\lambda/2
D. $$\lambda/2$$
E. $$\lambda$$
F. \lambda/4
G. 5\lambda/4
H. 5\lambda/8
I. 3\lambda/4
Answer: D
The wavelength (in $\mathring{A}$) of an emission line observed for $Li^{2+}$ during electronic transition from $n_2 = 2$ to $n_1 = 1$ is: ($R$ is Rydberg constant)

Options:
A. \dfrac { 4 }{ 2R }
B. \dfrac { 4 }{ 9R }
C. $$\dfrac { 27R }{ 4 } $$
D. \dfrac { 4 }{ 6 R}
E. \dfrac { 4 }{ 12R }
F. $$\dfrac { 3R }{ 4 } $$
G. $$\dfrac { 4 }{ 27R } $$
H. $$\dfrac { 4 }{ 3 R} $$
I. \dfrac { 4 }{ 16R }
Answer: H
A satellite sent into space samples the density of matter within the solar system and gets a value $2.5$ hydrogen atoms per cubic centimeter. What is the mean free path of the hydrogen atoms? Take the diameter of a hydrogen atoms as $d=0.24\ nm$.

Options:
A. 1.65 \times 10^{12}\ m
B. $$2.56\times 10^{12}\ m$$
C. 2.65 \times 10^{12}\ m
D. 3.65 \times 10^{12}\ m
E. 2.66 \times 10^{12}\ m
F. $$3.56\times 10^{12}\ m$$
G. 1.66 \times 10^{12}\ m
H. $$4.56\times 10^{12}\ m$$
I. $$1.56\times 10^{12}\ m$$
Answer: I
One mole of a mono-atomic ideal gas is expanded by a process described by $PV^3=C$ where $C$ is a constant. The heat capacity of the gas during the process is given by ($R$ is the gas constant).

Options:
A. \(\frac{8}{3}R\)
B. \frac{5}{3}R
C. $\displaystyle\frac{5}{2}$R
D. \(\frac{7}{3}R\)
E. \(\frac{6}{3}R\)
F. $$\displaystyle\frac{3}{2}R$$
G. \(\frac{4}{3}R\)
H. $$R$$
I. $$2R$$
J. \(\frac{2}{3}R\)
Answer: H
Ratio of solubilities of gases $N_2$ & $O_2$ in water from air at $25^o$ & 1 atm will be if air is 20% by volume of $O_2$ and 80% by volume of $N_2$.Given : $K_H (N_2) = 2 \times 10^4\ atm$              $K_H (O_2) = 10^4\ atm$

Options:
A. 1 : 3
B. $$8:1$$
C. $$3 : 1$$
D. 1 : 2
E. $$1 : 4$$
F. 4 : 1
G. 3 : 2
H. $$2 : 1$$
I. $$1 : 8$$
Answer: H
Suppose that two heat energies are connected in series, such that the heat exhaust of the first engine is used as the heat input of the second engine as shown in figure. The efficiencies of the engines are $ { \eta  }_{ 1 }$ and $ { \eta  }_{ 2 }$, respectively. The net efficiency of the combination is given by

Options:
A. { \eta }_{ net }={ \eta }_{ 2 }+\left( 1-{ \eta }_{ 2 } \right) { \eta }_{ 1 }
B. $${ \eta }_{ net }={ \eta }_{ 1 }+\left( 1-{ \eta }_{ 1 } \right) { \eta }_{ 2 }$$
C. $${ \eta }_{ net }=\dfrac { { \eta }_{ 1 } }{ \left( 1-{ \eta }_{ 1 } \right) { \eta }_{ 2 } }$$
D. { \eta }_{ net }={ \eta }_{ 2 }+\left( 1-{ \eta }_{ 2 } \right) { \eta }_{ 2 }
E. $${ \eta }_{ net }=\dfrac { 1-{ \eta }_{ 1 } }{ \left( 1-{ \eta }_{ 2 } \right) { \eta }_{ 2 } }$$
F. { \eta }_{ net }={ \eta }_{ 1 }+\left( 1-{ \eta }_{ 2 } \right) { \eta }_{ 1 }
G. { \eta }_{ net }={ \eta }_{ 1 }+\left( 1-{ \eta }_{ 2 } \right) { \eta }_{ 2 }
H. $${ \eta }_{ net }={ \eta }_{ 2 }+\left( 1-{ \eta }_{ 1 } \right) { \eta }_{ 2 }$$
Answer: H
Consider a Gaussian spherical surface covering a dipole of charge $q$ and $-q$, then :

Options:
A. $ \int ec{E}\cdot ec{d}s=0 $(surface integral of over the spherical surface)
B. $\phi_{net}=0$(net flux coming out the spherical surface)
C. $E=0$ at all points on the spherical surface
D. $\int \vec{E}\cdot \vec{d}s=q_{in}/\epsilon_0$ (surface integral of over the spherical surface)
E. \int \vec{E} \cdot \vec{d}s = q_{in} / \epsilon_0
F. $\int \vec{E} \cdot \vec{d}s = q_{in} \epsilon_0$
G. $q_{in}=0$(net charge enclosed by the spherical surface)
H. \int \vec{E} \cdot \vec{d}s = q_{in} / (2\epsilon_0)
I. \int \vec{E} \cdot \vec{d}s = q_{in} / (3\epsilon_0)
Answer: A
Two non conducting discs of same material, thickness and density have same charge uniformly distributed over their circumference are placed on their edge on a rough horizontal surface. A vertical time varying in magnetic field $B=kt^2$, where $t$ is time in seconds, is switched on at $t=0$. The radius of bigger dis is twice the smaller one. The bigger disc starts rotating after $2$ seconds. The smaller disc will start rotating after

Options:
A. $$6 \ seconds$$
B. $$1 \ second$$
C. 8 \ seconds
D. $$2 \ seconds$$
E. $$5 \ seconds$$
F. $$4 \ second$$
G. $$3 \ seconds$$
H. 7 \ seconds
Answer: F
De-broglie hypothesis states that every material particle is associated with a wave known as matter wave. The wavelength of matter wave is $\lambda = \dfrac{h}{mv},$ where h is Planck's constant. When we use this in Bohr's theory. It shows that the angular momentum of electron in a hydrogen atom is quantized.The same principle can be applied to a particle of mass m moving in a circular path on a smooth inverted circular cone as shown. The speed of the particle at a height H from vertex is v. Reference level for potential energy is at the ground.The total energy of particle is (n is an integer)

