MikaStars39/nano-eval huggingface dataset:
A battery of e.m.f. $E$ and internal resistance is connected to a variable resistor $R$ as shown. Which one of the following is true?

Options:
A. Current in the circuit is maximum when $R = \frac{3r}{4}$
B. Potential difference across the terminals of the battery is maximum where $R = r$
C. Current in the circuit is maximum when $R = 2r$
D. Current in the circuit is maximum when $R = r$
E. Current in the circuit is maximum when $R < r$
F. Power delivered to resistor is maximum when $R = r$
G. Current in the circuit is maximum when $R = \frac{3r}{2}$
H. Current in the circuit is maximum when $R > > r$
I. Current in the circuit is maximum when $R = \frac{r}{2}$
J. Current in the circuit is maximum when $R = \frac{r}{4}$
Answer: D
The mass of a rocket is 500 kg and the relative velocity of gases ejecting from it is 250 m/s with respect to the rocket. Calculate the rate of burning of the fuel to give the rocket an initial acceleration of $2{\text{0}}\;{\text{m/}}{{\text{s}}^{\text{2}}}$ in the vertically upward direction. ${\text{(Take}}\;{\text{g}} = {\text{10}}\;{\text{m/}}{{\text{s}}^{\text{2}}})$

Options:
A. 100 kg/s
B. 150 kg/s
C. 70 kg/s
D. 120 kg/s
E. 60 kg/s
F. 30 kg/s
G. 80 kg/s
H. 110 kg/s
I. 300 kg/s
J. 90 kg/s
Answer: J
A  solid element ,(metal) crystallises in ABABAB....... type packing in three dimension . If the density of the element and its atomic radus are 5.0gm/${cm}^{3}$ and $100\sqrt { 2 } $pm, the atomic mass of the element is ${ (N }_{ A }=6\times { 10 }^{ 23 })$

Options:
A. 72
B. 144
C. 60
D. None of these
E. 108
F. 54
G. 52
H. 96
I. 120
J. 48
Answer: J
Starting at time $t = 0$ from the origin with speed $1 ms^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$ plane so that its coordinates are related by the equation $y = \dfrac{x^2}{2}$. The x and y components of its acceleration are denoted by $a_x$ and $a_y$, respectively. Then

Options:
A. $a_x = 1 ms^{-2}$ implies that when the particle is at the origin, $a_y = 1 ms^{-2}$
B. $a_x = 0$ implies that at $t = 1 s$, the angle between the particles velocity and the x axis is $50^{\circ}$
C. $a_x = 0$ implies that at $t = 1 s$, the angle between the particle's velocity and the x axis is $30^{\circ}$
D. $a_x = 0$ implies that at $t = 1 s$, the angle between the particles velocity and the x axis is $60^{\circ}$
E. $a_x = 0$ implies that at $t = 1 s$, the angle between the particles velocity and the x axis is $75^{\circ}$
F. at $t = 0$, the particles velocity points in the x-direction
G. $a_x = 0$ implies that at $t = 1 s$, the angle between the particle's velocity and the x axis is $15^{\circ}$
H. $a_x = 0$ implies $a_y = 1 ms^{-2}$ at all times
I. $a_x = 0$ implies that at $t = 1 s$, the angle between the particles velocity and the x axis is $45^{\circ}$
J. $a_x = 0$ implies that at $t = 1 s$, the angle between the particle's velocity and the x axis is $55^{\circ}$
Answer: I
A rod of length l and negligible mass is suspended at its two ends by two wires of steel (wire A) and aluminium (wire B ) of equal lengths (Figure). The cross - sectional areas of wires A and B are $1.0 \ mm^2$ and $2.0 \ mm^2$, respectively. $\left(Y_{AI} = 70 \times 10^9 N.m^{-2} \,and\, Y_{steel} = 200 \times 10^9 Nm^{-2}\right)$

Options:
A. Mass m should be suspended close to wire B to have equal strain in both wires.
B. Mass m should be suspended close to wire A to have equal strain in both wires.
C. Mass m should be suspended close to wire A to have equal stresses in both the wires.
D. Mass m should be suspended close to wire B to have equal stresses in both the wires.
E. Mass m should be suspended close to wire B to have equal stress in both wires.
F. Mass m should be suspended close to B to have eqaul stresses in both the wires.
G. Mass m should be suspended at the middle of the wires to have equal strain in both wires.
H. Mass m should be suspended close to wire A to have equal stress in both wires.
I. Mass m should be suspended at the middle of the wires to have equal stresses in both the wires
J. Mass m should be suspended close to wire A to have equal stresses in both wires.
Answer: B
Surface of certain metal is first illuminated with light of wavelength $\lambda_1 =350\, nm$ and then, by light of wavelength $\lambda_2=54  nm$. It is found that the maximum speed of the photo electrons in the two cases differ by a factor of $2$. The work function of the metal (in $eV$) is close to:(Energy of photon $ = \dfrac{1240}{\lambda (in\, nm)}eV)$

Options:
A. 4.2
B. $$1.8$$
C. 2.4
D. $$2.5$$
E. 2.3
F. 2.0
G. 2.1
H. 2.2
I. $$1.4$$
J. $$5.6$$
Answer: B
Solar rays are incident at $45^o$ on the surface of water $(\mu =4/3)$. What is the length of the shadow of a pole of length $1.2\ m$ erected at the bottom of the pond, if the pole is verticle assuming that $0.2\ m$ of the pole is above the water surface?

Options:
A. 0.8\ m
B. $$0.75\ m$$
C. 0.9\ m
D. 0.875\ m
E. 0.925\ m
F. $$0.825\ m$$
G. $$1\ m$$
H. 0.815\ m
I. 0.85\ m
J. $$1.2\ m$$
Answer: F
Consider an infinite ladder network shown in the figure. A voltage $V$ is applied between the points $A$ and $B$. This applied value of voltage is halved after each section if:

Options:
A. \frac { R _ { 1 } } { R _ { 2 } } = \frac { 2 } { 3 }
B. $$\frac { R _ { 1 } } { R _ { 2 } } = 2$$
C. \frac { R _ { 1 } } { R _ { 2 } } = \frac { 1 } { 3 }
D. \frac { R _ { 1 } } { R _ { 2 } } = \frac { 2 } { 5 }
E. $$\frac { R _ { 1 } } { R _ { 2 } } = 1$$
F. \frac { R _ { 1 } } { R _ { 2 } } = \frac { 1 } { 4 }
G. $$\frac { R _ { 1 } } { R _ { 2 } } = \frac { 1 } { 2 }$$
H. $$\frac { R _ { 1 } } { R _ { 2 } } = 3$$
I. \frac { R _ { 1 } } { R _ { 2 } } = \frac { 1 } { 6 }
J. \frac { R _ { 1 } } { R _ { 2 } } = \frac { 1 } { 5 }
Answer: G
Assertion (A): Two metallic spheres of same size, one of copper and other of aluminium heated to the same temperatures, will cool at the same rate when they are suspended in the same enclosure.
Reason (R): The rate of cooling of a body is directly proportional to the excess of temperature of the body over the surroundings.

