MikaStars39/nano-eval huggingface dataset:
The x-coordinate of a particle moving on x-axis is given by $x = 3 sin 100t + 8 cos^2 50t$, where x is in cm and t is time in seconds. Which of the following is/are correct about this motion?

Options:
A. the amplitude of the resultant S.H.M. is $\sqrt{65}$ cm
B. the maximum displacement of the particle from the origin is $12$ cm.
C. the amplitude of the S.H.M. of the particle is $4$ cm
D. the amplitude of the resultant S.H.M. is  $\sqrt{73}$ cm
E. the maximum displacement of the particle from the origin is $9$ cm.
F. the maximum displacement of the particle from the origin is $7$ cm.
G. the maximum displacement of the particle from the origin is $10$ cm.
Answer: E
A flexible chain of length L = 20$\surd {2}$ m and mass W = 10 kg is initially placed at rest on a smooth frictionless wedge surface ABC. It is given a slight jerk on one side so that it will start sliding on the side. Find the speed of the chain when its one end will leave the vertex of the wedge

Options:
A. 8 m/s
B. $$(10\surd {2})^{1/2}m/s$$
C. 5\sqrt{2} m/s
D. $$5\sqrt{4} m/s$$
E. 5\sqrt{3}\ m/s
F. $$4 m/s$$
G. 5\sqrt{2}\ m/s
H. $$10\surd{2}\ m/s$$
I. 20 m/s
J. $$10 m/s$$
Answer: J
At $40^oC$, a brass wire of $1 mm$ radius is hung from the ceiling. A small mass, $M$ is hung from the free end of the wire. When the wire is cooled down from $40^oC$ to $20^oC$ it regains its original length of $0.2 m$. The value of $M$ is close to : (Coefficient of linear expansion and Young's modulus of brass are $10^{-5} / ^oC$ and $10^{11} \ N/m^2$, respectively; $g = 10 ms^{-2}$)

Options:
A. 0.8 \ kg
B. $$0.9 \ kg$$
C. 1.2 \ kg
D. $$0.5 \ kg$$
E. 1.1 \ kg
F. 1.3 \ kg
G. $$9 \ kg$$
H. 7.5 \ kg
I. 1.8 \ kg
J. $$1.5\ kg$$
Answer: G
A parallel plate capacitor of capacitance $10 \mu F$ is connected to a cell of e.m.f. 10 V and is fully charges. Now a dielectric slab (k = 3) of thickness equal to the gap between the plates is completely filled in the gap, keeping the cell connected. During the filings process,

Options:
A. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab + 125 \mu J
B. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab + 50 \mu J
C. the increase in charge on the capacitor is $200 \mu C$
D. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab
E. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab + 75 \mu J
F. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab + 150 \mu J
G. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab + 25 \mu J
H. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab + 100 \mu J
I. energy supplied by the cell = increase in stored potential energy + work done on the person who is filling the dielectric slab + heat produced
J. the heat produced is zero
Answer: D
It is observed that some of the spectral lines in hydrogen spectrum have wavelengths almost equal to those of the spectral lines in ${He}^{+}$ ion. Out of the following the transitions in ${He}^{+}$ that will make this possible is

Options:
A. $n=6$ to $n=4$
B. $n=4$ to $n=2$
C. $n=3$ to $n=1$
D. $n=5$ to $n=3$
E. $n=5$ to $n=2$
F. n=4 to n=1
G. n=6 to n=3
H. $n=3$ to $n=2$
Answer: C
A gas expands in a piston-cylinder device from ${V}_{1}$ to ${V}_{2}$, the process being described by $PV=a+bV$, where $P$ is in $N{m}^{-2}$, $V$ is in ${m}^{3}$ and $a$ and $b$ are constants. The work done in the process is:

Options:
A. a\log _{ e }{ \left( \cfrac { { V }_{ 1 } }{ { V }_{ 2 } } \right) } -b\left( { V }_{ 1 }-{ V }_{ 2 } \right)
B. -a\log _{ e }{ \left( \cfrac { { V }_{ 1 } }{ { V }_{ 2 } } \right) } +b\left( { V }_{ 1 }-{ V }_{ 2 } \right)
C. -a\log _{ e }{ \left( \cfrac { { V }_{ 1 } }{ { V }_{ 2 } } \right) } -b\left( { V }_{ 1 }-{ V }_{ 2 } \right)
D. $$-a\log _{ e }{ \left( \cfrac { { V }_{ 1 } }{ { V }_{ 2 } } \right) } -b\left( { V }_{ 2 }-{ V }_{ 1 } \right) $$
E. $$-a\log _{ e }{ \left( \cfrac { { V }_{ 2 } }{ { V }_{ 1 } } \right) } -b\left( { V }_{ 2 }-{ V }_{ 1 } \right) $$
F. -a\log _{ e }{ \left( \cfrac { { V }_{ 2 } }{ { V }_{ 1 } } \right) } -b\left( { V }_{ 1 }-{ V }_{ 2 } \right)
G. $$a\log _{ e }{ \left( \cfrac { { V }_{ 1 } }{ { V }_{ 2 } } \right) } +b\left( { V }_{ 2 }-{ V }_{ 1 } \right) $$
H. a\log _{ e }{ \left( \cfrac { { V }_{ 2 } }{ { V }_{ 1 } } \right) } -b\left( { V }_{ 2 }-{ V }_{ 1 } \right)
I. $$a\log _{ e }{ \left( \cfrac { { V }_{ 2 } }{ { V }_{ 1 } } \right) } +b\left( { V }_{ 2 }-{ V }_{ 1 } \right) $$
J. -a\log _{ e }{ \left( \cfrac { { V }_{ 1 } }{ { V }_{ 2 } } \right) } +b\left( { V }_{ 2 }-{ V }_{ 1 } \right)
Answer: E
To calculate the size of a hydrogen anion using the Bohr model, we assume that its two electrons move in an orbit such that they are always on diametrically opposite sides of the nucleus. With each electron having the angular momentum equal to $\dfrac{h}{ 2 \pi}$, and taking electron interaction into account the radius of the orbit in terms of the Bohr radius of hydrogen atom $a_B=\displaystyle\frac{4\pi \epsilon_0h^2}{me^2}$ is?

Options:
A. $$\displaystyle\frac{3}{2}a_B$$
B. \displaystyle\frac{5}{4}a_B
C. \dfrac{5}{3}a_B
D. \displaystyle\frac{3}{4}a_B
E. $$\displaystyle\frac{2}{3}a_B$$
F. $$\displaystyle\frac{4}{3}a_B$$
G. \displaystyle\frac{4}{5}a_B
H. $$a_B$$
I. \displaystyle\frac{3}{5}a_B
J. \displaystyle\frac{7}{4}a_B
Answer: F
An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston(as shown in the figure). Initially the gas is at temperature $T_1$, pressure $P_1$ and volume $V_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_2$, pressure $P_2$ and volume $V_2$. During this process the piston moves out by a distance x. Ignoring the friction between the piston and the cylinder, the correct statement(s) is(are).

Options:
A. If $V_2=3V_1$ and $T_2=4T_1$, then the work done by the gas is $\dfrac{5}{3}P_1V_1$
B. If $V_2=3V_1$ and $T_2=4T_1$, then the work done by the gas is $\dfrac{7}{3}P_1V_1$
C. If $V_2=3V_1$ and $T_2=4T_1$, then the work done by the gas is $\dfrac{6}{3}P_1V_1$
D. If $V_2=3V_1$ and $T_2=4T_1$, then the heat supplied to the gas is $\dfrac{17}{6}P_1V_1$
E. If $V_2=3V_1$ and $T_2=4T_1$, then the work done by the gas is $\dfrac{11}{3}P_1V_1$
F. If $V_2=3V_1$ and $T_2=4T_1$, then the work done by the gas is $\dfrac{5}{2}P_1V_1$
G. If $V_2=3V_1$ and $T_2=4T_1$, then the work done by the gas is $\dfrac{9}{3}P_1V_1$
H. If $V_2=3V_1$ and $T_2=4T_1$, then the work done by the gas is $\dfrac{8}{3}P_1V_1$
Answer: B
The cell reaction of an electrolyte cell is given below:$Fe + 2Fe^{3+} \rightarrow 3Fe^{2+}  $ with $E^o = 0.404\ V$ and $\Delta G^o = - 2.424\ kJ$. If the cell reaction is to be written as $2Fe + 4Fe^{3+}   \rightarrow 6Fe^{2+}$, then which of the following statements will be correct?