Options:
A. $$\dfrac{1}{2}m^{1/3}g^{2/3}n^{2/3} \left ( \dfrac{h}{2\pi}tan\theta \right)^{2/3}$$
B. \dfrac{2}{3}m^{1/3}g^{2/3}n^{2/3} \left ( \dfrac{h}{2\pi tan\theta} \right )^{2/3}
C. \dfrac{1}{2}m^{2/3}g^{2/3}n^{2/3} \left ( \dfrac{h}{2\pi tan\theta} \right )^{2/3}
D. \dfrac{3}{2}m^{1/3}g^{2/3}n^{2/3} \left ( \dfrac{h}{2\pi \tan\theta} \right )^{1/3}
E. $$\dfrac{2}{3}m^{1/3}g^{1/3} \times n^{2/3} \left ( \dfrac{h}{2\pi}tan\theta \right )^{2/3}$$
F. $$\dfrac{3}{2}m^{1/3}g^{2/3}n^{2/3} \left ( \dfrac{h}{2\pi tan\theta} \right )^{2/3}$$
G. \dfrac{3}{2}m^{1/3}g^{1/3}n^{2/3} \left ( \dfrac{h}{2\pi tan\theta} \right )^{2/3}
H. None of these
I. \dfrac{3}{2}m^{2/3}g^{2/3}n^{2/3} \left ( \dfrac{h}{2\pi \tan\theta} \right )^{2/3}
J. \dfrac{1}{3}m^{2/3}g^{2/3}n^{2/3} \left ( \dfrac{h}{2\pi \tan\theta} \right )^{2/3}
Answer: F
From the data of $\displaystyle \Delta H$ of the following reactions
$\displaystyle C(s)+1/2O_{2}(g)\rightarrow CO(g) ; \: \Delta H=-110$ kJ
and $\displaystyle C(s)+H_{2}O(g)\rightarrow CO(g)+H_{2}(g) ; \: \Delta H=132$ kJ
Calculate the mole composition of the mixture of steam and oxygen on being passed over coke at
1273 K, keeping the reaction temperature constant.

Options:
A. mole % $\displaystyle O_{2}(g)=54.6, \: H_{2}O(g)=45.5$
B. mole % $\displaystyle O_{2}(g)=37.5, \: H_{2}O(g)=57.5$
C. mole % $\displaystyle O_{2}(g)=37.5, \: H_{2}O(g)=54.6$
D. mole % $\displaystyle O_{2}(g)=37.5, \: H_{2}O(g)=62.5$
E. mole % $\displaystyle O_{2}(g)=42.5, \: H_{2}O(g)=57.5$
F. mole % $\displaystyle O_{2}(g)=45.5, \: H_{2}O(g)=54.6$
G. mole % $\displaystyle O_{2}(g)=37.5, \: H_{2}O(g)=67.5$
H. mole % $\displaystyle O_{2}(g)=45.5, \: H_{2}O(g)=62.5$
I. mole % $\displaystyle O_{2}(g)=47.5, \: H_{2}O(g)=52.5$
J. mole % $\displaystyle O_{2}(g)=52.5, \: H_{2}O(g)=47.5$
Answer: D
A circular beam of light of diameter $d = 2\ cm$ falls on a plane surface of glass. The angle of incidence is $60^{\circ}$ and refractive index of glass is $\mu = 3/2$. The diameter of the refracted beam is

Options:
A. 3.48 cm
B. $$2.52 cm$$
C. 2.62 cm
D. 3.74 cm
E. 3.38 cm
F. 3.14 cm
G. $$4.00 cm$$
H. $$3.26 cm$$
I. $$3.0 cm$$
J. 3.54 cm
Answer: H
Given the following bond enthalpies:
$$B.E.(N \equiv N)=942\ kJ/mol;BE(H-H)=436\ kJ/mol$$
$BE(N-N)=163\ kJ/mol, BE(N-H)=390\ kJ/mole$ Determine enthalpy change for the following polymerisation reaction per mole of $N_2(g)$ consumed-
$$nN_2(g)+nH_2(g) \rightarrow (NH- NH)_n$$

Options:
A. 306\ kJ
B. 365\ kJ
C. 322\ kJ
D. 344\ kJ
E. $$435\ kJ$$
F. $$200\ kJ$$
G. 265\ kJ
H. $$272\ kJ$$
I. $$136\ kJ$$
J. 235\ kJ
Answer: H
If the velocity of electron is 25% of the velocity of photon having same de-Broglie ${\text{wave}}\;{\text{length,}}\;{\text{then}}\;\dfrac{{Ele}}{{{E_{ph}}}}{ = _:}$

Options:
A. 1 : 10
B. 1 : 16
C. 1 : 9
D. 1 : 4
E. 1 : 5
F. 1 : 6
G. 1 : 8
H. 1 : 2
I. 1 : 12
J. 1 : 3
Answer: H
A solid uniform sphere rotating about its axis with kinetic energy $E_0$ is gently placed on a rough horizontal plane. The coefficient of friction on the plane varies from point to point after some time, the sphere begins pure rolling with total kinetic energy equal to E. Then

Options:
A. E = \dfrac{1E_0}{7}
B. $$E = \dfrac{5E_0}{7}$$
C. E cannot be expressed in term of $E_0$
D. E = \dfrac{6E_0}{7}
E. E = \dfrac{4E_0}{7}
F. $$E = \dfrac{2E_0}{5}$$
G. E = \dfrac{5E_0}{2}
H. E = \dfrac{3E_0}{5}
I. E = \dfrac{3E_0}{7}
J. $$E = \dfrac{2E_0}{7}$$
Answer: J
The cell $Pt|H_2(g, \space0.1\space bar)|H^+(aq), \space pH = X || Cl^-(aq, \space 1\space M)|Hg_2Cl_2|Hg|Pt$, 
has e.m.f. of $0.5755\space V$ at $25^{\small\circ}C$. The SOP of calomel electrode is $-0.28\space V$, then $pH$ of the solution will be:

Options:
A. $$4.5$$
B. 5.7
C. none of these
D. 4.7
E. 4.9
F. $$11$$
G. 5.3
H. 4.8
I. $$5.5$$
J. $$5.0$$
Answer: I
A smooth hoop lies on a smooth horizontal table and is fixed. A particle is projected on the table from a point $A$ on the inner circumference of the hoop at angle $\theta$ with radius vector. If $e$ be the coefficient of restitution and the particle returns to the point of projection after two successive impacts. The final angle $\theta'$ made by velocity vector with radius of hoop is

Options:
A. \displaystyle \tan\theta' = \frac{e \tan\theta}{e^2 + 3}
B. $$\displaystyle\tan\theta'=\frac{\tan\theta}{e^2}$$
C. $$\displaystyle \tan\theta'=\frac{e^2}{(1+e^2)}\tan\theta$$
D. \displaystyle \tan\theta' = \frac{\tan\theta}{e^2 + 1}
E. $$\displaystyle\tan\theta'=\frac{\tan\theta}{e}$$
F. \displaystyle \tan\theta' = \frac{e \tan\theta}{e^2 - 1}
G. \displaystyle \tan\theta' = \frac{\tan\theta}{e + 1}
H. \displaystyle \tan\theta' = \frac{e \tan\theta}{e^2 + 2}
I. $$\displaystyle\tan\theta'=e^2\tan\theta$$
J. \displaystyle \tan\theta' = \frac{e \tan\theta}{e^2 + 1}
Answer: B
A time varying magnetic field, $(\bar{B}=B_{0}t\hat{k})$, is confined in a cylindrical region and is cutting the xy plane on a circle x$^{2}$ + y$^{2}=$ 4 where x and y are in meters. A wire frame A$_{1}$ A$_{2}$ A$_{4}$ A$_{3}$ A$_{1}$ is placed in the magnetic field as shown. Segment A$_{1}$ A$_{2}$ and A$_{3}$ A$_{4}$ are identical quarter circles parallel to each other with axis along z-axis.The induced current flowing in the wire frame is equal to : (The total length of the loop of wire frame is 10m,radius of arc A$_{3}$ A$_{4}$ and arc A$_{1}$ A$_{2}$ is 1m each and resistance per unit length is $1\Omega /m)$:

Options:
A. $$\dfrac{B_{0}\pi }{5}$$
B. \dfrac{B_{0}\pi }{6}
C. \dfrac{B_{0}\pi }{8}
D. \dfrac{B_{0}\pi }{15}
E. \dfrac{B_{0}\pi }{4}
F. $$\dfrac{B_{0}\pi }{20}$$
G. zero
H. $$\dfrac{B_{0}\pi }{10}$$
I. \dfrac{B_{0}\pi }{12}
J. \dfrac{B_{0}\pi }{2}
Answer: G
${ Be }^{ +3 }$ and a proton are accelerated by the same potential,their de-Broglie wavelengths have the ratio:(Assume mass of proton = mass of neutron)

Options:
A. $$\dfrac{1}{2\sqrt{2}}$$
B. $$1:3\sqrt { 3 } $$
C. \dfrac{1}{3\sqrt{3}}
D. $$1:4$$
E. \dfrac{1}{4\sqrt{2}}
F. \dfrac{1}{2\sqrt{4}}
G. \dfrac{1}{3\sqrt{2}}
H. $$1:1$$
I. \dfrac{1}{2\sqrt{3}}
J. \dfrac{1}{2\sqrt{6}}
Answer: A
A mono-electronic species in energy level with energy X was provided with  excess of energy so that it jumps to higher energy level with energy Y. If it can emit 6 wavelengths originated from all possible transition between these groups levels, then which of the following relation is correct?(Here n is the principle quantum of energy level X)

Options:
A. X/Y = (n-1)^3
B. $$\sqrt{(X/Y)} =1+3/N$$
C. $$X/Y=N/6$$
D. $$X/Y = 1+3/N$$
E. X/Y = (n-2)^2
F. X/Y = (n-1)^4
G. $$X/Y = (n-1)^{2}$$
H. X/Y = (n-2)^{2}
I. $$X/Y = (n-4)^2$$
J. X/Y = (n-3)^2
Answer: G
The equivalent mass of $H_3BO_3(M=molar \:mass \:of\: H_3BO_3)$ in its reaction with $NaOH$ to form $Na_2B_4O_7$ is equal to:

Options:
A. \frac{M}{5}
B. $$\frac{M}{6}$$
C. $$M$$
D. \frac{M}{3}
E. $$\frac{M}{4}$$
F. \frac{M}{8}
G. \frac{M}{12}
H. $$\frac{M}{2}$$
I. \frac{M}{9}
J. \frac{M}{7}
Answer: C
Hydrogen atoms absorb radiation of wavelength $\lambda_0$ and consequently emit radiations of $6$ different wavelengths of which two wavelengths are shorter than $\lambda_0$. Then,

Options:
A. There are five transitions belonging to Lyman series
B. There are four transitions belonging to Lyman series
C. There are two transitions belonging to Balmer series
D. There are two transitions belonging to Lyman series
E. The initial state of the atoms may be $n=3$
F. There are three transitions belonging to Lyman series
G. There are three transitions belonging to Balmer series
H. There are four transitions belonging to Balmer series
Answer: F
Which of the following sets of quantum numbers represents the 19th electron of $Cr\ (Z = 24)$?

Options:
A. \left( 4, 1, 0, + \dfrac{1}{2} \right )
B. \left( 4, 1, 0, - \dfrac{1}{2} \right )
C. $$\left( 3, 2, -2, + \dfrac{1}{2} \right )$$
D. \left( 4, 0, 0, - \dfrac{1}{2} \right )
E. \left( 4, 0, 1, - \dfrac{1}{2} \right )
F. $$\left( 3, 2, 0, - \dfrac{1}{2} \right )$$
G. \left( 4, 0, 1, + \dfrac{1}{2} \right)
H. $$\left( 4, 1, -1, + \dfrac{1}{2} \right )$$
I. $$\left( 4, 0, 0, + \dfrac{1}{2} \right )$$
Answer: I
A uniform circular loop of radius $a$ and resistance $R$ is placed perpendicular to a uniform magnetic field $B$. One half of the loop is rotated about the diameter with angular velocity $\omega$ as shown in figure. Then, the current in the loop is:

Options:
A. \cfrac{3\pi {a}^{2}B\omega}{4R}, when \theta=\cfrac{\pi}{2}
B. $\cfrac{\pi {a}^{2}B\omega}{2R}$, when $\theta$ is zero
C. zero, when $\theta$ is zero
D. $\cfrac{\pi {a}^{2}B\omega}{R}$, when $\theta=\cfrac{\pi}{2}$
E. Zero, when $\theta=\pi /2$
F. $\cfrac{\pi {a}^{2}B\omega}{2R}$, when $\theta=\cfrac{\pi}{2}$
G. \cfrac{\pi {a}^{2}B\omega}{3R}
H. \cfrac{2\pi {a}^{2}B\omega}{3R}, when \theta=\cfrac{\pi}{2}
I. \cfrac{3\pi {a}^{2}B\omega}{8R}
J. \cfrac{3\pi {a}^{2}B\omega}{4R}
Answer: F
One end of a metal rod of length  $1.0\mathrm { m }$  and area of cross-section  $100\mathrm { cm } ^ { 2 }$  is maintained at  $100 ^ { \circ } C$  If the other end of the rod is maintained at  $0 ^ { \circ } \mathrm { C } ,$  the quantity of heat transmitted through the rod per minute is (coefficient of thermal conductivity of material of rod  $= 100\mathrm { Wkg } ^ { - 1 } \mathrm { K } ^ { - 1 } )$

Options:
A. 7 \times 10 ^ { 3 } \mathrm { J }
B. $$12 \times 10 ^ { 3 } \mathrm { J }$$
C. $$9 \times 10 ^ { 3 } \mathrm { J }$$
D. $$3 \times 10 ^ { 3 } \mathrm { J }$$
E. 4 \times 10 ^ { 3 } \mathrm { J }
F. $$6 \times 10 ^ { 3 } \mathrm { J }$$
G. 10 \times 10 ^ { 3 } \mathrm { J }
H. 8 \times 10 ^ { 3 } \mathrm { J }
I. 2 \times 10 ^ { 3 } \mathrm { J }
J. 5 \times 10 ^ { 3 } \mathrm { J }
Answer: D
In $Fig. 6.377$, both pulleys and the string are massless and all the surfaces are frictionless. 
Given ${ m }_{ 1 }=1 kg,{ m }_{ 2 }=2 kg.{ m }_{ 3 }=3 kg$.
Find the tension in the string.