Options:
A. A is true, but R is false.
B. A is true, but the R is false.
C. both A and R are true and the R is correct explanation of the A
D. both A and R are true, but R is the correct explanation of the A
E. both A and R are false, but R is not the correct explanation of the A
F. both A and R are true, but R is not the correct explanation of A
G. both A and R are true, but R is not the correct explanation of the A
H. A is true, and R is the correct explanation of A.
I. both A and R are true, but R is not correct explanation of the A
J. A is false, but the R is true.
Answer: C
A solid disc has a radius $R$. Density of its material is given by $\rho={\rho}_{0}(2-\cfrac{r}{R})$ where ${\rho}_{0}$ is a constant and $r$ is the radial distance measured from the centre. Then the moment of inertia of the disc about an axis passing through its centre and perpendicular to the plane of the disc is

Options:
A. \cfrac{4}{9}M{R}^{2}
B. \cfrac{5}{14}M{R}^{2}
C. \cfrac{3}{8}M{R}^{2}
D. $$\cfrac{5}{18}M{R}^{2}$$
E. \cfrac{13}{35}M{R}^{2}
F. $$\cfrac{7}{12}M{R}^{2}$$
G. \cfrac{3}{14}M{R}^{2}
H. $$\cfrac{9}{20}M{R}^{2}$$
I. \cfrac{11}{28}M{R}^{2}
J. $$\cfrac{2}{7}M{R}^{2}$$
Answer: J
In $YDSE$ experiment shown in the figure, a parallel beam of light of wavelength $(\lambda = 0.3\ mm)$ in medium $\mu_{1}$ is incident at an angle $\theta = 30^{\circ}$ as shown$(S_{1}O = S_{2}O)$. If the intensity due to each light wave at point $O$ is $I_{0}$ then the resultant intensity at point $O$ will be:

Options:
A. $$I_{0}$$
B. 2I_{0} [1 + \cos (60 \pi / 9)]
C. Zero
D. $$2I_{0} [1 + \cos (40 \pi / 9)]$$
E. 2I_{0} [1 + \cos (50 \pi / 9)]
F. 2I_{0} [1 + \cos (80 \pi / 9)]
G. 2I_{0} [1 + \cos (70 \pi / 9)]
H. 2I_{0} [1 + \cos (30 \pi / 9)]
I. 2I_{0} [1 + \cos (20 \pi / 9)]
J. $$3I_{0}$$
Answer: A
Let $ G_1,G_2, G_3$ be the centroids of the triangular faces OBC, OCA, OAB of a tetrahedron OABC. If $V_1$ denote the volume of the tetrahedron OABC and $V_2$ that of the parallelepiped with $OG_1, OG_2, OG_3$ as three concurrent edges, then

Options:
A. 8 V_1=9 V_2
B. $$3 V_2=2 V_1$$
C. 8 V_1 = 3 V_2
D. 6 V_1=9 V_2
E. 3 V_1=8 V_2
F. $$3 V_1=2 V_2$$
G. 2 V_1=3 V_2
H. 9 V_1=4 V_2
I. $$4 V_1=9 V_2$$
Answer: I
Two capacitors $C_{1} = C_{2} = \dfrac{1}{\pi^2} \times 10^{-2}F$ and inductor $L = 2 \times 10^{-2}H$ are connected in series as shown in the figure, initially charge on each capacitors are $4\sqrt{3}\mu C.$ At $t = 0$ switch $S_{1}$ is closed and $t = \dfrac{1}{300}sec,$ switch $S_{2}$ is also closed. The maximum charge on capacitor $C_{2}$ during LC oscillation is

Options:
A. $$3\sqrt{3}\mu C$$
B. 9\mu C
C. $$6\mu C$$
D. 2\mu C
E. $$3\mu C$$
F. $$2\sqrt{3}\mu C$$
G. 4\sqrt{3}\mu C
H. 5\mu C
I. $$4\mu C$$
Answer: E
Assuming that copper contains only iron, silver and gold as impurities. After passage of 12.4 A for 4825 s, the mass of anode decreased by 20.00 g and the cathode increased by 19.05 g. The percentages of iron and copper in the original sample are, respectively, (Cu = 63.5, Fe = 56)

Options:
A. 1.9%, 95.25%
B. 95.25%, 1.91%
C. 95.25%, 4.75%
D. 2.8%, 95.25%
E. 1.9%, 95.50%
F. 3.7%, 95.25%
G. 1.91%, 95.25%
H. 2.8%, 95.50%
I. 4.75%, 95.25%
J. 1.91%, 95.50%
Answer: D
A particle moves along $x$-axis as follows: It starts from rest at $t=0$ from a point $x=0$ and comes to rest at $t=1$ at a point $x=1$. No other information is available about its motion for the intermediate time $0<t <1$. If $lpha$ denotes the instantaneous acceleration of the particle, then

Options:
A. $\left| \alpha \right| $ cannot exceed 2.5 at any point in its path
B. $\left| \alpha \right| $ cannot exceed 3 at any point in its path
C. $lpha$ must change sign during the motion, but no other assertion can be made with the information given
D. $\left| lpha \right| $ must be $\ge 3$ at some point or points in its path
E. $lpha$ cannot remain positive for all $t$ in the interval $0\le t \le 1$
F. $\left| lpha \right| $ cannot exceed 2 at any point in its path
G. $|\alpha| \text{ must be } < 4 \text{ at some point in its path}
H. $\left| \alpha \right| $ must be $\ge 3.5$ at some point or points in its path
I. $\left| \alpha \right| $ must be $\ge 5$ at some point or points in its path
J. $\left| lpha \right| $ must be $\ge4$ at some point or points in its path
Answer: E
Initially two particle A and B are present at (0, 0) and (d, 0) respectively. They start moving with velocity $v_A  = v\widehati + v\widehatj\,{\text{and}}\,v_B  =  - v\widehatj$. If R is magnitude of relative separation between them and $T_o $ be the time when separation between them is a minimum. then

Options:
A. T_0 = \frac{d}{4\sqrt{5}v}
B. Graph of R versus time is straight line
C. T_0 = \frac{d}{3\sqrt{5}v}
D. T_0 = \frac{d}{2\sqrt{5}v}
E. $
T_0 = \frac{d}
{{\sqrt 5 v}}
$
F. $
R_{\min } \frac{{2d}}
{{\sqrt 5 }}
$
G. T_0 = \frac{d}{\sqrt{3}v}
H. Graph of R versus time is circle
I. T_0 = \frac{3d}{\sqrt{5}v}
J. T_0 = \frac{2d}{\sqrt{5}v}
Answer: E
$Zn\, +\, 2H^{\oplus}\, \rightarrow\, Zn^{2+}\, +\, H_2$The half-life period is independent of the concentration of zinc at constant pH. For the constant concentration of Zn, the rate becomes 100 times when pH is decreased from 3 to 2. Hence:

Options:
A. The rate becomes four times if the concentration of $H^{\oplus}$ ion is doubled at constant Zn concentration
B. The rate becomes six times if the concentration of \( H^{\oplus} \) ion is doubled at constant Zn concentration
C. The rate becomes sixteen times if the concentration of \( H^{\oplus} \) ion is doubled at constant Zn concentration
D. Rate is not affected if the concentration of zinc is made four times and that of $H^{\oplus}$ ion is halved
E. $$\displaystyle \frac{dx}{dt}\, =\, k[Zn][H^{\oplus}]^2$$
F. The rate becomes nine times if the concentration of \( H^{\oplus} \) ion is doubled at constant Zn concentration
G. The rate becomes three times if the concentration of \( H^{\oplus} \) ion is doubled at constant Zn concentration
H. The rate becomes two times if the concentration of $H^{\oplus}$ ion is doubled at constant Zn concentration
I. The rate becomes eight times if the concentration of $H^{\oplus}$ ion is doubled at constant Zn concentration
Answer: A
$$\dfrac{1+cotlpha+coseclpha}{1-cotlpha+cosec lpha}=$$

Options:
A. \dfrac{sin\alpha}{1+sin\alpha}
B. \dfrac{1-cos\alpha}{sin\alpha}
C. $$\dfrac{1-sinlpha}{coslpha}$$
D. \dfrac{1+cos\alpha}{sin\alpha}
E. \dfrac{1 - sin\alpha}{1 - cos\alpha}
F. $$\dfrac{1+coslpha}{sinlpha}$$
G. \dfrac{1 - sin\alpha}{1 + cos\alpha}
H. $$\dfrac{sinlpha}{1+coslpha}$$
I. $$\dfrac{sinlpha}{1-coslpha}$$
Answer: I
A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 $km h^{-1}$ passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 $ms^{-1}$ to hit the plane. (Take g= 10 $ms^{-2}$)

Options:
A. sin^{-1}(1/4)
B. sin^{-1}(\frac{1}{7})
C. sin$^{-1}$(1/2)
D. cos$^{-1}$(2/3)
E. sin^{-1}(\frac{1}{5})
F. sin^{-1}(\frac{1}{6})
G. cos$^{-1}$(1/3)
H. sin^{-1}(\frac{1}{4})
I. sin$^{-1}$(1/3)
J. sin$^{-1}$(2/3)
Answer: I
A uniform rod of mass m and length L performs small oscillations about a horizontal a is passing through its upper end. Find the mean kinetic energy of the rod during its oscillation period , If at t=0 it s deflected from its vertical by an angle $\theta_0$ and imparted an angular velocity $\omega_0$.