Options:
A. The value of $n$ changes from $6$ to $7$.
B. $\Delta G^o$ will be given by $- 2.424\ kJ$.
C. The value of $n$ changes from $6$ to $11$
D. The value of \( n \) changes from 6 to 10.
E. The value of \( n \) changes from 6 to 9.
F. The value of $n$ changes from 6 to 8.
G. The value of \( n \) changes from 6 to 3.
H. The value of $n$ changes from $6$ to $12$.
Answer: H
The kinetic energy of photo-electron emitted on irradiating a metal surface with frequency $v$ is related by $KE=hv-IE$. The plots of $KE$ $vs$. incidented frequency $v$ shows:

Options:
A. a straight line with extra polated intercept on $y$-axis equal to the product of the ionization energy and the threshold frequency
B. a straight line with extra polated intercept on $y$-axis equal to the product of ionization energy and Planck's constant.
C. a straight line with slope equal to Planck's constant.
D. a straight line with extra polated intercept on $y$-axis equal to the threshold frequency.
E. a straight line with extra polated intercept on $y$-axis equal to the product of the threshold frequency and Planck's constant.
F. a straight line with intercept on $x$-axis equal to threshold frequency.
G. a straight line with intercept on $x$-axis equal to the ratio of ionization energy to Planck's constant.
H. a straight line with extra polated intercept on $y$-axis equal to ionization energy.
I. a straight line with intercept on $x$-axis equal to the product of threshold frequency and Planck's constant.
J. a straight line with extra polated intercept on $y$-axis equal to the product of ionization energy and threshold frequency.
Answer: H
${ He }^{ + }$ ion is in the ground state, a photon of energy 48.36 eV has been absorbed by the electron. Find the KE and PE of electron __________________.

Options:
A. K = 6.04 eV, U = -60.44eV
B. K = 6.04 eV, U = -6.04eV
C. K = 6.04 eV, U = -54.4eV
D. K = 6.04 eV, U = -42.32eV
E. K = 6.04 eV, U = 12.08eV
F. K = 6.04 eV, U = -12.08eV
G. K = 6.04 eV, U = -24.16eV
H. K = 8.36 eV, U = -6.04eV
I. K = 6.04 eV, U = -48.36eV
J. K = 6.04 eV, U = -18.12eV
Answer: C
A photosensitive metallic surface is illuminated alternately with lights of wavelength $3100\overset {\circ}{A}$ and $6200\overset {\circ}{A}$. It is observed that maximum speeds of the photoelectrons in two cases are in ratio $2 : 1$. The work function of the metal is $(hc = 12400\ eV\overset {\circ}{A})$.

Options:
A. \dfrac{5}{4}eV
B. $$\dfrac {4}{3}eV$$
C. $$2 eV$$
D. $$1 e V$$
E. \dfrac {3}{4}eV
F. \dfrac{1}{2}eV
G. \dfrac{3}{2}eV
H. $$\dfrac {2}{3}ev$$
I. \dfrac{5}{3}eV
Answer: C
A fighter plane of length 20 m, wing span (distance from tip of one wing to the tip of the other wing) of 15 m and height 5 m is flying towards east over Delhi. Its speed is $240 ms^{-1}$. The earth's magnetic field over Delhi is $5 \times 10^{-5}$ T with the declination angle $\sim 0^o$ and dip of $\theta$ such that $sin \theta = \frac{2}{3}$. If the voltage developed is $V_B$ between the lower and upper side of the plane and $V_{W}$ between the tips of the wings then $V_{B}$ and $V_{W}$ are close to

Options:
A. V_B = 40 mV; V_W= 120 mV with right side of pilot at higher voltage
B. $V_B = 40 mV; V_W= 120 mV$ with left side of pilot at higher voltage
C. $V_B =45 mV; V_W= 120 mV $ with right side of pilot at higher voltage
D. V_B = 45 mV; V_W= 135 mV with left side of pilot at higher voltage
E. $V_B = 45 mV; V_W= 120 mV$ with left side of pilot at higher voltage
F. $V_B= 40 mV; V_W= 135 mV$ with right side of pilot at high voltage
G. V_B = 40 mV; V_W= 135 mV with right side of pilot at higher voltage
H. $V_B= 40 mV; V_W= 135 mV$ with left side of pilot at higher voltage
Answer: E
A thin spherical shell of mass $m$ and radius $R$ is completely filled with water of mass $m$. Find the angular frequency of oscillation of the shell about the smooth horizontal axis $P$.

Options:
A. $$\omega = \sqrt {\dfrac {15g}{46R}}$$
B. \omega = \sqrt {\dfrac {15g}{51R}}
C. $$\omega = \sqrt {\dfrac {15g}23{R}}$$
D. \omega = \sqrt {\dfrac {15g}{52R}}
E. $$\omega = \sqrt {\dfrac {3g}{46R}}$$
F. \omega = \sqrt {\dfrac {15g}{47R}}
G. \omega = \sqrt {\dfrac {15g}{49R}}
H. $$\omega = \sqrt {\dfrac {3g}{23R}}$$
I. \omega = \sqrt {\dfrac {15g}{59R}}
J. \omega = \sqrt {\dfrac {15g}{45R}}
Answer: A
The cell, $Pb | H_{2}(1\ atm)|| H^{+} (pH = x)|$normal calomel electrode, has an emf of $0.67$ volt as $25^{\circ}C$. Calculate the $pH$ of the solution. The oxidation potential of calomel electrode on the hydrogen scale is :$-0.28\ volt$.

Options:
A. $pH = 8.61$.
B. pH = 6.16
C. pH = 7.11
D. $pH = 7.61$.
E. pH = 6.11
F. none of these
G. pH = 5.16
H. pH = 5.61
I. $pH = 6.61$.
J. pH = 7.16
Answer: I
Suppose the density of air at Hyderabad is $\rho _ { 0 }$ and atmospheric pressure is $P _ { \mathrm { atm } }$.Suppose we wish to calculate the pressure $P$ at a height $10 \mathrm { km }$ above Hyderabad. If we use the equation $P _ { 0 } - P = \rho g z$ to calculate $P$ with $z = 10 km$. But if we take only variation of $\rho$ with height we get $P_1$. if we take only of $g$ with height we get $P_2$, if we take both $\rho$ and $g$ variations with height we get $P_3$. Then ?

Options:
A. P _ { 3 } > P _ { 2 }
B. P _ { 1 } > P
C. $$P _ { 0 } - P > P _ { 0 } - P _ { 3 }$$
D. $$P _ { 3 } < P _ { 1 }$$
E. P _ { 3 } > P
F. P _ { 2 } < P
G. $$P = P _ { 1 } = P _ { 2 } = P _ { 3 }$$
H. P _ { 0 } - P > P _ { 0 } - P _ { 2 }
I. $$P _ { 2 } > P$$
J. P _ { 0 } - P < P _ { 0 } - P _ { 3 }
Answer: I
The maximum intensity in Young's double-slit experiment is $I_{0}$. Distance between the slits is $d=5\lambda$, where $\lambda$ is the wavelength of monochromatic light is used in the experiment. What will be the intensity of light in front of one of the slits on a screen at a distance $D=d$ ?