Options:
A. $$\dfrac { 120 }{ 7 } N$$
B. \dfrac{220}{7} N
C. \dfrac { 110 }{ 7 } N
D. \dfrac { 160 }{ 7 } N
E. \dfrac { 100 }{ 7 } N
F. $$\dfrac { 130 }{ 7 } N$$
G. $$\dfrac { 240 }{ 7 } N$$
H. None of these
I. \dfrac{150}{7} N
J. \dfrac { 140 }{ 7 } N
Answer: A
A body cools in surrounding which is at a constant temperature of $ \theta_0 $ Assume that it obeys newton's law of cooling. its temperature  $ \theta $ is plotted against time t. tangents are drawn to the curve at the points $ P( \theta =\theta_1 ) $ and $ Q ( \theta =\theta_2 ) $. these tangents meet the time axis at angles of $ \phi_2 $ and $ \phi_1 $. as shown

Options:
A. ( tan \phi_2 / tan \phi_1 )= ( \theta_1 - \theta_0 / \theta_2 + \theta_0 )
B. $$ ( tan \phi_2 / tan \phi_1 )= ( \theta_2 - \theta_0 / \theta_1 - \theta_0 ) $$
C. $$ ( tan \phi_1 / tan \phi_2 )= ( \theta_1/ \theta_2 ) $$
D. ( tan \phi_2 / tan \phi_1 )= ( \theta_1 + \theta_0 / \theta_2 - \theta_0 )
E. $$ ( tan \phi_2 / tan \phi_1 )= ( \theta_1 - \theta_0 / \theta_2 - \theta_0 ) $$
F. ( tan \phi_2 / tan \phi_1 )= ( \theta_1 - \theta_0 / \theta_2 + 2\theta_0 )
G. ( tan \phi_2 / tan \phi_1 )= ( \theta_1 - \theta_0 / \theta_2 + 3\theta_0 )
H. $$ ( tan \phi_1 / tan \phi_2 )= ( \theta_2/ \theta_1 ) $$
I. ( tan \phi_2 / tan \phi_1 )= ( \theta_1 + \theta_0 / \theta_2 + \theta_0 )
Answer: E
Light falling on metallic surface of work function $2eV$ is given by $B = (100V/m)[\sin(5 \times 10^{15}s^{-1})t + \sin(8 \times 10^{15})t]$. Then maximum kinetic energy of the photo-electrons is:-

Options:
A. $$4.17\ eV$$
B. $$6.58 \ eV$$
C. 4.58 \ eV
D. 6.27 \ eV
E. 5.42 \ eV
F. 5.67 \ eV
G. 5.32 \ eV
H. $$3.27\ eV$$
I. 5.82 \ eV
J. $$5.2 \ eV$$
Answer: B
Let $E_n = \dfrac{-1me^4}{8arepsilon_0^{2}h^2}$ be the energy of the nth level of H - atom . If all the H - atoms are in the found state and radiation of frequency $\dfrac{\left(E_2 - E_1\right)}{h}$ falls on it,

Options:
A. all atoms will be excited to be $n - 2$ state.
B. some atoms will make a transition to the $n = 5$ state.
C. some atoms will make a transition to the $n = 1$ state.
D. some atoms will make a transition to the $n = 3$ state.
E. some of atoms will move to the first excited state.
F. some atoms will make a transition to the $n = 4$ state.
G. some atoms will make a transition to the $n = 2$ state.
H. no atoms will make a transition to the $n = 3$ state.
I. some atoms will make a transition to the $n = 6$ state.
J. it will not be absorbed at all
Answer: H
The moment of inertia of a solid cylinder of mass M, length 2R and radius R about an axis passing through the center of mass and perpendicular to the axis of the cylinder is $l_{1}$ and about an axis passing through one end of the cylinder and perpendicular to the axis of cylinder is $l_{2}$

Options:
A. \frac{l_{2}}{l_{1}} = \frac{16}{12}
B. \frac{l_{2}}{l_{1}} = \frac{11}{12}
C. $$l_{1}=l_{2}= MR^{2}$$
D. \frac{l_{2}}{l_{1}} = \frac{15}{12}
E. $$\frac{l_{2}}{l_{1}}=\frac{19}{12}$$
F. \frac{l_{2}}{l_{1}} = \frac{13}{12}
G. $$l_{2}=l_{1}= MR^{2}$$
H. $$l_{2}=l_{1}$$
I. \frac{l_{2}}{l_{1}} = \frac{17}{12}
J. \frac{l_{2}}{l_{1}} = \frac{14}{12}
Answer: G
There are $45$ number of cells with internal resistance of each cell is $0.5\Omega$. To get the maximum current through a resistance of $2.5\Omega$, one can use m rows of cells, each row having n cells. The values of m and n are?

Options:
A. m=8, n=5.625
B. m=4, n=11.25
C. m=7, n=6.43
D. $$m=15, n=3$$
E. $$m=9, n=5$$
F. m=10, n=4.5
G. $$m=5, n=9$$
H. $$m=3, n=15$$
I. m=6, n=7.5
J. m=5, n=9.2
Answer: H
A stick of length $L$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass $m$ moving with speed $v$ collides elastically with the stick as shown in figure. If after the collision the ball comes to rest, then what should be the mass of the ball?

Options:
A. m=\cfrac{M}{7}
B. m=\cfrac{M}{3}
C. m=\cfrac{M}{8}
D. m=\cfrac{M}{6}
E. $$m=\cfrac{M}{2}$$
F. m=\cfrac{M}{5}
G. $$m=\cfrac{M}{4}$$
H. $$m=M$$
I. m=\cfrac{M}{9}
J. $$m=2M$$
Answer: G
The Gibb's energy for the decmposition of $ AI_2O_3 $ at $ 500^oC $ is as follows 
$$ \frac { 2 }{ 3 } AI_{ 2 }O_{ 3 }\rightarrow \frac { 4 }{ 3 } AI+O_2 $$
$$ { \Delta  }_{ r }G=+960kJ\quad mol^{ -1 } $$
The potential difference needed for the electrolytic reduction of aluminium oxide $(AI_2O_3) $ at $ 500^oC $ is at least

Options:
A. 3.0 V
B. 4.5 V
C. 5.0 V
D. 3.5 V
E. 4.0 V
F. 4.7 V
G. 4.6 V
H. 2.5 V
I. 5.5 V
J. 4.3 V
Answer: B
An open pipe of sufficient length is dipping in water with a speed $v$ vertically. If at any instant $l$ is length of tube above water. Then the rate at which fundamental frequency of pipe changes, is (speed of sound $= c)$.

Options:
A. $$c/ 4v^{2} t^{2}$$
B. $$cv/ 8l$$
C. $$cv/ 16l^{2}$$
D. $$cv/ 6l^{2}$$
E. $$cv/ 4l^{2}$$
F. cv/ 8l^{2}
G. $$cv/2l^{2}$$
H. $$c/2v^{2}t^{2}$$
I. cv/ 12l^{2}
Answer: E
For a mixture of $100$mL of $0.3$ M $CaCl_2$ solution and $400$ mL of $0.1$M $HCl$ solution, select the correct options.