Options:
A. $$mgL[\dfrac{2L\omega^2}{3g}+\theta_0^2]$$
B. \dfrac{1}{4}mgL[\dfrac{2L\omega^2}{3g}+2\theta_0^2]
C. \dfrac{1}{4}mgL[\dfrac{L\omega^2}{2g}+2\theta_0^2]
D. $$\dfrac{1}{4}mgL[\dfrac{L\omega^2}{3g}+\theta_0^2]$$
E. $$\dfrac{1}{8}mgL[\dfrac{2L\omega^2_0}{3g}+\theta_0^2]$$
F. $$\dfrac{1}{4}mgL[\dfrac{2L\omega^2-0}{3g}+\theta_0^2]$$
G. \dfrac{1}{4}mgL[\dfrac{L\omega^2}{2g}+3\theta_0^2]
H. \dfrac{1}{4}mgL[\dfrac{L\omega^2}{3g}+3\theta_0^2]
I. \dfrac{1}{4}mgL[\dfrac{L\omega^2}{4g}+3\theta_0^2]
J. \dfrac{1}{4}mgL[\dfrac{L\omega^2}{6g}+2\theta_0^2]
Answer: D
An Uncharged capacitor of capacitance $C$ is connected to a battery of emf  $arepsilon $ at  $ t=0$ through a resistance $R$, then  (i) the maximum rate at which energy is stored in the capacitor is:

Options:
A. \dfrac {arepsilon^2}{3R}
B. $$ \dfrac {arepsilon^2 }{2R}$$
C. \dfrac {3 \arepsilon^2}{2R}
D. $$ \dfrac {arepsilon^2 }{ 4R } $$
E. \dfrac {cząrepsilon^2}{3R}
F. \dfrac {2
\t
arepsilon^2}{3R}
G. $$ \dfrac {arepsilon^2} {R}$$
H. \dfrac {4 \arepsilon^2}{3R}
I. \dfrac {3 \arepsilon^2}{4R}
J. $$ \dfrac {2 arepsilon^2}{R}$$
Answer: G
Consider following ionisation steps:$M(g)  \rightarrow M^+ (g) + e^- ;              \Delta H = 100 eV$$M(g)  \rightarrow M^{2+} (g) + 2e^-;          \Delta H = 250 eV$Select correct statement(s):

Options:
A. $(IE)_2$ or M(g) is 225 eV
B. $(IE)_1$ or M$^+$(g) is 250 eV
C. $(IE)_1$ or M(g) is 150 eV
D. $(IE)_2$ or M(g) is 150 eV
E. $(IE)_1$ or M(g) is 200 eV
F. $(IE)_1$ or M(g) is 175 eV
G. $(IE)_2$ or M(g) is 250 eV
H. $(IE)_2$ or M(g) is 200 eV
Answer: D
A quantity of 8.0 g of solid sulphur is first oxidized to $ SO_{2} $ and then it is divided into two equal parts. One part is sufficient for complete decolourization of 200 ml of acidified $ KMnO_{4} $ Solution. Another part is oxidized to $ SO_{3} $ and the $ SO_{3} $ formed is sufficient for complete precipitation of all $ BaCl_{2} $ present in 100 ml solution as $ BaSO_{4} $. Which of the following statements is/are correct? (S = 32, Ba  = 138 )

Options:
A. The weight of $ BaSO_{4} $ precipitated out is 27.75 g.
B. The molarity of $ KMnO_{4} $ solution is 0.25.
C. The weight of $ BaSO_{4} $ precipitated out is 28.25 g.
D. The same equivalents of $ KMnO_{4} $ and $ BaCl_{2} $ are reacted.
E. The weight of $ BaSO_{4} $ precipitated out is 29.25 g.
F. The weight of $ BaSO_{4} $ precipitated out is 28.75 g.
G. The weight of $ BaSO_{4} $ precipitated out is 29.75 g.
H. The weight of $ BaSO_{4} $ precipitated out is 27.25 g.
Answer: E
The condition for the coaxial system $x^2+y^2+2\lambda x+c=0$, where $\lambda$ is a parameter and c is a constant, to have distinct limiting points, is?

Options:
A. $$c=-1$$
B. $$c \leq 0$$
C. $$c=0$$
D. c \geq 1
E. c = -2
F. $$c > 0$$
G. c = 1
H. $$c \geq 0$$
I. c < 0
Answer: F
A disc of radius $R$ is rolling without sliding on a horizontal surface with a velocity of center of mass $v$ and angular velocity $\omega$ in a uniform magnetic field $B$ which is perpendicular to the plane of the disc as shown in figure. $O$ is the center of the disc and $P,Q,R$ and $S$ are the four points on the disc. Which of the following statements is true?

Options:
A. Due to rotation, induced emf across $PS=Bvr$
B. Due to translation, induced emf across $PS=0$
C. Due to rotation, induced emf across $QS=0$
D. Due to rotation, induced emf across $OQ=Bvr$
E. Due to translation, induced emf across $RO=BvR$
F. Due to translation, induced emf across $RO=0$
G. Due to rotation, induced emf across $RO=BvR$
H. Due to translation, induced emf across $PS=Bvr$
I. Due to translation, induced emf across $QS=0$
J. Due to rotation, induced emf across $PS=BvR$
Answer: F
Electron transition in ${ He }^{ + }$ ion takes from ${ n }_{ 2 }$ to ${ n }_{ 1 }$ shell such that:$2{ n }_{ 2 }+3{ n }_{ 1 }=18$                ....(i)$2{ n }_{ 2 }-3{ n }_{ 1 }=6$                   ....(ii)What will be the total number of photons emitted when electrons transit to ${ n }_{ 1 }$ shell?

Options:
A. $$20$$
B. 11
C. 18
D. 16
E. $$21$$
F. 13
G. 12
H. $$15$$
I. $$10$$
J. 14
Answer: I
Let the position of an object moving along a straight line at any instant t is given by $s = t^3  - 3t^2  - 9t$, where s is in meters and t in seconds then

Options:
A. The object is at the origin at three distinct instants t
B. The acceleration of the object is positive for all t > 2s
C. For t < [0, 3], the maximum magnitude of the velocity is 9 m/s
D. For t < [0, 3], the maximum magnitude of the velocity is 15 m/s
E. For t < [0, 3], the maximum magnitude of the velocity is 13 m/s
F. For t < [0, 3], the maximum magnitude of the velocity is 12 m/s
G. For t < [0, 3], the maximum magnitude of the velocity is 14 m/s
H. For t < [0, 3], the maximum magnitude of the velocity is 11 m/s
I. The velocity of the object is zero at t = 3s only
J. For t < [0, 3], the maximum magnitude of the velocity is 10 m/s
Answer: F
Two metal bars are fixed vertically and are connected on the top by a capacitor C. A sliding conductor AB of length L slides with its ends in contact with the bars. The arrangement is placed in a uniform horizontal magnetic field directed normal to the plane of the figure. The conductor is released from rest. The displacement (x) in meter of the conductor at time $ t = 2 sec$ is:(Given $ m = 100 gm , g = 10m/s^{2} , B = 100 Tesla , L = 1m ,  and  C = 10 \mu F$)

Options:
A. $$5$$
B. $$14$$
C. 9.8
D. 12
E. 11
F. 6
G. $$10$$
H. 8
I. 9
J. $$7$$
Answer: G
Assume that $  2 \times 10^{-1 7}  $ J of light energy is needed by the interior of the human eye to see an object. How many photons of yellow light with $  \lambda=595.2 \mathrm{nm}  $ are needed to generate this minimum energy?

