MikaStars39/nano-eval huggingface dataset:
A uniform solid cylinder is placed on a horizontal board. The coefficient of friction between them is 0.3. The board is imparted a constant acceleration of $ a = 6m/ s^2$ in a horizontal direction at right angles of the cylindrical axis. Then (take $g = 10m/s^2$)

Options:
A. The limiting value of 'a' for which there will be no slipping is $14 m/ s^2$
B. The acceleration of the cylindrical axis in the absence of slipping will be $ 2 m/ s^2$.
C. The limiting value of 'a' for which there will be no slipping is $15 m/ s^2$
D. The limiting value of 'a' for which there will be no slipping is $13 m/ s^2$
E. The limiting value of 'a' for which there will be no slipping is $17 m/ s^2$
F. The limiting value of 'a' for which there will be no slipping is $12 m/ s^2$
G. The limiting value of 'a' for which there will be no slipping is $10 m/ s^2$
H. The limiting value of 'a' for which there will be no slipping is $19 m/ s^2$
I. The limiting value of 'a' for which there will be no slipping is $11 m/ s^2$
J. The acceleration of the cylindrical axis in the absence of slipping will be $ 1.5 m/ s^2$.
Answer: F
Use the relationship $\Delta \,G^{\circ}\,=\,- nFE^{\circ}_{cell}$ to estimate the minimum voltage required to electrolyse $Al_2O_3$ in the Hall-Heroult process.$\Delta G^{\circ}_{f}(Al_2O_3)\,=\,-1520\,kJ\,mol^{-1}$$\Delta G^{\circ}_{f}(CO_2)\,=\,-394\,kJ\,mol^{-1}$The oxidation of the graphite anode to $CO_2$ permits the electrolysis to occur at a lower voltage than if the electrolysis reactions were :$Al_2O_3\,\rightarrow\,2Al\,+\,3O_2$.What is the approximate value of low voltage?

Options:
A. 2 V
B. 3.5 V
C. 3.7 V
D. 3 V
E. 2.5 V
F. 4V
G. 3.8 V
H. 2.3 V
I. 3.2 V
J. 5V
Answer: A
An analysis of the rock shows that the relative number of $ Sr^{87}(t_{1/2} = 7.4 \times 10^15 year) $ and $Rb^{87} (t_{1/2} = 4.7 \times 10^15 year) $ atoms is 0.05. What is the age of the rock? Assume all $ Sr^{87} $ to be formed $ Rb^{87} $ only.

Options:
A. 1.33 \times 10^9 year
B. $$ 3.28 \times 10^9 year $$
C. 2.43 \times 10^9 year
D. $$ 4.32 \times 10^8 year $$
E. 1.43 \times 10^{11} year
F. 5.43 \times 10^9 year
G. 1.38 \times 10^9 year
H. 1.43 \times 10^{10} year
I. $$ 7.62 \times 10^9 year $$
J. $$ 1.43 \times 10^9 year $$
Answer: J
The surface of a metal has work function 2.66eV. This is illuminated with photons of wavelength 450nm. The de Broglie wavelength of the emitted photoelectrons is:
(Mass of electron = $9 \times 10^{-31} \ kg$)

Options:
A. $$8.18 \times 10^{-9} \ m$$
B. 3.07 \times 10^{-9} \ m
C. $$4.09 \times 10^{-9} \ m$$
D. 2.045 \times 10^{-9} \ m
E. 5.045 \times 10^{-9} \ m
F. 6.13 \times 10^{-9} \ m
G. 5.13 \times 10^{-9} \ m
H. $$1.02 \times 10^{-9} \ m$$
I. 6.12 \times 10^{-9} \ m
Answer: C
A vessel is partly filled with a liquid. Co-efficients of voltmetric expansion of material of the vessel and liquid are $\gamma_V $ & $ \gamma_L $ respectively. If the system is heated then volume unoccupied by the liquid will necessarily :

Options:
A. increase if $\gamma_V $ = $ \gamma_L $
B. remain unchanged if $\gamma_V $ > $ \gamma_L $
C. remain unchanged if $\gamma_V $ < $ \gamma_L $
D. decrease if $\gamma_V $ < $ \gamma_L $
E. decrease if $\gamma_V $ > $ \gamma_L $
F. increase if $\gamma_V $ > $ \gamma_L $
G. remain unchanged if $\gamma_V $ = $ \gamma_L $
H. decrease if $\gamma_V $ = $ \gamma_L $
I. increase if $\gamma_V $ < $ \gamma_L $
Answer: A
A particle of charge q and mass m starts moving from the origin under the action of an electric field $ ec E=E_0\hat i$  and magnetic field $ec B=B_0\hat k$. Its velocity at (x,0,0) is $(6\hat i+8\hat j)$. The value of x is