Options:
A. \dfrac{7}{4}I_{0}
B. $$\dfrac{I_{0}}{4}$$
C. $$I_{0}$$
D. \dfrac{I_{0}}{3}
E. \dfrac{2I_{0}}{3}
F. $$\dfrac{I_{0}}{2}$$
G. \dfrac{3I_{0}}{4}
H. \dfrac{I_{0}}{6}
I. $$\dfrac{3}{4}I_{0}$$
J. \dfrac{5}{4}I_{0}
Answer: F
The relative density of a material of a body is found by weighing it first in air and then in water. If the weight of the body in air is $W_1 = 8.00 +0.05N$ and the weight in water is $W_2 = 6.00+ 0.05 N$, then the relative density $\rho_r = W_1/(W_1 - W_2)$ with the maximum permissible error is

Options:
A. $$4.00 + 3.2\%$$
B. $$4.00 + 0.82\%$$
C. $$4.00 + 0.62\%$$
D. $$4.00 + 2.62\%$$
E. 4.00 + 6.22\%
F. 4.00 + 4.2\%
G. 4.00 + 0.42\%
H. 4.00 + 7.22\%
I. 4.00 + 1.62\%
J. $$4.00 +5.62\%$$
Answer: J
If the velocity of an $e^-$ in the K shell of hydrogen atom is measured with an error of 10%, the uncertainty in its position measurement would be (velocity of electron in K-shell=$2.18\times 10^6\,m/s$)

Options:
A. $$>10 A^0$$
B. $$2.6\times 10^{-2} A^0$$
C. 2.5 \times 10^{-2} A^0
D. $$1 A^0$$
E. 2.6 \times 10^{-3} A^0
F. $$2.7 A^0$$
G. 2.9 \times 10^{-2} A^0
H. 2.7 \times 10^{-2} A^0
I. 2.4 \times 10^{-2} A^0
J. 2.8 \times 10^{-2} A^0
Answer: B
An $AC$ source is connected with a resistance $(R)$ and an uncharged capacitance $(C)$, in series. The potential difference across the resistor is in phase with the initial potential difference across the capacitor for the first time at the instant (assume that at $t=0$, emf is zero)

Options:
A. \dfrac { \pi }{ 6\omega }
B. \dfrac { \pi }{ 7\omega }
C. $$\dfrac { \pi }{ 2\omega }$$
D. \dfrac { 3\pi }{ 4\omega }
E. \dfrac { \pi }{ 5\omega }
F. \dfrac { \pi }{ 3\omega }
G. $$\dfrac { 2\pi }{ \omega }$$
H. \dfrac { 5\pi }{ 4\omega }
I. $$\dfrac { 3\pi }{ 2\omega }$$
J. $$\dfrac { \pi }{ 4\omega }$$
Answer: J
At $ 300^0 C $ sulphur trioxide decomposes into sulphur dioxide and oxygen as follows:
$ 2SO_3 \leftrightharpoons 2So_2 +O_2 : K_e = 1.6 \times 10^{-10} $ in a mixture of the three species at equilibrium , we can accurately predict that

Options:
A. the concentration of $ SO_2 $ will be exactly the same as the concentration of $ SO_3 $
B. The concentration of $ O_2 $ will be exactly twice the concentration of $ SO_3 $
C. the concentration of both $ SO_2 $ and $ O_2 $ will be much smaller than the concentration of $ SO_3 $
D. The concentration of $ O_2 $ will be exactly one half the concentration of $ SO_3 $
E. the concentration of $ SO_2 $ will be exactly half the concentration of $ SO_3 $
F. the concentration of both $ SO_2 $ and $ O_2 $ will be much than the concentration of $ SO_3 $
G. the concentration of $ O_2 $ will be much larger than the concentration of $ SO_2 $
H. the concentration of both $ SO_2 $ and $ O_2 $ will be much larger than the concentration of $ SO_3 $
I. the concentration of either $ SO_2 $ and $ O_2 $ ( and possibly both ) will be much smaller than the concentration of $ SO_2 $
J. the concentration of $ O_2 $ will be much larger than the concentration of $ SO_3 $
Answer: F
A dibasic acid containing $C,H$ and $O$ was found to contain $C=26.7$% and $H=22$%. The vapour density of its dimethyl ester was found to be $73$. What is the molecular formula?

Options:
A. C_2H_4O_3
B. C_2H_4O_4
C. C_2H_6O_4
D. $$C_2H_4O_2$$
E. C_2H_6O_2
F. $$HCHO$$
G. $$C_2H_2O_4$$
H. C_2H_4O_5
I. C_2H_2O_3
Answer: G
Two ideal carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, $T_1$ and $T_2$. The temperature of the hot reservoir of the first engine is $T_1$ and the temperature of the cold reservoir of the second engine is $T_2$. T is temperature of the sink of first engine which is also the source for the second engine. How is T related to $T_1$ and $T_2$, if both the engines perform equal amount of work?

Options:
A. T = \dfrac{T_1 + T_2}{3}
B. T = \dfrac{T_1 + T_2}{8}
C. $$T = \dfrac{T_1 + T_2}{2}$$
D. T = \dfrac{T_1 + T_2}{6}
E. $$T = \sqrt{T_1 T_2}$$
F. T = \dfrac{T_1 + T_2}{5}
G. T = \dfrac{T_1 + T_2}{4}
H. T = \dfrac{T_1 + T_2}{7}
I. $$T = 0$$
J. $$T = \dfrac{2 T_1 T_2}{T_1 + T_2}$$
Answer: C
An electron in hydrogen atom after absorbing energy photons can jump between energy states $n_{1}$ and $n_{2}(n_{2} > n_{1})$. Then it may return to ground state after emitting six different wavelengths is emission spectrum. The energy of emitted photons is either equal to, less than or greater than the absorbed photons. Then $n_{1}$ and $n_{2}$ are

Options:
A. n_{2} = 5, n_{1} = 1
B. $$n_{2} = 5, n_{1} = 3$$
C. n_{2} = 4, n_{1} = 4
D. n_{2} = 5, n_{1} = 2
E. $$n_{2} = 4, n_{1} = 1$$
F. n_{2} = 3, n_{1} = 2
G. n_{2} = 4, n_{1} = 2
H. n_{2} = 3, n_{1} = 1
I. $$n_{2} = 4, n_{1} = 3$$
Answer: I
When 5 litres of a gas mixture of methane and propane is perfectly combusted at and 1 atmosphere, 16 litre of oxygen at the same temperature and pressure is consumed. The amount of heat released from this combustion in KJ $\left( {{\text{\Delta }}{{\text{H}}_{{\text{Comd}}\;}}{\text{(C}}{{\text{H}}_{\text{4}}}} \right){\text{ = 890}}\;{\text{KJ}}\;{\text{mo}}{{\text{l}}^{{\text{ - 1}}}}\;{\text{,}}\;{\text{\Delta }}{{\text{H}}_{{\text{comb}}}}\left( {{{\text{C}}_{\text{3}}}{{\text{H}}_{\text{8}}}} \right){\text{ = 2220}}\;{\text{KJ}}\;{\text{mo}}{{\text{l}}^{{\text{ - 1}}}}{\text{)}}\;{\text{is}}$

Options:
A. 477
B. 371
C. 327
D. 38
E. 312
F. 387
G. 317
H. 32
I. 417
J. 377
Answer: G
A pure Si crystal has $5 \times 10^{28} \ atoms \ m^{-3}$. It is doped by 1 ppm concentration of pentavalent As. The number of holes is (approx) (Given $n_i = 1.5 \times 10^{16} \ m^{-3}$)

Options:
A. $$4.5 \times 10^9 \ m^{-3}$$
B. 2.2 \times 10^9 \ m^{-3}
C. $$6.2 \times 10^6 \ m^{-3}$$
D. 2.2 \times 10^8 \ m^{-3}
E. $$8.1 \times 10^9 \ m^{-3}$$
F. 3.2 \times 10^6 \ m^{-3}
G. 2.2 \times 10^7 \ m^{-3}
H. 3.5 \times 10^6 \ m^{-3}
I. 1.2 \times 10^7 \ m^{-3}
J. $$2.2 \times 10^6 \ m^{-3}$$
Answer: J
If dimensions of length are expressed as ${ G }^{ x }{ C }^{ y }{ h }^{ z }$ where $G$, $C$ and $h$ are universal gravitational constant, speed of light and Planck's constant respectively. Then