Options:
A. Moles of \( Ca^{2+} \) in mixture \( =0.025 \)
B. $[Cl^-]=0.2$M
C. Moles of $Ca^{2+}$ in mixture $=0.06$
D. Moles of $Ca^{2+}$ in mixture $=0.045$
E. None of these
F. Moles of $Ca^{2+}$ in mixture $=0.015$
G. Moles of $Ca^{2+}$ in mixture $=0.03$
H. Moles of \( Ca^{2+} \) in mixture \( =0.035 \)
I. Moles of $Ca^{2+}$ in mixture $=0.02$
Answer: G
Two gear wheels of the same thickness (as shown), but one with twice the diameter of the other, are mounted on parallel light axles far enough so that the teeth of the wheels do not mesh. The larger wheel is spun with angular velocity $\Omega$  and the wheels are then moved together so that they mesh. For the meshed rotating gears, if $\displaystyle \frac{Kinetic\  energy\, lost}{Original\, Kinetic\, energy} = \frac{1}{3+X}$, find the value of $X$.

Options:
A. 2
B. 12
C. 7
D. 3
E. 4
F. 8
G. \frac{1}{4}
H. 1
I. 6
J. 5
Answer: A
Five rods of the same dimensions are arranged as shown in the figure. They have thermal conductivities $K_{1},K_{2},K_{3},K_{4}  and  K_{5}$.When points A and B are maintained at different temperatures no heat flows through the central rod CD if:

Options:
A. $$K_{1}K_{2}=K_{3}K_{4}$$
B. K_{1}K_{4}=K_{2}K_{5}
C. K_{1}=K_{5} and K_{2}=K_{3}
D. $$K_{1}K_{4}=K_{2}K_{3}$$
E. $$K_{1}=K_{4} and K_{2}=K_{3}$$
F. $$K_{1}K_{3}=K_{2}K_{4}$$
G. K_{1}=K_{3} and K_{2}=K_{4}
H. K_{1}=K_{2} and K_{3}=K_{4}
I. $$K_{1}=K_{5} and K_{2}=K_{4}$$
Answer: E
There are a number of hydrogen like atoms is free space. Initially all the atoms are stationary and in a particular excited level $A$. They undergo transitions from the initial level $A$ to the ground state. The recoiling atom enters a region of constant magnetic field. If the radii of the paths of the recoiling atoms is observed to have only three different values, then

Options:
A. The angular momenta of the electrons in the initial state of the emitting atoms is $\dfrac {5h}{2\pi}$
B. The angular momenta of the electrons in the initial state of the emitting atoms is $\dfrac {2h}{2\pi}$
C. The angular momenta of the electrons in the initial state of the emitting atoms is $\dfrac {3h}{2\pi}$
D. The angular momenta of the electrons in the initial state of the emitting atoms is $\dfrac {7h}{2\pi}$
E. The principal quantum number of the level $A$is given by $n = 3$
F. The angular momenta of the electrons in the initial state of the emitting atoms is $\dfrac {4h}{2\pi}$
G. Magnetic moment of the atoms in the initial state $A$ is $\dfrac {5eh}{4\pi m}$
H. The radii must be in the ratio $5 : 27 : 32$
I. The angular momenta of the electrons in the initial state of the emitting atoms is $\dfrac {9h}{2\pi}$
J. The angular momenta of the electrons in the initial state of the emitting atoms is $\dfrac {6h}{2\pi}$
Answer: C
The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left( x \right) = \frac{a}{{_x12}} - \frac{b}{{_x6}}$ , where a nad b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is $D = \left[ {U\left( {x = \infty } \right) - {U_{{\text{at}}\;{\text{equilibrium}}}}} \right]$, D is

Options:
A. \dfrac{{{b^2}}}{{8a}}
B. \dfrac{{{b^2}}}{{10a}}
C. $$\dfrac{{{b^2}}}{{2a}}$$
D. \dfrac{{{b^2}}}{{9a}}
E. $$\dfrac{{{b^2}}}{{4a}}$$
F. \dfrac{{{b^2}}}{{5a}}
G. $$\dfrac{{{b^2}}}{{6a}}$$
H. \dfrac{{{b^2}}}{{3a}}
I. \dfrac{{{b^2}}}{{7a}}
J. $$\dfrac{{{b^2}}}{{12a}}$$
Answer: C
A block is attached to the ceiling by a spring that has a force constant $k=200\ N/m$. A conducting rod is rigidly attached to the block. The combined mass of the block and the rod is $m=0.3\ kg$. The rod can slide without friction along two vertical parallel rails, which are a distance $L=1\ m$ apart. A capacitor of known capacitance $C=500\ \mu m$ is attached to the rails by the wires. The entire system is placed in a uniform magnetic field $B=20 \ T$ directed as shown. Find the angular frequency $(in\ rad/sec)$ of the vertical oscillations of the block. Neglect the self-inductance and electrical resistance of the rod and all wires.

Options:
A. \dfrac{20\sqrt{5}}{6}
B. $$\dfrac{20\sqrt{5}}{3}$$
C. \dfrac{10\sqrt{5}}{4}
D. \dfrac{20\sqrt{5}}{5}
E. 10 \sqrt{5}
F. $$20 \sqrt{5}$$
G. $$none\ of\ these$$
H. \dfrac{10\sqrt{5}}{3}
I. $$20$$
J. \dfrac{10\sqrt{5}}{5}
Answer: I
Two waves of light in air have the same wavelength and are initially in phase. They then travel through plastic layers with thickness of $L_{1}=3.5mm$ and $L_{2}=5.0mm$ and indices of refraction $n_{1}=  1.7$ and $n_{2}=1.25$ as shown in the figure. The rays later arrive at a common point. The longest wavelength of light for which constructive interference occurs at the point is

Options:
A. 1.8\mu m
B. $$1.7 \mu m$$
C. 1.5 \mu m
D. 1.6 \mu m
E. 1.3\mu m
F. 1.4 \mu m
G. $$2.9 \mu m$$
H. 1.9 \mu m
I. $$1.2\mu m$$
J. $$0.8\mu m$$
Answer: I
A boy of mass $30kg$ starts running from rest along a circular path of radius $6m$ with constant  tangential acceleration of magnitude $2m\quad { s }^{ -1 }$. After $2s$ from start he feels that his shoes started slipping on the ground. The friction between his shoes and ground is (Take $g=10m\quad { s }^{ -1 }$)

Options:
A. $$\cfrac { 1 }{ 2 } $$
B. \cfrac{1}{6}
C. $$\cfrac { 1 }{ 3 } $$
D. \cfrac{2}{7}
E. \cfrac{1}{7}
F. $$\cfrac { 1 }{ 5 } $$
G. \cfrac{1}{4}
H. \cfrac{2}{5}
Answer: C
A 0.50g sample of rock was found to have $ 2.5 \times 10^4 mol$ of $ ^{ 40 }_{ 19 } K (t_1/2 = 13 \times 10^9 ) $ and $ 7.5 \times 10^8 $ mol of  $^{ 40 }_{ 20 }  Ca$. How old is the rock?