Options:
A. $$6$$
B. 12
C. $$54$$
D. $$45$$
E. $$30$$
F. 18
G. 36
H. $$60$$
I. 42
J. 48
Answer: A
A 2 m long tube closed at one end is lowered vertically into water the until the closed end is flush with the water surface. See figure. Calculate the water level height in the tube. (Barometric pressure = 1 atm = 10 m of hydrostatic water head. Temperature = $25^0$C, Density of water 1.00 g/ml. Neglect water vapour pressure)

Options:
A. 1.01 m
B. 1.02 m
C. 0.99 m
D. 1.03 m
E. 1.71 m
F. 1.10 m
G. 1.92 m
H. 0.92 m
I. 0.97 m
J. 0.29 m
Answer: A
A wire of cross-sectional are A forms three sides of a square and is free to rotate about axis OO'. If the structure is deflected by an angle $\theta$ from the vertical when current i is passed through it in a magnetic field B acting vertically upwards and density of the wire is $\rho$, then the value of $\theta$ is given by

Options:
A. \dfrac {2A\rho g}{iB} = \sin\theta
B. \dfrac {A\rho g}{iB} = \cos\theta
C. \dfrac {A\rho g}{iB} = \sin\theta
D. $$\dfrac {2A\rho g}{iB}=cot\theta$$
E. $$\dfrac {A\rho g}{iB}=sin\theta$$
F. \dfrac {A\rho g}{2iB} = \cot\theta
G. $$\dfrac {A\rho g}{2iB}=cos\theta$$
H. $$\dfrac {2A\rho g}{iB}=tan\theta$$
I. \dfrac {A\rho g}{2iB} = \tan\theta
J. \dfrac {A\rho g}{iB} = \tan\theta
Answer: D
Select the right expression for determining packing fraction (P.F) of NaCl unit cell (assume ideal), if the ions along an edge diagonal are absent.

Options:
A. \(\displaystyle P.F = \frac{\frac{4}{3} \pi\left(\frac{5}{2}r_+^3 + 3r_-^3\right)}{16 \sqrt{2} r_-^3} \)
B. \(\displaystyle P.F = \frac{\frac{4}{3} \pi\left(\frac{5}{2}r_+^3 + 6r_-^3\right)}{16 \sqrt{2} r_-^3} \)
C. \(\displaystyle P.F = \frac{\frac{4}{3} \pi\left(\frac{3}{2}r_+^3 + 4r_-^3\right)}{16 \sqrt{2} r_-^3} \)
D. \(\displaystyle P.F = \frac{\frac{4}{3} \pi\left(\frac{5}{2}r_+^3 + 7r_-^3\right)}{16 \sqrt{2} r_-^3} \)
E. $$\displaystyle P.F = \frac{\frac{4}{3} \pi(\frac{5}{2}r_+^3 + 4r_-^3)}{16 \sqrt{2} r_-^3}$$
F. $$\displaystyle P.F = \frac{\frac{4}{3} \pi(r_+^3 + r_-^3)}{16 \sqrt{2} r_-^3}$$
G. \(\displaystyle P.F = \frac{\frac{4}{3} \pi\left(\frac{5}{2}r_+^3 + 5r_-^3\right)}{16 \sqrt{2} r_-^3} \)
H. $$\displaystyle P.F = \frac{\frac{4}{3} \pi(\frac{7}{2}r_+^3 + r_-^3)}{16 \sqrt{2} r_-^3}$$
I. $$\displaystyle P.F = \frac{\frac{4}{3} \pi(\frac{5}{2}r_+^3 + r_-^3)}{16 \sqrt{2} r_-^3}$$
J. \(\displaystyle P.F = \frac{\frac{4}{3} \pi\left(\frac{5}{2}r_+^3 + 2r_-^3\right)}{16 \sqrt{2} r_-^3} \)
Answer: E
A metal ball immersed in water weighs ${ w }_{ 1 }$ at 5 ${ \circ  }_{ C }$  and ${ W }_{ 2 }$ at ${ 50 }^{ \circ  }$. The coefficient of cubical expansion of metal is less than that of water. Then

Options:
A. ${ w }_{ 1 }$ < ${ w }_{ 2 }$
B. ${ w }_{ 1 } > { w }_{ 2 }$
C. { w }_{ 1 } \geq { w }_{ 2 }
D. ${ w }_{ 1 } = { w }_{ 2 }$
E. { w }_{ 1 } \leq { w }_{ 2 }
F. $$data\ is\ insufficient$$
G. { w }_{ 1 } \neq { w }_{ 2 }
Answer: A
A copper bar of length L and area of cross-section A is placed in a chamber at atmospheric pressure. If the chamber is evacuated, the percentage change in its volume will be: (compressibility of copper is $8 \times 10^{-12}m^2/N$ 1 atm = $10^5 N/m^2$)

Options:
A. 8 x $10^{-5}$
B. 1.25 x $10^{-5}$
C. 1.25 x $10^{-4}$
D. 8 x $10^{-7}$
E. 1.2 x $10^{-6}$
F. 1.2 x 10^{-5}
G. 1.2 x $10^{-7}$
H. 8 x $10^{-8}$
I. 8 x 10^{-6}
Answer: D
In a nautral decay, series starts with $_{90}Th^{232}$ and finally terminates at $_{82}Pb^{208}$. A thorium one sample was found to contain $8\times 10^{-5} $ml of helium at $1$ atm and $273$ K and $5\times 10^{-7}$ gm of $Th^{232}$. Find the age of ore sample assuming that source of $He$ to be only due to decay of $Th^{232}$. Also assume complete retention of helium within the ore.(Half-life of $Th^{232}=1.39\times 10^{10}Y)$

Options:
A. $$t=4.89 \times 10^{10} years$$
B. t=4.89 \times 10^{12} years
C. $$t=4.89 \times 10^{8} years$$
D. t=4.89 \times 10^{14} years
E. $$t=4.89 \times 10^{9} years$$
F. t=4.89 \times 10^{11} years
G. t=4.89 \times 10^{7} years
H. t=4.89 \times 10^{6} years
I. t=4.89 \times 10^{13} years
Answer: E
A steel tape is placed around the earth at the equator. When the temperature is  $0 ^ { \circ } { C }$  neglecting the expansion of the earth, the clearance between the tape and the ground if he temperature of the tape rises to  $30 ^ { \circ } { C } ,$  is nearly  $\left( lpha _ {steel} = 11 \times 10 ^ { - 6 } / K \right)$

Options:
A. 1.2 { km}
B. $$0 ) 5 { km }$$
C. $$2 ) 1 { km }$$
D. $$1 ) 1 { km }$$
E. 1.5 { km }
F. 1.1 { km }
G. 1.3 { km}
H. 0.1 { km}
I. $$6400 km $$
J. 0.5 { km }
Answer: D
The height "y" and the distance x along horizontal for a body projected in the vertical plane are given by $2y = 6t - 3t^2$ and $x = 4t$. Then the initial velocity and angle of projection are

Options:
A. $$5, 30^o$$
B. 5, 60^o
C. 4, \pi / 4
D. $$5, \pi / 6$$
E. 5, \pi / 4
F. 5, 3 \pi / 6
G. $$5,45^o$$
H. 5, \pi / 3
I. $$5. \pi/3$$
J. $$Tan^{-1}(3/4)$$
Answer: G
Two circular coils X and Y , having equal number of turns, carry equal currents in the same sense and subtend same solid angle at point O. If the smaller coil , X is midway between O and Y, then if we represent the magnetic induction due to bigger coil Y at O as $B_{Y}$ and that due to smaller coil X at O as $B_{X}$ , then

Options:
A. \cfrac{B_{Y}}{B_{X}}=\cfrac{3}{2}
B. $$\cfrac{B_{Y}}{B_{X}}=1$$
C. $$\cfrac{B_{Y}}{B_{X}}=\cfrac{1}{6}$$
D. \cfrac{B_{Y}}{B_{X}}=\cfrac{1}{5}
E. \cfrac{B_{Y}}{B_{X}}=\cfrac{3}{4}
F. $$\cfrac{B_{Y}}{B_{X}}=\cfrac{2}{3}$$
G. $$\cfrac{B_{Y}}{B_{X}}=\cfrac{1}{2}$$
H. $$\cfrac{B_{Y}}{B_{X}}=2$$
I. $$\cfrac{B_{Y}}{B_{X}}=\cfrac{1}{4}$$
J. \cfrac{B_{Y}}{B_{X}}=\cfrac{1}{3}
Answer: G
For a process which follows the equation $PV^3=C$, the work done when 1 mole of ideal gas was taken from one atmosphere to 2 under root 2 atmosphere starting from initial pressure at 300 kelvin is