Options:
A. $$z = 3/2$$
B. $$z = 1/2$$
C. $$y = -3/2$$
D. $$x = z$$
E. $$x = y$$
F. z = -3/2
G. z = 1/4
H. $$z = -1/2$$
Answer: B
A horizontal is string fixed at two ends, is vibrating in its fifth harmonic according o the equation $y(x,t)=0.01m\ \sin [(62.8m^{-1})x]\ \cos [(628s^{-1})t]$. Assuming $\pi =3.14$, the correct statement(s) is (are)

Options:
A. The fundamental frequency is $130\ Hz$.
B. The fundamental frequency is $110\ Hz$.
C. The fundamental frequency is $200\ Hz$.
D. The fundamental frequency is $110\ Hz$
E. The number of nodes is $5$.
F. The fundamental frequency is $100\ Hz$.
G. The fundamental frequency is $125\ Hz$.
H. The fundamental frequency is $150\ Hz$.
Answer: F
An electric dipole moment $ \overrightarrow P = (2.0 \hat i + 3.0 \hat j) \mu $ is placed in a uniform electric field $ \overrightarrow E = (3.0 \hat i+ 2.0 \hat k) \times 10^5 NC^{-1} $ .

Options:
A. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.4 J
B. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.5 J
C. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.7 J
D. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.6 J
E. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.8 J
F. The potential energy of the dipole is 0.9 J
G. The potential energy of the dipole is 0.6 J
H. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.2 J
I. The torque that $ \overrightarrow E $ exerts on P is $ (0.6 \hat i - 0.4 \hat j - 0.9 \hat k)$ Nm
J. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.3 J
Answer: J
Sodium (atomic mass $=23$ amu) crystallizes in a bcc arrangement with the interfacial separation between the atoms at the edge $53.6$ pm. The density (in g cm$^{-3}$) of sodium crystal is:

Options:
A. 1.29
B. $$2.07$$
C. 2.19
D. 1.97
E. 2.32
F. none of the above
G. 1.91
H. 1.21
I. $$2.46$$
J. $$1.19$$
Answer: J
A point moves in a straight line so that its displacement $x$ m at time t sec is given by $ x^2= 1+t^2$. Its acceleration in $ m/s^2 $  at a time t sec is:

Options:
A. $$\cfrac {1} {x} - \cfrac {t^2} {x^3}$$
B. \cfrac {1} {x} - \cfrac {4t^2} {x^3}
C. \cfrac {1} {x} - \cfrac {2t^2} {x^3}
D. $$\cfrac {1} {x} - \cfrac {1} {x^2}$$
E. $$1/x^3$$
F. \cfrac {1} {x} - \cfrac {2t^2} {x^2}
G. \cfrac {1} {x} - \cfrac {t^2} {x^4}
H. \cfrac {1} {x} - \cfrac {3t^2} {x^3}
I. \cfrac {1} {x} - \cfrac {2t} {x^3}
J. $$-t/x^3$$
Answer: A
A homogeneous rod of length $l = \eta x$ and mass M is lying on a smooth horizontal floor. A bullet of mass m hits the rod at a distance x from the middle of the rod at a velocity $\bar v_0$ perpendicular to the rod. If the velocity of the farther end of the rod just after impact is in the opposite direction of $\bar v_0$ then.

Options:
A. \eta > 5
B. \eta < 7
C. \eta < 3
D. $$\eta > 6$$
E. \eta < 5
F. $$\eta > 3$$
G. \eta < 4
H. $$\eta < 6$$
I. \eta < 8
Answer: H
Magnetic field at p on the axis of two coplanar, concentric current carrying loop is zero if:

Options:
A. \dfrac {i_1}{i_2} = \dfrac {5}{2} \sqrt {\dfrac {7}{5}}
B. $ \dfrac {i_2}{i_1}$ = $\dfrac {5}{2}$  $ \sqrt {\dfrac {5}{8}} $
C. $ \dfrac {i_2}{i_1} = \dfrac {5}{2} \sqrt {\dfrac {7}{8}} $
D. $ \dfrac {i_1}{i_2}$ = $\dfrac {5}{2}$ $ \sqrt {\dfrac {5}{8}} $
E. $ \dfrac {i_1}{i_2}$ = $\dfrac {5}{2}$ $ \sqrt {\dfrac {8}{5}} $
F. $ \dfrac {i_2}{i_1}$ = $\dfrac {5}{2}$ $ \sqrt {\dfrac {8}{5}} $
G. \dfrac {i_1}{i_2} = \dfrac {5}{2} \sqrt {\dfrac {6}{5}}
Answer: E
In Young's double slit experiment, the separation between slits is $0.2\ mm\, ,\  \lambda =600\ $ and the distance of the screen from the slit is 2.0 m. If intensity at the centre of the central maximum is $0.2\quad W/m^2$, the intensity at a point 0.5cm away from this center along the which of fringes will be

Options:
A. $$0.15 \quad W/m^2$$
B. 0.125 \quad W/m^2
C. $$0.05 \quad W/m^2$$
D. $$0.10 \quad W/m^2$$
E. 0.055 \quad W/m^2
F. 0.06 \quad W/m^2
G. 0.07 \quad W/m^2
H. 0.075 \quad W/m^2
I. 0.045 \quad W/m^2
J. $$0.20 \quad W/m^2$$
Answer: C
$2.4$ mole of $H_2$ sample was taken. In one experiment $60\%$ of the sample was exposed to continuous radiation frequency $4.47\times 10^{15} Hz$, of which all the electron are removed from the atom. In another experiment, remaining sample was irradiated with a light of wavelength $600 \mathring{A}$, when all the electrons are removed from the surface. The ratio of the maximum velocity if the ejected electrons in the two cases. Assume that ejected electrons do not interact with any photon(Ionization potential of $H=36eV$).

Options:
A. 1
B. 1.76
C. 1.14
D. 1.44
E. 1.09
F. 1.39
G. 1.19
H. 1.29
I. 2.44
J. 1.36
Answer: E
Three gears A, B and C are in contact with each other and rotate about their respective axes passing through the their centre of mass. Radius of gear 'A' is 'R' that of 'B' is 2R and that of 'C' is 3R. When 'A'  is given angular velocity $'\omega '$ it rotates 'B' and 'B' in turn rotates 'C' with angular. velocity $\omega _{ 1 }$. Now B is replaced with another gear 'B' of radius 4 R Now 'A' is again rotated with angular velocity $\omega $, it rotates 'B' which in turn rotates 'C' with angular velocity $\omega _{ 2 }$

Options:
A. $${ { \omega }_{ 1 }>\omega }_{ 2 }$$
B. \(\omega_{1} = \frac{7}{8} \omega_{2} \)
C. $${ { \omega }_{ 1 }=\omega }_{ 2 }$$
D. \(\omega_{1} = \frac{6}{7} \omega_{2} \)
E. $$\omega _{ 1 }=4/3{ \omega }_{ 2 }$$
F. $${ { \omega }_{ 1 }<\omega }_{ 2 }$$
G. \(\omega_{1} = \frac{5}{6} \omega_{2} \)
H. \(\omega_{1} = \frac{1}{2} \omega_{2} \)
I. \(\omega_{1} = \frac{3}{4} \omega_{2}\)
J. \(\omega_{1} = \frac{2}{3} \omega_{2} \)
Answer: F
To what series does the spectral lines of atomic hydrogen belong of its wave number is equal to the difference between the wave numbers of the following two lines of the Balmer series $486.1\ nm$ and $410.2\ nm$? What is the wavelength of this line?