Options:
A. 1.63 \times 10^9 yr
B. $$ 2.6 \times 10^9 yr $$
C. 3.9 \times 10^9 yr
D. $$ 1.3 \times 10^9 yr $$
E. $$ 5.2 \times 10^9 yr $$
F. $$ 6.5 \times 10^8 yr $$
G. 1.95 \times 10^9 yr
H. 3.95 \times 10^9 yr
I. 4.55 \times 10^8 yr
J. 3.25 \times 10^9 yr
Answer: F
An electron and a proton of equal momentum enter a uniform magnetic field normal to the lines of force. If the radii of circular paths be $r_e$ and $r_p$, respectively, then

Options:
A. \left (\dfrac {r_e}{r_p} \right )=\left (\dfrac {m_e}{m_p} \right )
B. \left (\dfrac {r_e}{r_p} \right )=\sqrt[3]{\dfrac {m_e}{m_p}}
C. \left (\dfrac {r_e}{r_p} \right )=\left (\dfrac {m_p}{m_e} \right )
D. \left (\dfrac {r_e}{r_p} \right )=\left (\dfrac {2m_e}{2m_p} \right )
E. \left (\dfrac {r_e}{r_p} \right )=\left (\dfrac {1}{2} \right )
F. \left (\dfrac {r_e}{r_p} \right )=\sqrt[4]{\dfrac {m_e}{m_p}}
G. $$\left (\dfrac {r_e}{r_p}\right )=\sqrt {\dfrac {m_p}{m_e}}$$
H. $$\left (\dfrac {r_e}{r_p}\right )=\sqrt {\dfrac {m_e}{m_p}}$$
I. $$\left (\dfrac {r_e}{r_p}\right )=\left (\dfrac {1}{1}\right )$$
Answer: I
The thermo emf of a thermocouple varies with the temperature $  \theta  $ of the hot junction as $  E=a \theta+b \theta^{2}  $ in volts where the ratio $a   b  $ is $  700^{\circ} \mathrm{C.}  $ If the cold junction is kept at$ 0^{\circ} \mathrm{C} $ then the neutral temperature is

Options:
A. 1750^{\circ} \mathrm{C}
B. 2100^{\circ} \mathrm{C}
C. No neutral temperature is possible for this termocouple
D. $$ 1400^{\circ} \mathrm{C} $$
E. 2800^{\circ} \mathrm{C}
F. 1000^{\circ} \mathrm{C}
G. 1050^{\circ} \mathrm{C}
H. 700^{\circ} \mathrm{C}
I. $$ 350^{\circ} \mathrm{C} $$
Answer: D
In the circuit shown the resistance R has a value that depends on the current. Specifically, R is 20 ohms when I is zero and the increase in resistance in ohms is numerically equal to one half of the current in amperes. What is the value of current I in circuit ?

Options:
A. 8.33 amp
B. 8.75 amp
C. 9.75 amp
D. 8.5 amp
E. 9.5 amp
F. 8.25 amp
G. 10 amp
H. 9.25 amp
I. 18.5 amp
J. 12.5 amp
Answer: G
The magnetic field existing in a region is given by $ec B=B_0 \left [1+\dfrac {x}{l}\right]\hat k$.A square loop of edge $l$ and carrying current $I$ is placed with its edges parallel to the $x-y$ axis. The magnitude of the net magnetic force experienced by the loop is :

Options:
A. Zero
B. \dfrac{B_0 Il}{8}
C. \dfrac{B_0 Il}{4}
D. $$B_0 Il$$
E. \( \dfrac{3B_0 Il}{2} \)
F. $$\dfrac{B_0 Il}{2}$$
G. $$2B_0Il$$
H. $$B_0 Il^2$$
I. $$3B_0 I l$$
Answer: D
In a parallel plate capacitor connected to a constant voltage source, a dielectric is introduce such that its surface area is same as that of plates but the thickness of the dielectric is half the separation between the plates. If dielectric constant of the dielectric between the plates is 6, then

Options:
A. the potential difference across the plates of the capacitor decreases by about 33.33%
B. percentage loss of energy of the capacitor, for introducing the dielectric between the plates is 41.67%
C. the potential difference across the plates of the capacitor decreases by about 62.5%
D. the potential difference across the plates of the capacitor decreases by about 41.67%
E. the potential difference across the plates of the capacitor decreases by about 37.5%
F. the charge of the capacitor remains conserved while introducing the dielectric between the plates.
G. the capacitors decreases by about 71%
H. the potential difference across the plates of the capacitor decreases by about 25%
I. the potential difference across the plates of the capacitor decreases by about 50%
J. the potential difference across the plates of the capacitors decreases by about 71%
Answer: F
Direction: Question number 5-7 are based on the following paragraph An initially parallel cylindrical beam travels in a medium of refractive in dex $ \mu(I) = \mu_o + \mu_2I $ Where $ \mu_o and \mu_2 $ are positive constan and I is the intensity of the light bem. The intensity of the beam is decreasing with increasing radius 
 (1) How does beam travel after entering in medium?

Options:
A. Converge
B. Diverge near the axis, diverge near the periphery
C. Converge near the periphery, diverge near the axis
D. Converge near the axis diverge near the periphery
E. Travel as a cylindrical beam
F. Converge near the axis, converge near the periphery
G. Converge near the axis, diverge near the periphery
H. Diverge near the axis, converge near the periphery
I. Diverge
J. Diverge near the axis converge near the periphery
Answer: I
An ideal gas has molar heat capacity at constant pressure : $C_P$ = 5R/2. The gas is kept in a cylindrical vessel fitted with a piston which is free to move. Mass of the frictionless piston is 9 kg. Initial volume of the gas is $0.0027m^3$ and cross-section area of the piston is $0.09m^2$. The initial temperature of the gas is 300 K. Atmospheric pressure $P_0 = 1.05 \times 10^5 N/m^2$. An amount of $2.5 \times 10^4J$ of heat energy is supplied to the gas, then:

Options:
A. Final pressure of the gas is $1.058 \times 10^5 N/m^2$
B. Initial pressure of the gas is $1.06 \times 10^5 N/m^2$
C. Final pressure of the gas is $1.04 \times 10^5 N/m^2$
D. Final pressure of the gas is $1.06 \times 10^5 N/m^2$
E. Final pressure of the gas is $1.055 \times 10^5 N/m^2$
F. Final pressure of the gas is $1.065 \times 10^5 N/m^2$
G. Work done by gas is $9.94 \times 10^3J$
H. Final pressure of the gas is $1.05 \times 10^5 N/m^2$
I. Final pressure of the gas is $1.052 \times 10^5 N/m^2$
J. Final temperature of the gas is 1000 K
Answer: D
The half-cell reactions for the corrossion are 
${2H^+}$ + $1/2  O_2$ + $2e^-$ $\rightarrow$ $H_2O$ ; $E^o$ = - 1.23 V
$Fe^{2+}$ + $2e^-$ $\rightarrow$ Fe ; $E^o$ = -0.44 V
Find the  $\triangle {G^o}$ (in kJ) for the overall reaction