Options:
A. $$\displaystyle \frac{100 m}{qE_0}$$
B. \frac{60 m}{qE_0}
C. \frac{70 m}{qE_0}
D. $$\displaystyle \frac{50 m}{qE_0}$$
E. \frac{55 m}{qE_0}
F. \frac{75 m}{qE_0}
G. $$\displaystyle \frac{25 m}{qE_0}$$
H. \frac{40 m}{qE_0}
I. $$\displaystyle \frac{14 m}{qE_0}$$
J. \frac{45 m}{qE_0}
Answer: D
The bob of a simple pendulum is given a velocity in horizontal direction when the bob is at lowest position ,so that the bob describes vertical circle of radius equal to length of pendulum and tension in the string is $10 \ N$ when the bob is at an angle $60^0$ from lowest position of vertical circle. The tension in the string when the bob reaches highest position is (The mass of bob is $ 100$ gram. $g = 10\ ms^{-2}$):

Options:
A. 7.5\ N
B. 6.5 \ N
C. $$5.5\  N$$
D. 4.5 \ N
E. $$7 \ N$$
F. 8.0\ N
G. 6.0 \ N
H. $$9\  N$$
I. $$3.5 \ N$$
J. 8.5\ N
Answer: C
A capacitor is initially connected to a battery of EMF $3$V. At $t=0$, switch is thrown to B state. Now charge on capacitor at any instant is given by.

Options:
A. $$q=C(V-2V e^{-t/2RC})$$
B. $$q=C(V-2V e^{-t/RC})$$
C. q=C(V+2V e^{-t/RC})
D. $$q=C(V+2V e^{-t/2RC})$$
E. $$q=C(V+2V e^{-2t/RC})$$
F. q=C(V-2V e^{-t/3RC})
G. q=C(V-2V e^{-2t/RC})
H. q=C(V-2V e^{-t/4RC})
Answer: D
A mass of $3.9kg$ suspended from a string of length $0.5m$ is at rest. Another body of mass $0.1kg$ moving horizontally with a velocity $200m{s}^{-1}$ strikes and sticks to it. The tension in the string when it males ${60}^{o}$ with the vertical is 
($g=10m{s}^{-2}$)

Options:
A. $$100N$$
B. $$220N$$
C. 170N
D. 210N
E. 140N
F. $$180N$$
G. $$80N$$
H. 200N
I. 190N
J. 160N
Answer: F
A sinusoidal voltage of peak value $283\, V$ and angular frequency $320/s$ is applied to a series LCR circuit. Given that $R=5 \Omega, L=25 \,mH$ and $C=1000\mu F$. The total impedance, and phase difference between the voltage across the source and the current will respectively be:

Options:
A. 7\Omega \, and \, \tan^{-1}\left(\dfrac{5}{3}\right)
B. $10\Omega$ and $\tan^{-1}\left(\dfrac{8}{3}\right)$
C. 7\Omega \, and \, \tan^{-1}\left(\dfrac{3}{4}\right)
D. $7\Omega $ and $45^o$
E. 10\Omega \, and \, \tan^{-1}\left(\dfrac{4}{3}\right)
F. 7\Omega \, and \, \tan^{-1}\left(\dfrac{4}{3}\right)
G. $10\Omega$ and $\tan^{-1}\left(\dfrac{5}{3}\right)$
H. 7\Omega \, and \, \tan^{-1}\left(\dfrac{6}{3}\right)
I. 7\Omega \, and \, \tan^{-1}\left(\dfrac{7}{3}\right)
J. $7\Omega$ and $\tan^{-1}\left(\dfrac{5}{3}\right)$
Answer: D
In a photoelectric effect setup, a point source light of power $\left( \dfrac { P }{ 4\pi { R }^{ 2 } }  \right) =3.2\times { 10 }^{ -3 }W$ emits monoenergetic photons of energy $5.0\ eV$. These photon falling on a metallic sphere of radius $r=6\ mm$ and of work function $3.0\ eV$ at a distance of $0.6\ m$. The efficiency of photo-electron emission is one for every ${ 10 }^{ 5 }$ incident photons. Assume that the sphere is located and initially neutral and that photoelectrons are instantly swept away after emission. $[Given\ hc = 1240\ eV-nm]$
Calculate the accumulated charge on the metallic sphere in one sec.