Options:
A. Brackett, $\lambda=125\times 10^{-4}cm$
B. Paschen, $\lambda=26.3\times 10^{-4}cm$
C. Paschen, $\lambda=2.63 \times 10^{-6}cm$
D. Lyman, $\lambda=9.1\times 10^{-5}cm$
E. Paschen, $\lambda=263 \times 10^{-6}cm$
F. Paschen, $\lambda=263 \times 10^{-7}cm$
G. Paschen, $\lambda=263 \times 10^{-5}cm$
H. Paschen, $\lambda=2.63 \times 10^{-5}cm$
I. Paschen, $\lambda=2.63 \times 10^{-3}cm$
J. Paschen, $\lambda=2.63\times 10^{-4}cm$
Answer: J
The roots of $6x^{4}-35x^{3}+62^{2}-35x+6=0$ are.......

Options:
A. 2,1/2,-1/3,-3
B. 2,1/2,3,1/3
C. -2,-1/2,-3,-1/3
D. 2,-1/2,-1/3,3
E. 2,1/2,-1/3,3
F. 2,1/2,-3,-1/3
G. 2,1/2,-1/3,1/3
H. 2,1/2,-1/3,-1/2
I. 2,1/2,1/3,-3
Answer: D
If $y_1 = 5 (mm)\ \sin \pi t$ is equation of oscillation of source $S_1$ and $y_2 = 5\ (mm) \sin (\pi t + \pi /6)$ be that of $S_2$ and it takes $1\ sec$ and $1/2\ sec$ for the transverse waves to reach point $A$ from sources $S_1$ and $S_2$ respectively then the resulting amplitude at point $A$. is :-

Options:
A. $$5\sqrt 3/ 2\ mm$$
B. 5\sqrt{2} + 5\ mm
C. $$5\sqrt{2} - \sqrt{3}\ mm$$
D. 5\sqrt{2} + 3\ mm
E. $$\frac{5\sqrt{3}}{2} + 2\ mm$$
F. 10\ mm
G. $$5\ mm$$
H. 5\sqrt{2} - 2\ mm
I. $$5\sqrt 2\ mm$$
J. $$5\sqrt2 + \sqrt3\ mm$$
Answer: G
A small particle of mass $m$ moves in such a way that $P.E=-\displaystyle \frac{1}{2}mkr^{2}$, where $k$ is a constant and $r$ is the distance of the particle from origin. Assuming Bohr's model of quantization of angular momentum and circular orbit, $r$ is directly proportional to:

Options:
A. n^{1/3}
B. none of these
C. $$n^{2}$$
D. n^{1/4}
E. n^{3/2}
F. $$n^{1/2}$$
G. $$\sqrt {n}$$
H. $$n$$
I. $$n^{3/4}$$
J. $$\sqrt[4]{n}$$
Answer: G
The electron is a hydrogen atom  makes a transition from $n_1$ to $n_2$ where $n_1$ and $n_2$ are the principal quantum numbers of the two states. Assume Bohr's model to be valid. The time period of the electron in the initial state  is eight times that in the final state. The posible value of $n_1$ and $n_2$ are

Options:
A. $$n_1=6, \: n_2=1$$
B. $$n_1=4, \: n_2=2$$
C. n_1=8, \: n_2=3
D. n_1=8, \: n_2=4
E. $$n_1=6, \: n_2=3$$
F. n_1=8, \: n_2=1
G. n_1=6, \: n_2=2
H. $$n_1=8, \: n_2=2$$
I. n_1=6, \: n_2=4
Answer: E
The magnitude of the average electric field normally present in the atmosphere just above the surface of the Earth is about 150 N/C, directed inward towards the contre of the Earth. This gives the total net surface charge carried by the Earth to be :$[given arepsilon _{0}= 8.88\times 10^{-12}C^{2}/N m^{2],R_{E}=6.37\times 10^{6}m$

Options:
A. $$+670KC $$
B. +680KC
C. +660KC
D. $$-680KC $$
E. -670KC
F. -660KC
G. +665KC
H. +690KC
Answer: A
In standard Young's double slit experiment, visible light $( 390 nm, 700 nm)$ is used in both the slits. Distance between the slits is $2 mm$, and distance of screen from the slits is $1 m$. A thin glass slab ($\mu =$ 1.5) of thickness $6$ $\mu m$ is placed in front of slit $S_{2}$. Light of which wavelength (s) is missing at point $P$, which is directly in front of slit $S_{2}$. (Neglect dispersion of light)

Options:
A. \dfrac{1800}{3} nm
B. \dfrac{1900}{3} nm
C. \dfrac{1400}{3} nm
D. \dfrac{2200}{3} nm
E. $$500 nm $$
F. $$\dfrac{2000}{3} nm$$
G. $$\dfrac{1600}{3} nm$$
H. \dfrac{2100}{3} nm
Answer: F
A point sound source is situated in a medium of bulk modulus $ 1.6b \times 10^{5} \, N/m^2 $ . The equation for the wave emitted it is given by $ y = A \,sin (7.5 \, \pi x - 300 \, \pi t ) $ . Velocity of wave is $
u $ and the displacement amplitude of the waves received by the observer standing at a distance $ 5 \,m $ from the source is A . The density of medium is $ \rho $ . The pressure amplitude at the observer ear is $ 30\,Pa $ . The intensity of wave received by the observer is I . Then

Options:
A. A = \dfrac{10^{-4}}{3 \pi}
B. $$ \rho = 1 \,kg/m^3 $$
C. $$ I = 1\,W/m^2 $$
D. A = \dfrac{10^{-4}}{8 \pi}
E. $
u = 400 \,m/s $
F. A = \dfrac{10^{-4}}{5 \pi}
G. A = \dfrac{10^{-4}}{6 \pi}
H. $$ A = \dfrac{10^{-4}}{4 \pi} $$
I. A = \dfrac{10^{-4}}{2 \pi}
J. A = \dfrac{10^{-4}}{7 \pi}
Answer: H
A steel wire of length $20$ cm and cross-section $1  mm^2$ is ried rigidly at both ends at room temperature $30^0$. If the temperature falls to $10^0C$, what will be the change in tension? Given that $Y= 2\times 10^{11}Nm^2$ and $lpha= 1.1\times 10^5/^0C$

Options:
A. 24.2
B. $$2.42$$
C. $$22.2$$
D. 22.42
E. 24.42
F. $4.4$ N
G. $$4.42$$
H. $2.2$ N
I. $22$ N
J. $$44$$
Answer: J
If $4x-3y=\sqrt{3}$ is a tangent to $4{x}^{2}-9{y}^{2}=1$ then the eccentric angle of the point of contact is

Options:
A. $$\dfrac{\pi}{4}$$
B. $$\dfrac{\pi}{2}$$
C. \dfrac{\pi}{5}
D. \dfrac{\pi}{8}
E. \dfrac{\pi}{10}
F. $$\dfrac{\pi}{3}$$
G. \dfrac{\pi}{12}
H. $$\dfrac{\pi}{6}$$
I. \dfrac{\pi}{9}
J. \dfrac{\pi}{15}
Answer: H
Ice crystallises in a hexagonal lattice. At low temperature at which the structure was determined, the lattice constant are : $a$ = $4.53$ $\overset { o }{ A } $, and c (height) = $7.41\overset { o }{ A } $. How many $H_2O$ molecules are contained in a unit cell ? (Density of ice = $0.92$${ gcm }^{ -3 }$).