Options:
A. 77
B. 152
C. -161
D. -322
E. -76
F. -38
G. 322
H. -152
Answer: E
Dry air at $15C$ and $10$ $atm$ is suddenly released in atmospheric pressure. Find the final temperature of the air.  (for air, $\cfrac{Cp}{Cv}=1.41$)

Options:
A. 168K
B. 138K
C. $$298K$$
D. 188K
E. $$148K$$
F. $$108K$$
G. 178K
H. $$198K$$
I. 128K
J. 158K
Answer: E
A ring of resistance $10 \Omega $ , radius $10cm$ and $100$ turns is rotated $100$ revolutions per second about a fixed axis which is perpendicular to a uniform magnetic field of induction $ 10 m T $ . The amplitude of the current in the loop will be nearly (Take $ \pi^2 = 10 $)

Options:
A. None of these
B. $$2 A$$
C. 2000 A
D. 0.2 A
E. 0.0002 A
F. 20 A
G. 0.002 A
H. 0.02 A
I. $$200 A$$
Answer: A
In the circuit shown, ${{\rm{I}}_1}$ is the current through the battery just after switch is closed and ${{\rm{I}}_2}$ is the current through the battery after a long time, then

Options:
A. $${{{I_2}} \over {{I_1}}} = 2$$
B. $${{{I_1}} \over {{I_2}}} = \infty $$
C. $${{{I_1}} \over {{I_2}}} = 2$$
D. $${{{I_1}} \over {{I_2}}} = 1$$
E. \frac{{I_1}}{{I_2}} = \frac{3}{2}
F. \frac{{I_1}}{{I_2}} = \frac{1}{2}
G. {{{I_1}} \over {{I_2}}} = 4
H. \frac{I_1}{I_2} = \frac{4}{3}
I. \frac{I_1}{I_2} = \frac{2}{3}
J. \frac{I_1}{I_2} = \frac{5}{2}
Answer: D
Electromagnetic radiations having $\lambda = 310 \overset{o}{A}$ am subjected to a metal sheet having work functions $=12.8\, eV$. What will be the velocity of photoelectrons with maximum Kinetic Energy.

Options:
A. 2.18\sqrt{5} \times 10^6 \, m/s
B. $$2.18\sqrt{2} \times 10^6 \, m/s$$
C. 2.18\sqrt{6} \times 10^6 \, m/s
D. 2.18\sqrt{8} \times 10^6 \, m/s
E. $$8.72 \times 10^6 \, m/s$$
F. $$2.18 \times 10^6 \, m/s$$
G. 2.18\sqrt{7} \times 10^6 \, m/s
H. 0. no emission will occur
I. $$2.18\sqrt{3} \times 10^6 \, m/s$$
J. 2.18\sqrt{4} \times 10^6 \, m/s
Answer: B
Four vessels $1, 2, 3$ and $4$ contain respectively: $10\ g-atom\ (t_{1/2}=10\ h),\ 1\ g-atom\ (t_{1/2}=5\ h),\ 5\ g-atom\ (t_{1/2}=2\ h)$ and $2\ g-atom\ (t_{1/2}=1\ h)$ of different radioactive nuclides. In the beginning, the maximum radioactive would be exhibited by the vessel

Options:
A. 6
B. $$5$$
C. 1
D. $$3$$
E. $$4$$
F. 2
Answer: E
The capacitor C is initially without charge. X is now joined to Y for a long time, during which $H_1$ heat is produced in the resistance R. X is now joined to Z for a long time, during which $H_2$ heat is produced in R

Options:
A. the maximum energy stored in C at any time is \(3H_1\)
B. $$H_1=H_2$$
C. the maximum energy stored in C at any time is \(\frac{1}{4}H_1\)
D. the maximum energy stored in C at any time is \(\frac{3}{4}H_1\)
E. $$H_1=2H_2$$
F. the maximum energy stored in C at any time is $H_1$
G. the maximum energy stored in C at any time is \(\frac{1}{6}H_1\)
H. the maximum energy stored in C at any time is \(\frac{1}{3}H_1\)
I. $$H_1=\frac {1}{2}H_2$$
J. the maximum energy stored in C at any time is \(\frac{1}{2}H_1\)
Answer: F
The displacement of $S$ of a particle at time $t(O< t< \pi$) is given by $S=\sin 2t-6\cos t$. Then the acceleration for the value of $t$ for which its velocity is zero is

Options:
A. 1\ unit /\sec^{2}
B. -4\ unit /\sec^{2}
C. -3\ unit /\sec^{2}
D. $$0\ unit /\sec^{2}$$
E. -2\ unit /\sec^{2}
F. $$4\ unit /\sec^{2}$$
G. $$3\ unit /\sec^{2}$$
H. 5\ unit /\sec^{2}
I. $$2\ unit /\sec^{2}$$
J. -1\ unit /\sec^{2}
Answer: D
On a warm summer day, the transpiration pull is the main force that drives from root parenchyma into the root xylem. The table shows values of $\psi_p$(pressure potential) and $\psi_s$ (solute potential) in root xylem and root parenchyma, in kPa. In which of the options(a-d) would transpiration pull cause water to move from root parenchyma into the root xylem?

Options:
A. Root parenchyma- $\psi_p = 200, \psi_s=-190$, Root xylem- $\psi_p=-200, \psi_s=5$
B. Root parenchyma- $\psi_p=200, \psi_s=-220$, Root xylem- $\psi_p=-65, \psi_s=-5$
C. Root parenchyma- $\psi_p=-200, \psi_s=220$, Root xylem- $\psi_p=65, \psi_s=-5$
D. Root parenchyma- $\psi_p=200, \psi_s=-220$, Root xylem- $\psi_p=65, \psi_s=-5$
E. Root parenchyma- $\psi_p=200, \psi_s=-225$, Root xylem- $\psi_p=-65, \psi_s=-10$
F. Root parenchyma- $\psi_p=200, \psi_s=-215$, Root xylem- $\psi_p=-65, \psi_s=-5$
G. Root parenchyma- $\psi_p=200, \psi_s=-220$, Root xylem- $\psi_p=-65, \psi_s=-15$
H. Root parenchyma- $\psi_p=200, \psi_s=-220$, Root xylem- $\psi_p=65, \psi_s=-15$
I. Root parenchyma- $\psi_p=200, \psi_s=-220$, Root xylem- $\psi_p=-60, \psi_s=-10$
J. Root parenchyma- $\psi_p=200, \psi_s=-210$, Root xylem- $\psi_p=-65, \psi_s=-5$
Answer: B
A large cylindrical tank of cross-sectional area $1m$ is filled with water. It has a small hole at a height of $1m$ from the bottom. A movable piston of mass $5 kg$ is fitted on the top of the tank such that it can slide in the tank freely without friction. A load of $45 kg$ is applied on the top of water by piston, as shown in figure. The value of v when piston is $7m$ above the bottom is (g = $10$ $m/s^2$) :-