Options:
A. 300 R
B. 1200 R
C. 200 R
D. 350 R
E. 450 R
F. 250 R
G. 600 R
H. 400 R
I. 150 R
J. 900 R
Answer: A
An uncharged capacitor of capacitance $100\mu F$ is connected to a battery of emf $20V$ at $t=0$ through a resistance $10\Omega$ , the maximum rate at which energy is stored in the capacitor is

Options:
A. 2Js
B. $$20Js$$
C. $$10Js$$
D. 15Js
E. 12Js
F. 25Js
G. $$5Js$$
H. 3Js
I. $$40Js$$
Answer: G
The planes $4x-5y-2z=2,5x-4y+2z+2=0,2x+2y+8z-1=0$

Options:
A. Form a sphere
B. Meet at a curve
C. Meet in a line
D. Form a cube
E. meet at a point
F. Form a tetrahedron
G. None of these
H. Form a cylinder
I. Form a cone
J. Form a prism
Answer: J
The locus of point of trisections of the focal chords of the parabola, $y^2=4x$ is?

Options:
A. $$y^2=x-1$$
B. $$y^2=2(1-x)$$
C. 8y^2=4ax
D. 9y^2=4x+2
E. 8y^2=4x
F. $$9y^2=4ax$$
G. $$9y^2=4ax+1$$
H. 9y^2=4x
I. 9y^2=4x+1
J. None of these
Answer: F
Calculate the maximum and minimum number of electrons. Which may have magetic quantum number m = +1 and spin quantum $s=\cfrac { 1 }{ 2 } $ in Chromium (Cr) :

Options:
A. 5,3
B. 3,2
C. 4,3
D. 4,1
E. 5,2
F. 2,1
G. 4,2
H. 6,4
I. 2,3
J. 3,1
Answer: B
Consider a cylinder filled with an ideal gas closed by a frictionless and massless movable piston. The gas undergoes different processes. If $\Delta Q$ represents heat supplied to system, $\Delta W$ work done by gas on surroundings and $\Delta U$ is change in internal energy of the gas then choose the correct statement(s).

Options:
A. In free expansion $\Delta U = 0$ and $\Delta W = \Delta Q \neq 0$
B. In free expansion \(\Delta U = 0\) and \(\Delta W = \Delta Q \neq 0\)
C. In adiabatic expansion $\Delta W > 0$ and $\Delta U = 0$
D. In free expansion \(\Delta U = \Delta W = 0\) and \(\Delta Q \neq 0\)
E. In free expansion $\Delta U = \Delta W = \Delta Q = 0$
F. For an isothermal expansion and compression $\Delta Q$ is always greater than zero and $\Delta U = 0$
G. In free expansion \(\Delta U = 0\) and \(\Delta W = 0\) but \(\Delta Q \neq 0\)
Answer: E
The reactor produced a power of $10^{10}\ MW$ with $50$% efficiency. Then the number of deutrons required to run the reactor for one year will be approximately:    
$$m\left( _{ 1 }^{  }{ { H }_{  }^{ 2 } } \right) =2.014\ u,m\left( p \right) =1.007\ u,m\left( n \right) =1.008\ u,m\left( _{ 2 }^{  }{ { H }e_{  }^{ 4 } } \right) =4.001\ u$$

Options:
A. $$5\times 10^{35}$$
B. 5 \times 10^{39}
C. $$5\times 10^{38}$$
D. 5 \times 10^{37}
E. $$5\times 10^{32}$$
F. 5 \times 10^{36}
G. 5 \times 10^{34}
H. 5 \times 10^{33}
I. $$5\times 10^{40}$$
Answer: C
An electron in a hydrogen like species is in the excited state $n_2$. The wavelength for the transition $n_2$ to $n_1=2$ is $48.24nm$. The corresponding wavelength for the transition from $n_2$ to $n_1=3$ is $142.46nm$. Find the value of $n_2$ and $Z$ and report the $H$ like atom.

Options:
A. $$n_2=4, Z=1$$
B. $$n_2=5, Z=4$$
C. $$n_2=7, Z=2$$
D. n_2=6, Z=2
E. n_2=6, Z=1
F. $$n_2=7, Z=3$$
G. $$n_2=4, Z=2$$
H. $$n_2=6, Z=3$$
I. n_2=5, Z=3
J. n_2=5, Z=2
Answer: F
One gram of an alloy of aluminium and magnesium when heated with excess of dilute $HCl$ forms magnesium chloride, aluminium chloride and hydrogen. The evolved hydrogen collected over mercury at $0^{o}C$ has volume of $1.2$ litres at $0.92$ atm pressure. Calculate the composition of the alloy.

Options:
A. Mg = 0.452 g
B. $Al = 0.546$ g
C. Mg = 0.545 g
D. $Mg = 0.454$ g
E. Mg = 0.453 g
F. Mg = 0.455 g
G. $Al = 0.454$ g
H. $Mg = 0.546$ g
I. Mg = 0.456 g
J. $Mg = 0.544$ g
Answer: D
Magnetic moments of $V (Z = 23), Cr(Z = 24)$ and $Mn(Z = 25)$ are $x, y, z$. Hence

Options:
A. $$x = y = z$$
B. x < z < y
C. z < x < y
D. z < x = y
E. $$x < y =z$$
F. z = y < x
G. $$z < y < x$$
Answer: E
The air entering the lungs passes through tiny sacs called alveoli, from which oxygen diffuses into the blood. The average radius of the alveoli is 0.0050 cm, and the air inside contains 14 mole percent oxygen. Assuming that the pressure in the alveoli is 1.0 atm and the temperature is $37^{O}C$; calculate the number of oxygen molecules in one of the alveoli. $(R = 0.08206\ L- atm\ mol^{-1} \ K^{-1},\  6.023 \times  10^{23} molecules\ mole^{-1})$

Options:
A. 1.7 \times 10^{10} oxygen molecules
B. 1.7 \times 10^{15} oxygen molecules
C. $ 1.7 \times 10^{12}$ oxygen molecules
D. 1.7 \times 10^{13} oxygen molecules
E. 1.7 \times 10^{16} oxygen molecules
F. 1.7 \times 10^{11} oxygen molecules
G. 1.7 \times 10^{14} oxygen molecules
Answer: C
Statement 1: The temperature dependence of resistance is usually given as $\mathrm{R}=\mathrm{R}_{0} (1 +lpha \Delta \mathrm{t})$. The resistance of a wire changes from 100 $\Omega$ to 150 $\Omega$ when its temperature is increased from $27^{\mathrm{o}}\mathrm{C}$ to $227^{\mathrm{o}}\mathrm{C}$. This implies that $lpha =2.5\times 10^{-3}/^{\mathrm{o}}\mathrm{C}$.
Statement 2: $\mathrm{R}=\mathrm{R}_{0} (1+lpha \Delta \mathrm{t})$ is valid only when the change in the temperature $\Delta \mathrm{T}$ is small and $\Delta \mathrm{R}=(\mathrm{R}- \mathrm{R}_{0})<<\mathrm{R}_{0}$.