Options:
A. 1
B. 3
C. 2
D. 7
E. 9
F. 5
G. 8
H. 4
I. 6
J. 12
Answer: A
When $100 V$ dc is applied across a coil, a current of $1 A$ flows through it. When $100 V$ ac of $50 Hz$ is applied across the same coil, only $0.5 A$ flows. The resistance and inductance of the coil are ( $take^{\pi^2} = 10)$

Options:
A. $$50\Omega,\sqrt { (0.3) } H$$
B. $$100\Omega,\sqrt { (0.3) } H$$
C. 50\Omega, 0.2 H
D. $$100\Omega, 0.3 H$$
E. 50\Omega, 0.4 H
F. 100\Omega, \sqrt{ (0.4) } H
G. 50\Omega, 0.6 H
H. 100\Omega, 0.4 H
I. 50\Omega, 0.5 H
J. $$50\Omega, 0.3 H$$
Answer: J
On a two road a car A is travelling with a speed of v=5 $m{s^{ - 1}}$.Two car B and C approach car A in opposite direction with a speed u=10 $m{s^{ - 1}}$ each.At a certain instant when the B and C are equidistant from A each being $l$=1500 m, B decides to  overtake A before C does.What minimum  acceleration of car B is required to avoid an accident with c:

Options:
A. -1/10 $m{s^{ - 2}}$
B. -0.2 $m{s^{ - 2}}$
C. 0.15 $m{s^{ - 2}}$
D. 0.2 $m{s^{ - 2}}$
E. -0.1 $m{s^{ - 2}}$
F. 0.1 $m{s^{ - 2}}$
G. -1/15 $m{s^{ - 2}}$
H. 1/15 $m{s^{ - 2}}$
I. -0.15 $m{s^{ - 2}}$
J. -0.12 $m{s^{ - 2}}$
Answer: B
Energy levels A, B, C of an atom are shown below
If $E_A < E_B < E_C$, then correct statements of the following is

Options:
A. \lambda_3 = \lambda_1 + 2\lambda_2
B. $$\lambda_3 = \frac{\lambda_1 \times \lambda_2}{\lambda_2 + \lambda_1}$$
C. \lambda_3 = \lambda_1 - \lambda_2
D. \lambda_3 = \lambda_1 + \lambda_2 + 1
E. \lambda_3 = 2\lambda_1 + \lambda_2
F. $$\lambda_3 = \lambda_1 + \lambda_2$$
G. \lambda_3 = \lambda_1 \cdot \lambda_2
H. $$\lambda_1 + \lambda_2 + \lambda_3 = 0$$
I. \lambda_3 = \lambda_2 - \lambda_1
J. $$\lambda_3^2 = \lambda_1^2 + \lambda_2^2$$
Answer: F
When light of $3000\mathop {\text{A}}\limits^{\text{0}} $ is incident on sodium chloride, the stopping potential is 1.85 volt and when light of $4000\mathop {\text{A}}\limits^{\text{0}} $ is incident, then the stopping potential becomes 0.82 volt. The threshold wavelength for sodium is :$\left( {h = 6.6 \times {{10}^{ - 34}}{\text{J - sec}}} \right)$

Options:
A. 5210\mathop {A}\limits^{0}
B. $$5451\mathop {\text{A}}\limits^{\text{0}} $$
C. 5211\mathop {A}\limits^{0}
D. $$5211\mathop {\text{A}}\limits^{\text{0}} $$
E. $$5320\mathop {\text{A}}\limits^{\text{0}} $$
F. $$5154\mathop {\text{A}}\limits^{\text{0}} $$
G. 5251\mathop {A}\limits^{0}
H. 5155\mathop {A}\limits^{0}
I. 5153\mathop {A}\limits^{0}
J. 5156\mathop {A}\limits^{0}
Answer: F
An acid-base indicator has $K_{a} = 3.0 \times 10^{-5}$. The acid form of the indicator is red and the basic form is blue. The $[H^{+}]$ required to change the indicator from $75$% blue to $75$% red is

Options:
A. $$8 \times 10^{-5}$$
B. 1.1 \times 10^{-5}
C. $$1 \times 10^{-5}$$
D. 7 \times 10^{-5}
E. $$9 \times 10^{-5}$$
F. $$3.33 \times 10^{-5}$$
G. 3.01 \times 10^{-5}
H. 1.2 \times 10^{-4}
I. 2.99 \times 10^{-5}
J. 6 \times 10^{-5}
Answer: E
Air (ideal gas with $\gamma = 1.4$) at $1 bar$ and $300K$ is compressed till the final volume is one-sixteenth of the original volume, following a polytropic process $PV^{1.25} =$ const. Calculate the heat transferred to the system during the process.

Options:
A. $$96.0 J$$
B. 91.5 J
C. 85.2 J
D. 89.4 J
E. 88.3 J
F. 92.3 J
G. $$90.6 J$$
H. 84.1 J
I. $$69 .0 J$$
J. $$60.9 J$$
Answer: G
A cylinder steel plug is inserted into a circular hole of diameter $2.60 cm$ in a brass plate. When the plug and the plates are at a temperature of $20^\circ C$, the diameter of the plug is $0.010 cm$ smaller than that of the hole. The temperature at which the plug will just fit in it is:
[Given, $lpha_ \text{steel} = \dfrac{11 \times 10^{-6}}{^\circ C}$ and $lpha_ \text{brass} = \dfrac{19 \times 10^{-6}}{^\circ C}$]

Options:
A. $$-458^\circ C$$
B. $$-48^\circ C$$
C. -18^\circ C
D. $$-20^\circ C$$
E. -28^\circ C
F. -38^\circ C
G. -25^\circ C
H. $$-10^\circ C$$
I. -30^\circ C
J. -58^\circ C
Answer: H
In Young's experiment $\lambda =$ 4000Å fringes observed have a width $\beta $ . The light illuminating with new $\lambda =$ 6000Å and the separation between the interfering sources is halved. The ratio of the distance between the screen and the interfering sources if the fringe width remains unaltered respectively is:

Options:
A. 1/2
B. 3/1
C. 2/1
D. 4/3
E. 3/4
F. 4/1
G. 2/3
H. 1/3
I. 3/2
J. 1/4
Answer: B
If $E_e, E_{lpha}$ and $E_P$ represents the kinetic energy of an electron, $lpha$ - particle and proton respectively each moving with same de Broglie wavelength then :

Options:
A. $$E_{lpha} > E_e > E_{lpha}$$
B. $$E_e > E_P > E_{lpha}$$
C. E_{\alpha} = E_e > E_P
D. E_{\alpha} = E_P > E_e
E. E_{\alpha} > E_P > E_e
F. $$E_e = E_{lpha} = E_P$$
G. $$E_e = E_{\alpha} > E_P$$
H. E_e < E_{\alpha} < E_P
I. E_P > E_e > E_{\alpha}
J. $$E_e > E_{lpha} > E_P$$
Answer: F
The electric field associated with a light wave is given by $E=E_0\sin[(1.57\times 10^7)(x-ct)]$(where, x and t are in metre and second). The stopping potential when its light is used in an experiment on photoelectric effect with the emitter having work function $\phi =19$eV is?

Options:
A. 2.2$eV
B. 1.8
C. $0.6$eV
D. $2.4$eV
E. $1.4$eV
F. $1.2$eV
G. $1.0$eV
H. $3.0$eV
I. 1.6$eV
J. $1.8$eV
Answer: F
A small spherical monoatomic ideal gas bubble $\left(\gamma =\dfrac{5}{3}\right)$ is trapped inside a liquid of density $\rho_l$(see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is $T_0$, the height of the liquid is H and the atmospheric pressure is $P_0$(Neglect surface tension).
The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant).