Options:
A. 4 \times 10^{-13}
B. $$1.6\times { 10 }^{ -13 }$$
C. $$8\times { 10 }^{ -13 }$$
D. 3.2 \times 10^{-13}
E. $$None\ of\ these$$
F. $$3.2\times { 10 }^{ -12 }$$
G. 2.4 \times 10^{-13}
H. 1.2 \times 10^{-12}
I. 6.4 \times 10^{-13}
J. 1.6 \times 10^{-12}
Answer: C
Velocity and acceleration vectors of a charged particle moving in a magnetic field at some instant are $ec{v}=3\hat{i}+4\hat{j}$ and $ec{a}=2\hat{i}+x\hat{j}$. Select the correct alternative(s) :

Options:
A. $$ x = 3$$
B. magnetic field is along x-direction
C. x = 2.5
D. kinetic energy of the particle is constant
E. magnetic field is along y-direction
F. $$x = 1.5$$
G. magnetic field is along the negative x-direction
Answer: D
A tunnel is dug in the earth across one of its diameter. Two masses $m$ and $2m$ are dropped from the two ends of the tunnel. The masses collide and stick each other. They perform $SHM,$ the amplitude of which is $(R = $ radius of earth)

Options:
A. $$R$$
B. R/8
C. $$R/2$$
D. $$R/5$$
E. $$2R/3$$
F. R/4
G. $$R/9$$
H. $$R/3$$
I. R/6
J. R/7
Answer: H
The effective radius of the iron atom is $1.42\dot {A}$ . It has FCC structure. Calculate its density (Fe = 56 amu)

Options:
A. $$5.74 gm/ cm^{3}$$
B. 9.36 gm/ cm^{3}
C. $$11.48 gm/ cm^{3}$$
D. 7.24 gm/ cm^{3}
E. 7.21 gm/ cm^{3}
F. 6.21 gm/ cm^{3}
G. $$2.87 gm/ cm^{3}$$
H. $$1.435 gm/ cm^{3}$$
I. 8.62 gm/ cm^{3}
J. 10.36 gm/ cm^{3}
Answer: C
A buggy of mass $100 kg$ is free to move on a frictionless horizontal track. Two men, each of mass $50 kg$, are standing on the buggy, which is initially stationary. The men jump off the buggy with velocity $=100m/s$ relative to the buggy. In one situation, the men jump one after the other. In another situation, the men jump simultaneously. What is the ratio of recoil velocities of the buggy in two cases ?

Options:
A. 6:7
B. $$5:4$$
C. 5:6
D. $$6:5$$
E. $$7:5$$
F. $$5:3$$
G. 5:7
H. $$7:6$$
Answer: H
A certain meal when irradiated by light $(v=3.2\times 10^{16}Hz)$ emits photoelectrons with twice K.E. as did photoelectrons when the same metal is irradiated by light $(v= 2.1\times10^{16}Hz)$. The $v_0$ of the metal is:

Options:
A. $$8 \times 10^{15}Hz$$
B. $$4\times 10^{12}Hz$$
C. $$1.2 \times 10^{14}Hz$$
D. 1.0 \times 10^{16}Hz
E. 6 \times 10^{15}Hz
F. 7.2 \times 10^{15}Hz
G. $$1.2 \times 10^{16}Hz$$
H. 1.2 \times 10^{15}Hz
I. 1.8 \times 10^{15}Hz
J. 1.6 \times 10^{15}Hz
Answer: A
Question 19:- 
 Fastest electrons ejected from a surface by incident light of wavelength 330 nm are bent into circle of radius 1m as they pass through magnetic field of $ 2.5 \times 10^{-6} T $ . Find the work function of the surface.

Options:
A. $$ arphi = 4.3 eV $$
B. $$ arphi = 3.2 eV $$
C. \varphi = 3.0 eV
D. \varphi = 2.7 eV
E. $$ arphi = 2.2 eV $$
F. \varphi = 2.3 eV
G. \varphi = 3.1 eV
H. $$ arphi = 2.5 eV $$
I. \varphi = 2.8 eV
J. \varphi = 2.1 eV
Answer: E
A small particle of mass $m/10$ is moving horizontally at a height of $3R/2$ from ground with velocity $10\ m/s$. A perfectly inelastic collision occurs at point $P$ of sphere is $R. (m = 10\ kg$ and $R = 0.1\ m)$ (Assume all surfaces to be smooth).
Speed of sphere just after collision is