Options:
A. Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1.
B. Statement 1 is True, Statement 2 is False
C. Statement 1 is False, Statement 2 is True.
D. Statement 1 is True, Statement 2 is True; Statement 2 is not the correct explanation for Statement 1.
E. Statement 1 is False, Statement 2 is False.
F. Statement 1 is True, Statement 2 is False.
Answer: C
The solar constant is the amount of heat energy received per second per unit area of a perfectly black surface placed at a mean distance of the Earth from the Sun, in the absence of Earths atmosphere, the surface being held perpendicular to the direction of Suns rays. Its value is $1388\ W/m^{2}$.If the solar constant for the earth is $s$. The surface temperature of the sun is $TK$, $D$ is the diameter of the Sun, $R$ is the mean distance of the Earth from the Sun. The sun subtends a small angle $\theta$ at the earth. Then correct options is/are :

Options:
A. s=\dfrac{\sigma T^{4}}{4}\left(\dfrac{R}{D}^{2}\right)
B. s=\dfrac{\sigma T^{4}}{4}\left(\dfrac{R}{D}\right)
C. $$s=\dfrac{\sigma T^{4}}{4}\left(\dfrac{D}{R}^{2}\right)$$
D. s=\dfrac{\sigma T^{4}}{4}\left(\dfrac{R}{2D}\right)
E. $$s=\sigma T^{4}\left(\dfrac{D}{R}\right)$$
F. s=\dfrac{\sigma T^{4}}{4}\left(\dfrac{D}{2R}\right)
G. s=\dfrac{\sigma T^{4}}{8}\left(\dfrac{D}{R}^{2}\right)
H. s=\dfrac{\sigma T^{4}}{2}\left(\dfrac{D}{R}^{2}\right)
I. $$s=\dfrac{\sigma T^{4}}{4}\left(\dfrac{R}{D}\right)^{2}$$
Answer: C
A uniform conducting rectangular loop of sides $l,  b$ and mass $m$ carrying current $i$ is hanging horizontally with the help of two vertical strings as shown in figure. There exists a uniform horizontal magnetic field $B$, which is parallel to the longer side of loop $l$. The least value of tension in strings is (Given acceleration due to gravity $=g$):

Options:
A. \dfrac {mg-iBb}{2}
B. \dfrac {mg-4iBb}{2}
C. \dfrac {mg-3iBb}{2}
D. $$\dfrac {mg-2iBb}{2}$$
E. $$\dfrac {mg-5iBb}{2}$$
F. $$\dfrac {mg+iBb}{2}$$
G. \dfrac {mg-iBb}{3}
H. $$\dfrac {mg+2iBb}{2}$$
I. \dfrac {mg-6iBb}{2}
Answer: D
A small piece of $Cs$ (work function $=1.9eV$) is placed 22 cm away from a large metal plate. The surface charge density on the metal plate is $1.21\times 10^{-9}Cm^{-2}$. Now, light of $460nm$ wavelength is incident on the piece of $Cs$. Find the maximum and minimum energies of photo electrons on reaching the plate. Assume that no change occurs in electric field produced by the plate due to the piece of $Cs$.

Options:
A. Minimum energy $=22.83eV$,Miximum  energy $=31.63eV$
B. Minimum energy $=23.83eV$,Maximum energy $=34.63eV$
C. Minimum energy $=24.83eV$,Miximum  energy $=33.63eV$
D. Minimum energy $=21.83eV$,Miximum  energy $=30.63eV$.
E. Minimum energy $=26.83eV$,Maximum energy $=34.63eV$
F. Minimum energy $=25.83eV$,Miximum  energy $=35.63eV$.
G. Minimum energy $=20.83eV$,Miximum  energy $=32.63eV$.
H. Minimum energy $=28.83eV$, Maximum energy $=30.63eV$
I. Minimum energy $=29.83eV$,Miximum  energy $=30.63eV$.
J. Minimum energy $=29.83eV$,Miximum  energy $=31.63eV$
Answer: I
A particle execute in S.H.M whose displacement equation is $Y= 5sin(10t- \pi/3)$ . minimum time after which the K.E of the particle will be maximum is

Options:
A. $$\frac{\pi}{90} sec$$
B. $$\frac{\pi}{120} sec$$
C. \frac{\pi}{100} sec
D. \frac{\pi}{105} sec
E. \frac{\pi}{60} sec
F. \frac{\pi}{75} sec
G. \frac{\pi}{15} sec
H. $$\frac{\pi}{30} sec$$
I. \frac{\pi}{45} sec
Answer: A
An atom is modelled as a point charge of +e at the nucleus with the electron charge distribution, described by the charge density$\rho=\dfrac{-15 e}{8 \pi a^3}\left ( 1-\dfrac{r^2}{a^2} \right )$  for  $r  \leq  a  <  0$$\rho \simeq 0  $ for $ r>a, $ where 'a' is a constant.The net charge contained within a sphere of radius r < a is :

Options:
A. \left [ \frac{3r^5}{2a^5} - \frac{5r^3}{4a^3} + 1 \right ]e
B. $$\displaystyle \left [ \frac{3r^5}{2a^5}-\frac{5r^3}{2a^3}+1 \right ]e$$
C. \left [ \frac{3r^5}{2a^5} - \frac{5r^3}{3a^3} + 1 \right ]e
D. \displaystyle \left [ \frac{3r^5}{2a^5} - \frac{5r^3}{2a^3} - 1 \right ]e
E. $$\text Zero$$
F. $$\displaystyle \left [ \frac{3r^5}{2a^5}-\frac{4r^4}{3a^4} \right ]e$$
G. \left [ \frac{3r^5}{2a^5} - \frac{5r^3}{3a^3} - 1 \right ]e
H. \left [ \frac{3r^5}{2a^5} - \frac{5r^3}{3a^3} - 2 \right ]e
I. $$\displaystyle \left [ \frac{3r^5}{2a^5}-\frac{5r^3}{2a^3} \right ]e$$
J. \left [ \frac{3r^5}{2a^5} - \frac{5r^3}{2a^3} - 1 \right ]e
Answer: B
The rate equation for a reaction is r =  $K[A]^{\circ}[B]^3$. Which of the following statements are true?

Options:
A. If the volume of the reaction vessel is decreased to $\frac{1}{3}$, the rate of reaction is $\frac{1}{9}$ of the original rate
B. If the volume of the reaction vessel is decreased to $\frac{1}{3}$, the rate of reaction is $ \frac{1^th}{27}$ of the original rate
C. If the volume of the reaction vessel is decreased to $\frac{1}{3}$, the rate of reaction is $\frac{1}{27}$ of the original rate
D. If the volume of the reaction vessel is decreased to $\frac{1}{4}$, the rate of reaction is $\frac{1}{32}$ of the original rate
E. If the volume of the reaction vessel is decreased to $\frac{1}{3}$, the rate of reaction is $\frac{1}{81}$ of the original rate
F. The plot of concentration of A Vs time is a straight line with a negative slope
G. If the volume of the reaction vessel is decreased to $\frac{1}{4}$, the rate of reaction is $\frac{1}{64}$ of the original rate
H. The plot of concentration of A Vs time is parallel to the time axis
I. Doubling the concentration of B quadruples the rate of reaction
Answer: H
The correct set of four quantum number for the valence electron of Kr$\left( Z=36 \right) $ having highest value of $\left( n+1 \right) $

Options:
A. 5,1,0,-\dfrac{1}{2}
B. $$4,1,-1,0$$
C. $$4,2,-2,-\dfrac { 1 }{ 2 } $$
D. 5,0,0,-\dfrac{1}{2}
E. $$4,1,-1,\dfrac { 1 }{ 2 } $$
F. 5,0,1,-\dfrac { 1 }{ 2 }
G. 5,0,1,\dfrac { 1 }{ 2 }
H. $$5,0,0,\dfrac { 1 }{ 2 } $$
I. 5,1,0,\dfrac { 1 }{ 2 }
Answer: H
The U-tube acts as a water siphon. The bend in the tube is lm above the water surface. The tube outlet is 7m below the water surface. The water issues from the bottom of the siphon as a free jet at atmospheric pressure. Determine the speed of the free jet and the minimum absolute pressure of the water in the bend. Given atm pressure $=1.01\times 10^5 \,N / m^2.g =9.8m  /s^2, p_w =10^3\, kg/g/m^3) $

Options:
A. 8 \, m/s, 1.3 \times 10^4 \, N/m^2
B. 10 \, m/s, 2.1 \times 10^4 \, N/m^2
C. 10 \, m/s, 1.5 \times 10^4 \, N/m^2
D. 9 \, m/s, 1.7 \times 10^4 \, N/m^2
E. $$12 \, m/s, 3\times 10^4 \, N/m^2$$
F. $$11.7 \, m/s, 2.27\times 10^4 \, N/m^2$$
G. 10 \, m/s, 1.9 \times 10^4 \, N/m^2
H. $$5 \, m/s, 4\times 10^4 \, N/m^2$$
I. 10 \, m/s, 2.5 \times 10^4 \, N/m^2
J. $$10 \, m/s, 2\times 10^4 \, N/m^2$$
Answer: J
The small dense stars rotate about their common centre of mass as a binary system, each with a period of $1$ year. One star has mass double than that of the other, while mass of the lighter star is one-third the mass of the Sun. The distance between the two stars is r and the distance of the earth from the Sun is R; find the relation between r and R.