Options:
A. \dfrac{\rho_l n R g T_0}{(P_0 + \rho_l g H)^{3/5}[P_0 + \rho_l g (H - y)]^{2/5}}
B. \dfrac{𫞁ho_lnRgT_0}{(P_0+ffield_lgH)^{2/5}[P_0+ffield_lg(H-y)]^{4/5}}
C. \dfrac{\rho_l n R g T_0}{(P_0 + \rho_l g H)^{2/5}[P_0 + \rho_l g (H - y)]^{4/5}}
D. \dfrac{\rho_l n R g T_0}{(P_0 + \rho_l g H)^{3/5}[P_0 + \rho_l g (H - y)]^{3/5}}
E. $$\rho_lnRgT_0\dfrac{(P_0+\rho_lgH)^{3/5}}{(P_0+\rho_lgy)^{8/5}}$$
F. $$\rho_lnRgT_0\dfrac{(P_0+\rho_lgH)^{2/5}}{(P_0+\rho_lgy)^{7/5}}$$
G. \dfrac{\rho_l n R g T_0}{(P_0 + \rho_l g H)^{4/5}[P_0 + \rho_l g (H - y)]^{1/5}}
H. $$\dfrac{\rho_lnRgT_0}{(P_0+\rho_lgH)^{2/5}[P_0+\rho_lg(H-y)]^{3/5}}$$
Answer: H
Determine the heat of the following reaction:
$$FeO(s)+{ Fe }_{ 2 }{ O }_{ 3 }(s)\longrightarrow { Fe }_{ 3 }{ O }_{ 4 }(s)\quad $$
Given informations:
$$2Fe(s)+{ O }_{ 2 }(g)\longrightarrow 2FeO(s);\quad \Delta { H }^{ o }=-544kJ$$
$$4Fe(s)+3{ O }_{ 2 }(g)\longrightarrow 2{ Fe }_{ 2 }{ O }_{ ;3 }(s);\Delta { H }^{ o }=-1648.4kJ$$
$${ Fe }_{ 3 }{ O }_{ 4 }(s)\longrightarrow 3Fe(s)+2{ O }_{ 2 }(g);\Delta { H }^{ o }=+1118.4kJ$$

Options:
A. $$-22.2kJ$$
B. -111.1kJ
C. $$-1074kJ$$
D. -122.2kJ
E. $$-2214.6kJ$$
F. -24.2kJ
G. -24.4kJ
H. -23.2kJ
I. -22.1kJ
J. $$-249.8kJ$$
Answer: A
Three charges $Q, +q, -q$ are placed at the vertices of an isosceles right angled triangle as shown. The net electrostatic potential energy of the system is $0$ if $Q=$

Options:
A. $$\dfrac{-2q}{1+\sqrt2}$$
B. $$-2q$$
C. \dfrac{-2q}{1-\sqrt2}
D. \dfrac{-2q}{\sqrt2-1}
E. $$2q$$
F. \dfrac{-2q}{2-\sqrt2}
G. \dfrac{2q}{1+\sqrt2}
H. $$\dfrac{-q}{1+\sqrt2}$$
I. \dfrac{-2q}{2+\sqrt2}
J. \dfrac{2q}{1-\sqrt2}
Answer: A
General solution of $cotx+tanx=2cosecx$ is

Options:
A. 2n\pi\pm\dfrac{7\pi}{6},n\inZ
B. $$2n\pi\pm\dfrac{4\pi}{3},n\in Z$$
C. $$2n\pi\pm\dfrac{2\pi}{3},n\inZ$$
D. 2n\pi\pm\dfrac{5\pi}{6},n\in Z
E. 2n\pi\pm\dfrac{7\pi}{3},n\in Z
F. 2n\pi\pm\dfrac{\pi}{4},n\inZ
G. $$2n\pi\pm\dfrac{\pi}{3},n\in Z$$
H. $$2n\pi\pm\dfrac{5\pi}{3},n\in Z$$
I. 2n\pi\pm\dfrac{\pi}{6},n\in Z
Answer: C
In a common emitter transistor amplifier circuit $\beta=100$. Input resistance $R_{1}=1k\Omega$ out put resistance $R_{2}=10k\Omega$. The power gain is given by

Options:
A. 10^ {8}
B. 10^ {2}
C. $$10^ {7}$$
D. $$10^ {3}$$
E. $$10^ {5}$$
F. $$10^ {9}$$
G. $$10^ {6}$$
H. 10^ {4}
Answer: D
A spring with no mass attached to it hangs from a rigid support. A mass m is now hung on the lower end to the spring. The mass is supported on a platform so that the spring remains relaxed. The supporting platform is then suddenly removed and the mass begins to oscillate. The lowest position of mass during the oscillation is 5 cm below the place where it was resting on the platform. What is the angular frequency of oscillation? Take $g=10 ms^{-2}$

Options:
A. 10 rad s^{-1}
B. $$30 rad s^{-1}$$
C. 15 rad s^{-1}
D. $$18 rad s^{-1}$$
E. 25 rad s^{-1}
F. 35 rad s^{-1}
G. $$40 rad s^{-1}$$
H. $$20 rad s^{-1}$$
Answer: H
The value of $n$ for which de Broglie wavelength of $H$ like atom corresponds to $n$th orbit is equal to the wavelength of $n$th line of Lyman series?(Given $Z=11$)

Options:
A. 20
B. 10
C. 15
D. $$5$$
E. 35
F. 30
G. $$25$$
Answer: G
There is a small air bubble inside a glass sphere ($\mu =1.5)$ of radius $5cm$. The bubble is at $O$ at $7.5cm$ below the surface of the glass. The sphere is placed inside the water ($\mu =\cfrac{4}{3}$) such that the top surface of the glass is $10cm$ below the surface of water. The bubble is viewed normally from air. Then the apparent depth of the bubble is

Options:
A. $$\cfrac{31}{2}cm$$
B. \cfrac{35}{2}cm
C. \cfrac{33}{2}cm
D. $$\cfrac{27}{2}cm$$
E. $$\cfrac{23}{2}cm$$
F. \cfrac{28}{2}cm
G. \cfrac{26}{2}cm
H. \cfrac{29}{2}cm
I. $$\cfrac{25}{2}cm$$
Answer: D
The photon radiated from hydrogen corresponding to $2nd$ line of Lyman series is absorbed by a hydrogen like atom $'X'$ in $2nd$ excited state. As a result the hydrogen like atom $'X'$ makes a transition to $n^{th}$ orbit. Then,

Options:
A. $$X=Li^{++}, n=9$$
B. X=Li^{++}, n=8
C. X=He^+, n=5
D. X=He^+, n=2
E. X=He^+, n=3
F. $$X=He^+, n=7$$
G. $$X=He^+, n=4$$
H. X=Li^{++}, n=7
I. $$X=He^+, n=6$$
J. $$X=Li^{++}, n=6$$
Answer: G
In YDSE, maximum for two wavelengths $\lambda_1$ & \Lambda_2$ belonging to visible reagion ( $380 nm- 740nm$), is formed at angular position $\left( \frac{1}{40}\right)$ radian on the screen. If the slit seperation is $d= 10^{-4}m$, then find $\lambda_1 $ & $ \lambda_2$.

Options:
A. 500 nm, 650 nm
B. 500 nm, 575 nm
C. $$400 nm, 600 nm$$
D. $$400 nm, 625 nm$$
E. 500 nm, 600 nm
F. 500 nm, 525 nm
G. $$500 nm, 625 nm$$
H. $$500 nm, 400 nm$$
I. 500 nm, 550 nm
J. 500 nm, 425 nm
Answer: G
In a Young's double slit experiment the slit is illuminated by a source having two wavelengths of 400 nm and 600 nm. If distance between slits, d=1 mm, and distance between the plane of the slit and screen. D=10 m then the smallest distance from the central maximum where there is complete darkness is:

Options:
A. 2 mm
B. 3 mm
C. 12 mm
D. 4 mm
E. there is no such point
F. 1.5 mm
G. 5 mm
H. 6 mm
I. 1 mm
J. 8 mm
Answer: A
A boy of mass $m$ is at one end of a flat boat of mass $M$ and length $l$ which floats stationary on water. If boy moves to the other end,

Options:
A. The displacement of boy with respect to ground is $ = \dfrac { - 7Ml} { ( M + m ) }$
B. The displacement of boy with respect to ground is $ = \dfrac { - 5Ml} { ( M + m ) } $
C. The displacement of boy with respect to ground is $ = \dfrac { - Ml} { ( M + m ) }$
D. The displacement of boy with respect to ground is $ = \dfrac { - 3Ml} { ( M + m ) }$
E. The displacement of boy with respect  to ground is $=\dfrac{-ml}{(M+m)}$
F. The displacement of boy with respect to ground is $ = \dfrac { - 4Ml} { ( M + m ) } $
G. The boat moves in opposite direction through a distance $=\dfrac{Ml}{(m+M)}$
H. The boat moves in opposite direction through a distance $ = \dfrac { m l } { ( M + m ) }$
I. The displacement of boy with respect to ground is $ = \dfrac { - 2Ml} { ( M + m ) }$
J. The displacement of boy with respect to ground is $ = \dfrac { - 6Ml} { ( M + m ) }$
Answer: C
A rectangular coil of conductor ABCD with the length $AD=3\ $ and breadth $DC=2\ m $is hung from one of the arms of a balance, as shown in the Figure. A current $I=8\ A$ is passed through the coil. When the magnetic Field turned off, a  mass M kg is added to the pan of other arm to balance the coil. When the magnetic field of strength $0.4\ T$ is switched on then, the system is