Options:
A. $$11 m/s$$
B. 9 m/s
C. $$14 m/s$$
D. $$\sqrt{110} m/s$$
E. 13 m/s
F. $$1 m/s$$
G. $$\sqrt{120} m/s$$
H. $$10 m/s$$
I. 12 m/s
Answer: A
At some instant, a particle is moving along a straight line $2x-3y=2$ and its coordinates on that line are $(4,2)$. Now, at another instant, the same particle is moving along a straight line $3x+4y=7$ and its coordinates are $(1,1)$. Find the coordinates of that point about which it is in pure rotation

Options:
A. \dfrac{40}{17}, \dfrac{40}{17}
B. \dfrac{40}{17}, \dfrac{45}{17}
C. $$\dfrac{20}{17}, \dfrac{25}{17}$$
D. $$\dfrac{50}{17}, \dfrac{61}{17}$$
E. \dfrac{40}{17}, \dfrac{35}{17}
F. $$\dfrac{30}{17}, \dfrac{35}{17}$$
G. \dfrac{30}{17}, \dfrac{55}{17}
H. \dfrac{30}{17}, \dfrac{45}{17}
I. $$\dfrac{8}{9}, \dfrac{16}{9}$$
J. none of these
Answer: F
138 gm of $N_{2}O_{4} (g)$ is placed in a $8.2L$ container at $300 K$. If the equilibrium vapour density of mixture was found to be $30.67$, then: $[R = 0.082 \ L.atm .mol^{-1}. K^{-1}]$

Options:
A. The density of equilibrium mixture is $16.83 gm/litre$
B. The density of equilibrium mixture is $17.23 gm/litre$
C. $K_{p}$ of $N_{2}O_{4}\rightleftharpoons 2NO_{2}(g)$ is $9$ atm
D. Total pressure at equilibrium is $6.75$ atm
E. The density of equilibrium mixture is $16.93 gm/litre$
F. The density of equilibrium mixture is $15.73 gm/litre$
G. The density of equilibrium mixture is $16.63 gm/litre$
H. The density of equilibrium mixture is $15.83 gm/litre$
I. The density of equilibrium mixture is $17.93 gm/litre$
Answer: A
A rod of length L has non uniformly distributed mass along its length. For its mass per unit length varying with distance x from one end as $\dfrac{m_0}{L^2}(L+x)$. Find the position of centre of mass of this system. Discuss the case, when $x=0$.

Options:
A. $$L/5$$
B. $$L/2$$
C. $$L/4$$
D. L/12
E. $$L/3$$
F. L/10
G. L/7
H. L/9
I. L/6
J. L/8
Answer: B
A mixture of $_{}^{239}\textrm{Pu}$ and $_{}^{240}\textrm{Pu}$ has a specific activity of $6\times10^9\:dis/s/g$. The half lives of the isotopes are $2.44\times10^4y$ and $6.08\times10^3\:y$ respectively. Calculate the isotopic composition of this sample.

Options:
A. $_{239}Pu=45.1$%, $_{240}Pu=54.9$%
B. $_{239}Pu=54.1$%, $_{240}Pu=45.9$%
C. $_{239}Pu=54.9$%, $_{240}Pu=45.1$%
D. $_{239}Pu=45.0$%, $_{240}Pu=55.0$%
E. $_{239}Pu=55.1$%, $_{240}Pu=44.9$%
F. $_{239}Pu=44.9$%, $_{240}Pu=55.1$%
G. $_{239}Pu=50.0$%, $_{240}Pu=50.0$%
H. $_{239}Pu=45.2$%, $_{240}Pu=54.8$%
I. None of these
J. $_{239}Pu=54.0$%, $_{240}Pu=46.0$%
Answer: A
An electron and a proton have the same De Broglie wavelength. Then the kinetic energy of the electron is :

Options:
A. \(\frac{3}{4}\) the kinetic energy of proton
B. Equal kinetic energy of proton
C. \(\frac{1}{2}\) the kinetic energy of proton
D. Greater than the kinetic energy of proton
E. Zero
F. \(\frac{1}{4}\) the kinetic energy of proton
G. Infinity
H. Twice the kinetic energy of proton
I. Less than the kinetic energy of proton
J. Half the kinetic energy of proton
Answer: B
If one mole of an ideal gas $\displaystyle \left ( \gamma =\frac{4}{3} \right )$ is expanded reversibly and adiabatically from 800 K to 200 K, which of the following statement's are correct ?

Options:
A. ratio of final pressure to initial pressure is \(\displaystyle \frac{1}{512} \)
B. ratio of final pressure to initial pressure is \(\displaystyle \frac{1}{128}\)
C. \(\displaystyle \frac{1}{4096}\)
D. ratio of final volume to initial volume is $32$
E. ratio of final pressure to initial pressure is \(\displaystyle \frac{1}{32} \)
F. ratio of final pressure to initial pressure is $\displaystyle \frac{1}{256}$
G. ratio of final pressure to initial pressure is $\displaystyle \frac{1}{16}$
H. ratio of final pressure to initial pressure is $\displaystyle \frac{1}{1024}$
I. ratio of final pressure to initial pressure is \(\displaystyle \frac{1}{768}\)
Answer: F
In a diploid plant with $2n=14$, all the chromosomes form a ring structure during meiotic prophase. The reason for this is?

Options:
A. Multiple single translocations
B. Gene mutations
C. Multiple deletions
D. Multiple single deletions
E. Multiple inversions
F. Multiple duplications
G. Single deletions
H. Multiple reciprocal translocations
I. Duplications
J. Single reciprocal translocations
Answer: C
A uniform thin flat isolated disc is floating in space. It has radius $R$ and mass $m$. A force $F$ is applied to it at a distance $d=(R/2)$ from the centre in the y-direction. Treat this problem as two dimensional. Just after the force is applied:

Options:
A. acceleration of leftmost point on the disc is \( \frac{F}{4m} \)
B. acceleration of leftmost point on the disc is \( \frac{F}{2m} \)
C. acceleration of leftmost point on the disc is \(\frac{F}{6m}\)
D. point which is instantaneously unaccelerated is the rightmost point
E. acceleration of leftmost point on the disc is \(\frac{F}{3m}\)
F. acceleration of the centre of the disc is $F/m$
G. angular acceleration of the disk is $F/mR$
H. acceleration of leftmost point on the disc is zero
I. acceleration of leftmost point on the disc is \(\frac{F}{5m}\)
J. acceleration of leftmost point on the disc is \(\frac{F}{7m}\)
Answer: H
A(2,3).B(3, -5) are two vertices of $\triangle ABC.C$ is a point in the line $x+4y-5=0.$ Then the locus of the centroid of $\triangle ABC$ is a line parallel to

Options:
A. L=2
B. L=1
C. AC
D. L=5
E. L=0
F. L=4
G. BC
H. L=o
I. L=3
J. AB
Answer: G