Options:
A. r$=$R/2.
B. r$=$R.
C. r$=$3R/4
D. r$=$R/6
E. r$=$R/3.
F. r$=$2R.
G. r$=$R/4.
H. r$=$2R/3
I. r$=$R/5
J. 3r$=$R.
Answer: E
The attractive potential for an atom is given by  $V = V _ { 0 } \ln \dfrac { r } { r _ { 0 } } , v _ { 0 }$  and  $r _ { 0 }$  are constant and  $r$  is the radius of orbit. The radius  $\mathrm { r } _ { \mathrm { n } }$  of the  $\mathrm { n } ^ { \text { th } }$  Bohr's orbit depends upon principal quantum number  $\mathrm { n }$  as -

Options:
A. $$\mathrm { r } _ { \mathrm { n } } \propto \mathrm { n }$$
B. \mathrm { r } _ { \mathrm { n } } \propto \mathrm { n }^6
C. $$r _ { n } \propto \dfrac { 1 } { n }$$
D. \mathrm { r } _ { \mathrm { n } } \propto \mathrm { n ^ { 1/2 }}
E. $$\mathrm { r } _ { \mathrm { n } } \propto \mathrm { n ^ { 2 }}$$
F. \mathrm { r } _ { \mathrm { n } } \propto \mathrm { n }^4
G. $$r _ { n } \propto \dfrac { 1 } { n ^ { 2 } }$$
H. \mathrm { r } _ { \mathrm { n } } \propto \mathrm { n ^ { 3 }}
I. \mathrm { r } _ { \mathrm { n } } \propto \mathrm { n }^3
J. \mathrm { r } _ { \mathrm { n } } \propto \mathrm { n ^ { 5 }}
Answer: E
A plot of the density versus the pressure for air at $27^o$C has the limiting slope$=1.2\times 10^{-5}$kg $m^{-3}Pa^{-1}$. The composition of air(contains only $N_2$ and $O_2$) in volume ratio of $N_2$ and $O_2$ is:($R=8.0$ J/K-mol)

Options:
A. 45:55
B. $$90:10$$
C. 65:35
D. $$75:25$$
E. $$80:20$$
F. $$50:50$$
G. 60:40
H. 70:30
I. 55:45
Answer: F
When $10$ ml of $0.2\ M$ solution of an acid is added to $250$ ml of a buffer solution with $pH = 6.34$, the pH of the solution becomes $6.32$. The buffer capacity of the solution is:

Options:
A. 0.35
B. $$0.2$$
C. 0.15
D. $$0.25$$
E. 0.5
F. $$0.4$$
G. $$0.3$$
H. $$0.1$$
Answer: F
The standard reduction potential for the half-cell reaction, $Cl_2+2e^-\rightarrow 2Cl^-$ will be:
$Pt^{2+}+2Cl^-\rightarrow Pt+Cl_2, E^o_{cell}=-0.15$V;
$Pt^{2+}+2e^-\rightarrow Pt, E^o=1.20$V

Options:
A. +1.15 V
B. $-1.05$V
C. $+1.35$V
D. +1.40 V
E. +1.45V
F. $-1.35$V
G. $+1.05$V
H. $+1.25$V
I. +1.30 V
J. +1.20 V
Answer: C
The constant 'a' in Mosely's law $\sqrt v  = a(Z - b)\,for\,{K_lpha }X - ray$ is related to Rydberg constant R as ('c' is the speed of light)

Options:
A. $$R = \dfrac{{4{a^2}}}{{3c}}$$
B. R = \dfrac{{4{c^2}}}{5}
C. R = \dfrac{{4c}}{3a}
D. R = \dfrac{{3c}}{{2a}}
E. R = \dfrac{{4{c^2}}}{3}
F. $$R = \dfrac{{3c}}{{4a}}$$
G. $$R = \dfrac{{3{c^2}}}{4}$$
H. R = \dfrac{{3{c^2}}}{5}
I. $$R = \dfrac{{2c}}{a}$$
J. R = \dfrac{{4{c^2}}}{9}
Answer: G
Two wheels of radii $R$ and 3$R$ are coupled by using a conveyor belt. At $t = 0$ small wheel starts which angular acceleration $\pi \mathrm { rad } / \mathrm { s } ^ { 2 } .$ The time after which bigger wheel has angular speed of 100$\mathrm { rpm }$ is

Options:
A. 25$\mathrm { s }$
B. 5$\mathrm { s }$
C. 10\mathrm { s}
D. 15$\mathrm { s }$
E. 18\mathrm { s}
F. 20$\mathrm { s }$
G. 30$\mathrm { s }$
H. 12\mathrm { s}
Answer: B
'$n$' droplets of equal size of radius $r$ coalesce to form a bigger drop of radius $R$. The energy liberated is equal to _________.
($T=$ Surface tension of water)

Options:
A. 4\pi { r }^{ 2 }T\left[ { n }^{ { 1 }/{ 2 } }-1 \right]
B. $$4\pi { R }^{ 2 }T\left[ { n }^{ { 1 }/{ 3 } }-1 \right] $$
C. 4\pi { r }^{ 2 }T\left[ { n }^{ { 3 }/{ 2 } }-1 \right]
D. $$4\pi { r }^{ 2 }T\left[ { n }^{ { 2 }/{ 3 } }-1 \right] $$
E. 4\pi { R }^{ 2 }T\left[ { n }^{ { 1 }/{ 4 } }-1 \right]
F. 4\pi { R }^{ 2 }T\left[ { n }^{ { 1 }/{ 5 } }-1 \right]
G. 4\pi { R }^{ 2 }T\left[ { n }^{ { 2 }/{ 3 } }-1 \right]
H. 4\pi { R }^{ 2 }T\left[ { n }^{ { 1 }/{ 2 } }-1 \right]
I. $$4\pi { r }^{ 2 }T\left[ { n }^{ { 1 }/{ 3 } }-1 \right] $$
Answer: B
An iron ring measuring $15.00\;cm$ in diameter is to be shrunk on a pulley which is $15.05\;cm$ in diameter. All measurements refer to the room temperature $20^{\circ}C$. To what minimum temperature should the ring be heated to make the job possible? Calculate the strain developed in the ring when it comes to the room temperature. Coefficient of linear expansion of iron $=12\times10^{-6}/^\circ C$.