Options:
A. in equilibrium by adding a mass of 0.64 kg to the left pan.
B. in equilibrium by reversing the current and taking a mass of 0.80 kg from the left pan.
C. in equilibrium by reversing the current and taking a mass of 0.56 kg from the left pan.
D. in equilibrium by reversing the current and taking a mass of 0.48kg from the left pan.
E. in equilibrium by reversing the current and taking a mass of 0.96 kg from the left pan.
F. in equilibrium by reversing the current and adding a mass of 0.32kg to the left pan.
G. in equilibrium by reversing the current and adding a mass of 0.64 kg to the left pan.
H. in equilibrium by reversing the current and taking a mass of 0.64kg from the left pan.
I. in equilibrium by reversing the current and adding a mass of 0.48kg to the left pan.
J. in equilibrium by taking a mass of 0.32 kg from left pan.
Answer: H
A particle free to move along the x-axis has potential energy given by $ U(x) = k $ for $ -\infty \le x \le + \infty $ where k is a positive constant of appropriate dimensions. then

Options:
A. for any finite nonzero value of x, there is a force directed away from the origin
B. for small displacements from x=0, the motion is exponentially growing
C. at points away from the origin , the particle is in unstable equilibirum
D. for small displacements from x=0, the particle experiences a linearly increasing force
E. for any finite nonzero value of x, there is a force directed towards the origin
F. for small displacements from x=0 , the motion is simple harmonic
G. if its total mechanical energy is k/2, it has its minimum kinetic energy at the origin.
H. for small displacements from x=0 , the motion is exponentially damped
I. for any finite nonzero value of x, there is no force directed towards or away from the origin
J. for small displacements from x=0, the particle experiences a constant force
Answer: F
When a train is approaching the stationary observer, the apparent frequency of the whistle observed as $100$ Hz, while when it has passed away from the observer with same speed, it is $50$ Hz. Calculate the frequency of the whistle when the observer moves with the train $(V=330 m/s)$.

Options:
A. $50$Hz
B. $33.3$Hz
C. 41.7 Hz
D. 25.0 Hz
E. 44.4 Hz
F. $75$Hz
G. 83.3 Hz
H. $66.6$Hz
I. 55.5 Hz
Answer: B
A loudspeaker that produces signals from 50 to 500 Hz is placed at the open end of a closed tube of length 1.1 m. The lowest and the highest frequency that excites resonance in the tube are $f_1$ and $f_2$ respectively.The velocity of sound is 330 m/s. Then

Options:
A. \( f_h = 425 \, Hz \)
B. f_h = 400 Hz
C. $$f_1 = 75 Hz$$
D. $$f_h = 500 Hz$$
E. \( f_h = 465 \, Hz \)
F. $$f_h = 450 Hz$$
G. \( f_h = 430 \, Hz \)
H. $$f_1 = 50 Hz$$
I. \( f_h = 525 \, Hz \)
J. \( f_h = 475 \, Hz \)
Answer: F
The wavelength of incident radiation on photo sensitive surface is $4000\mathring{A}$. If the wavelength of this light is $3600\mathring{A}$, what will be the change in the kinetic energy of emitted photoelectron?($h=6.62\times {10}^{-34}J.s,c=3\times {10}^{8}m{s}^{-1},1eV=1.6\times {10}^{-19}J$)

Options:
A. $$4.16eV$$
B. 3.12eV
C. 1.52eV
D. 1.08eV
E. $$1.04eV$$
F. $$2.08eV$$
G. $$0.52eV$$
H. 1.56eV
I. 3.68eV
Answer: E
Three identical blocks each of mass $m\ =\ 1\ kg$ and volume $3\ \times \ { 10 }^{ -4 }\ { m }^{ 3 }$ are suspended by massless string from a support as shown. Underneath are three identical containers containing the same amount of water that are placed over the scales. In Fig.4222 (a), the block is completely out of the water; in Fig. 4.222(b), the block in completely submerged but not touching the beaker and in Fig. 4.222(c), the block rests on the bottom of the beaker. The scale in Fig. 4.222(a). The scale in Fig. 4.222(a) reads $14\ N$.

Options:
A. The reading of the scale in (c) is $23\ N$
B. The reading of the scale in (c) is $21\ N$
C. The reading of the scale in (c) is $24\ N$
D. The tension in the string in (b) is $10\ N$
E. The reading of the scale in (b) is $17\ N$
F. The reading of the scale in (c) is $25\ N$
G. The tension in the string in (b) is $7\ N$
H. The reading of the scale in (c) is $26\ N$
I. The reading of the scale in (c) is $20\ N$
J. The reading of the scale in (c) is $22\ N$
Answer: C
For a dilute solution containing 2.5 g of a non-volatile non-electrolyte solute in 100 g of water, the elevation in boiling point at 1 atm pressure is $2^oC$. Assuming concentration of solute is much lower than the concentration of solvent, the vapour pressure (mm of Hg) of the solution is:[$K_b=0.76 K Kg mol^{-1}$]

Options:
A. 728
B. 722
C. 732
D. $$740$$
E. 716
F. 720
G. $$718$$
H. $$724$$
I. $$736$$
J. 712
Answer: H
A solution contains $0.09\ M \ HCl, 0.09\ M \ CCl_2HCOOH$, and $0.1\ M \ CH_3COOH$. If total $[H^+] = 0.1$ and $K_a$ for $CH_3COOH = 10^{-5}. \ K_a$ for $CCl_2HCOOH$ is:

Options:
A. 0.63 \times 10^{-4}
B. $$1.35\times 10^{-4}$$
C. 1.2 \times 10^{-4}
D. 1.17 \times 10^{-4}
E. $$0.9\times 10^{-4}$$
F. 1.8 \times 10^{-4}
G. $$0.18\times 10^{-5}$$
H. 0.45 \times 10^{-4}
I. $$0.18\times 10^{-2}$$
J. 1.08 \times 10^{-4}
Answer: E
Ram and Ali have been fast friends since childhood. Ali neglected studies and now has no means to earn money other than a camel whereas Ram has become an engineer. Now both are working in the same factory. Ali uses camel to transport the load within the factory.
  Due to low salary and degradation in health of camel, Ali becomes worried and meets his friends Ram and discusses his problems. Ram collected some data and with some assumptions concluded the following:
i. The load used in each trip is 1000 kg and has friction coefficient $\mu_k$ = 0.1 and $\mu_s$ = 0.2.
ii. Mass of camel is 500 kg.
iii. Load is accelerated for first 50 m with constant acceleration, then it is pulled at a constant speed of 5$ms^{-1}$ for 2 km and at last stopped with constant retardation in 50 m.
iv. From biological data, the rate of consumption of energy of camel can be expressed as $P = 18 X 10^3v + 10^4 js^{-1}$ where P is the power and v is the velocity of the camel. After calculations on different issues, Ram suggested proper food, speed of camel, etc.to his friend. For the welfare of Ali, Ram wrote a letter to the management to increase his salary.
(Assuming that the camel exerts a horizontal force on the load): 
The ratio of magnitude of work done by camel on the load during accelerated motion to retarded motion is

Options:
A. 5 : 4
B. 22 : 1
C. 3 : 5
D. 3 : 4
E. 4 : 5
F. 1 : 1
G. 2 : 3
H. 3 : 2
I. 4 : 3
J. 5 : 3
Answer: J