Options:
A. Temperature $=298^{\circ}C$, Strain $=3.33\times10^{-5}$.
B. Temperature $=268^{\circ}C$, Strain $=3.33 \times10^{-4}$
C. Temperature $=278^{\circ}C$, Strain $=3.33\times10^{-5}$.
D. Temperature $=288^{\circ}C$, Strain $=3.33 \times10^{-4}$
E. Temperature $=278^{\circ}C$, Strain $=3.33\times10^{-3}$.
F. Temperature $=278^{\circ}C$, Strain $=3.33 \times10^{-4}$.
G. Temperature $=268^{\circ}C$, Strain $=3.33 \times10^{-3}$.
H. Temperature $=288^{\circ}C$, Strain $=3.33 \times10^{-3}$
I. Temperature $=298^{\circ}C$, Strain $=3.33\times10^{-3}$.
J. Temperature $=268^{\circ}C$, Strain $=3.33 \times10^{-3}$
Answer: E
If gravitational attraction between two points masses be given by $F = G\dfrac{m_1 m_2}{r^n}$. Then the period of a satellite in a circular orbit will be proportional to

Options:
A. independent of n
B. r^\dfrac{1}{3}
C. r^\dfrac{3}{2}
D. $$r^\dfrac{1}{2}$$
E. r^\dfrac{1}{4}
F. r^\dfrac{2}{3}
G. r^1
H. r^3
I. r
J. $$r^2$$
Answer: I
At a height of $15\ m$ from ground velocity of a projectile $\displaystyle ec{v}= \left ( 10\hat{i}+10\hat{j} \right )$ and  ($g = 10\ ms^{-2}$ )

Options:
A. horizontal range of projectile is $100\ m$
B. maximum height of projectile form ground is $17.5\ m$
C. maximum height of projectile form ground is $22\ m$
D. maximum height of projectile form ground is $19.5\ m$
E. particle was projected at an angle of $\displaystyle 45^{\circ}$ with horizontal
F. maximum height of projectile form ground is $15\ m$
G. time of flight of projectile is $4\ s$
H. maximum height of projectile form ground is $20\ m$
I. maximum height of projectile form ground is $25\ m$
J. maximum height of projectile form ground is $21\ m$
Answer: H
Perimeter of the locus represented by $arg\ \left(\dfrac {z+i}{z-i}\right)=\dfrac {\pi}{4}$ is equal to

Options:
A. $$\dfrac {3\pi}{\sqrt 2}$$
B. $$None\ of\ these$$
C. $$\dfrac {\pi}{\sqrt 2}$$
D. \dfrac{2\pi}{\sqrt{2}}
E. \dfrac {5\pi}{\sqrt 2}
F. \dfrac{6\pi}{\sqrt{2}}
G. \dfrac{8\pi}{\sqrt{2}}
H. \dfrac{4\pi}{\sqrt{2}}
I. $$\dfrac {3\pi}{2}$$
J. \dfrac{7\pi}{\sqrt{2}}
Answer: B
The length of the latus rectum of the parabola $169\left \{ \left ( x-1 \right )^2+\left ( y-3 \right )^2 \right \}=\left ( 5x-12y+17 \right )^2$ is

Options:
A. \displaystyle \frac{30}{13}
B. \displaystyle \frac{14}{13}
C. \displaystyle \frac{20}{13}
D. \frac{26}{13}
E. \frac{16}{13}
F. $$\displaystyle \frac{28}{13}$$
G. \frac{24}{13}
H. none of these
I. $$\displaystyle \frac{12}{13}$$
Answer: F
A liquid flows along a horizontal pipe AB of uniform cross-section. The difference between the level of the liquid in tubes P and Q is $10\  cm$. The diameter of the tubes P and Q are the same. Then: ($\displaystyle \ g=9.8{ ms }^{ -2 }$)

Options:
A. level in Q is greater than that of P and velocity of flow is $1.2\  m/s$
B. level in Q is greater than that of P and velocity of flow is $1.4\  m/s$
C. level in P is greater than that of Q and velocity of flow is $1.0\ m/s$
D. level in Q is greater than that of P and velocity of flow is $1.0\  m/s$
E. level in Q is greater than that of P and velocity of flow is $1.3\  m/s$
F. level in P is greater than that of Q and velocity of flow is $1.4\  m/s$
G. level in Q is greater than that of P and velocity of flow is $0.7\ m/s$
H. level in P is greater than that of Q and velocity of flow is $0.7\  m/s$
Answer: B
One end of a $0.25m$ long metal bar is in steam and the other is in contact with ice .If $12g$ of ice melts per minute, then the thermal conductivity of the metal is (Given cross section of the bar$=5\times 10^{-4}m^{2}$ and latent heat of ice is $80\,cal \,g^{-1}$)

Options:
A. 25\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}
B. 35\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}
C. 20\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}
D. $$10\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}$$
E. $$40\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}$$
F. 15\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}
G. 32\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}
H. 31\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}
I. $$30\,cal\,s^{-1}\,m^{-1}\,^oC^{-1}$$
Answer: I
A ray of light travelling along the line $x + \sqrt {3y} = 5$ is incident on the x-axis and after refraction it enters the other side of the x-axis by turning $\dfrac {\pi}{6}$ away from the x-axis. The equation of the line along which the refracted ray travels is

Options:
A. $$\sqrt {3} - y - 5\sqrt {3} = 0$$
B. x + \sqrt{3} y - 10 = 0
C. \sqrt{3}x + y + 5\sqrt{3} = 0
D. $$x - \sqrt {3}y - 5\sqrt {3} = 0$$
E. x + \sqrt{3} y + 5 = 0
F. x + \sqrt{3} y + 5\sqrt{3} = 0
G. x - \sqrt{3} y + 5\sqrt {3} = 0
H. $$x + \sqrt{3} y - 5\sqrt {3} = 0$$
I. $$\sqrt {3}x + y - 5\sqrt {3} = 0$$
J. x + \sqrt{3} y - 5 = 0
Answer: H
A liquid is kept in a cylindrical vessel which is rotated along its axis. The liquid is rises at the sides. If the radius of the vessel is $r$ and the speed of revolution is $n$ rotations/second, find the difference in height of the liquid at the centre of vessel and its sides.

Options:
A. $$ h=\dfrac { 2{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g } $$
B. None of these
C. h=\dfrac { 8{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g }
D. $$ h=\dfrac {{ \pi }^{ 2 }{ r }^{ 2 }{ n }^{ 2 } }{ g } $$
E. $$ h=\dfrac { 4{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g } $$
F. h=\dfrac { 3{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g }
G. h=\dfrac { 7{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g }
H. h=\dfrac { 5{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g }
I. h=\dfrac { 6{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g }
J. h=\dfrac { 9{ \pi }^{ 2 }{ n }^{ 2 }{ r }^{ 2 } }{ g }
Answer: A
At $27^{o}C$ a gas compressed suddenly such that its pressure becomes $\dfrac{1}{32}$ of original pressure. Final temperature in Kelvin will be $\left ( \gamma =5/3 \right )$

Options:
A. 82.63
B. 75.5
C. 75.83
D. 74.63
E. 76
F. 100
G. 80.63
H. 74.83
I. 75
J. 81.63
Answer: I
A point particle of mass 0.1 kg is executing SHM with amplitude of 0.1 m. When the particle passes through the mean position, its kinetic energy is $8\times { 10 }^{ -3 }\quad J$. If the initial phase of oscillation is $4\mathring { 5 } $, the equ. of motion of this particle is

Options:
A. y=0.1sin\left( 4T+\frac { \pi }{ 2 } \right)
B. y=0.1sin\left( 4T+\frac { \pi }{ 3 } \right)
C. y=0.1sin\left( 3T+\frac { \pi }{ 5 } \right)
D. y=0.1sin\left( 2T+\frac { \pi }{ 4 } \right)
E. y=0.1sin\left( 3T+\frac { \pi }{ 3 } \right)
F. $$y=0.4sin\left( 6T+\frac { \pi }{ 4 } \right) $$
G. y=0.1sin\left( 4T+\frac { \pi }{ 5 } \right)
H. $$y=0.1sin\left( 3T+\frac { \pi }{ 4 } \right) $$
I. $$y=0.1sin\left( 4T+\frac { \pi }{ 4 } \right) $$
Answer: H
In the circuit shown the capacitor and inductor do not have energy initially. Then

Options:
A. the current flowing through the battery at t = $\infty$ is $2arepsilon /5R$
B. the current flowing through the battery at t = $\infty$ is $3\epsilon /7R$
C. the current flowing through the battery at t = $\infty$ is $4ớtarepsilon /5R$
D. the current flowing through the battery at t = 0 is $arepsilon /2R$
E. the current flowing through the battery at t = $\infty$ is $3arepsilon /5R$
F. the current flowing through the battery at t = $\infty$ is $arepsilon /2R$
G. the current flowing through the battery at t = 0 is $2arepsilon /3R$
H. the current flowing through the battery at t = $\infty$ is $4\epsilon /5R$
I. the current flowing through the battery at t = $\infty$ is $5\epsilon /7R$
J. the current flowing through the battery at t = $\infty$ is $\epsilon /3R$
Answer: A