Options:
A. $$\dfrac{27}{43}\ m/s$$
B. $$\dfrac{40}{43}\ m/s$$
C. \dfrac{29}{43}\ m/s
D. \dfrac{34}{43}\ m/s
E. \dfrac{33}{43}\ m/s
F. \dfrac{25}{43}\ m/s
G. \dfrac{32}{43}\ m/s
H. \dfrac{31}{43}\ m/s
I. $$\dfrac{35}{43} \ m/s$$
J. $$\dfrac{30}{43}\ m/s$$
Answer: J
Sinusoidal waves $5.00 cm$ in amplitude are to be transmitted along a string having a linear mass density equal to $4.00 \times 10^{-2} kg/m$. If the source can deliver an average power of $90 W$ and the string is under a tension of $100 N$, then the highest frequency at which the source can operate is (take $\pi^2 = 10)$:

Options:
A. $$62 Hz$$
B. 35 Hz
C. 42 Hz
D. 32 Hz
E. 40 Hz
F. $$45 Hz$$
G. 33 Hz
H. $$50 Hz$$
I. 37 Hz
J. $$30 Hz$$
Answer: J
The acceleration of the moon with respect to earth is $0.0027 m/s^{2}$. If suppose moon is suddenly stopped and then released, (given: radius of moon = $1/4^{th}$ the radius of earth) the acceleration of the moon just before striking the earth's surface is :

Options:
A. 6.0 m/s^{2}
B. $$10 m/s^{2}$$
C. 6.6 m/s^{2}
D. $$6.4 m/s^{2}$$
E. 6.2 m/s^{2}
F. $$5 m/s^{2}$$
G. 4.8 m/s^{2}
H. $$0.0027 m/s^{2}$$
I. 5.8 m/s^{2}
J. 5.6 m/s^{2}
Answer: D
The acceleration of a particle is $a=-100x+50.$ It is released from $x=2$ Here $a$ and $x$ are in SI units:

Options:
A. the particle will perform SHM of time period $0.63\ s$
B. the particle will have a maximum velocity of $17.5\ m/s$
C. the particle will have a maximum velocity of $15\ m/s$
D. the particle will have a maximum velocity of $25\ m/s$
E. the particle will have a maximum velocity of $20\ m/s$
F. the particle will have a maximum velocity of $22.5\ m/s$
G. the particle will have a maximum velocity of $12.5\ m/s$
H. the particle will perform SHM of amplitude $2\ m$
I. the particle will have a maximum velocity of $10\ m/s$
Answer: C
An air bubble in sphere having $4$ cm diameter appears $1$cm from surface nearest to eye when loaded along diameter. If $_{a}\mu_g=1.5$, the distance of bubble from refracting surface is?

Options:
A. 1.4 cm
B. $3.2$cm
C. 1.4
D. 1.0 cm
E. $2.8$cm
F. $1.8$cm
G. 2.2 cm
H. 2.6 cm
I. $1.6$cm
J. $1.2$cm
Answer: J
What voltage must  be applied to an electron mocroscope to produce electrons of $\lambda=1.0\mathring{A}$?

Options:
A. $160$ volt
B. $180$ volt
C. 140 volt
D. 185 volt
E. $150$ volt
F. $190$ volt
G. 192 volt
H. 195 volt
I. 200 volt
J. 170 volt
Answer: F
The figure shows an insulated cylinder divided into three parts A, B and C. Piston I and II are connected by a rigid rod and can move without friction inside the cylinder. Piston I is perfectly conducting while piston II is perfectly insulating. The initial state of gas $(\gamma = 1.5)$ present in each compartment A, B and C is as shown. Now, compartment A is slowly given heat through a heater H such that the final volume of C becomes $\displaystyle{\frac{4V_0}{9}}$. Assume the gas to be ideal and find the heat supplied by the heater.

Options:
A. 24P_oV_o
B. 20P_oV_o
C. $$18P_oV_o$$
D. 22P_oV_o
E. $$9P_oV_o$$
F. $$25P_oV_o$$
G. 17P_oV_o
H. 15P_oV_o
I. $$12P_oV_o$$
J. 16P_oV_o
Answer: C
Calculate the density of NaCl type ionic solid AB ($M_{w}$=60) in $gm/cm^{3}$ if shortest distance between two nearest neighbours is 100 pm & solid having $20\%$ schottky defect$(N_{A}=6\times10^{23}$)

Options:
A. 10
B. 36
C. 30
D. 33
E. 32
F. 38
G. 20
H. 40
I. 35
J. 34
Answer: H
A railway compartment is $16\ m$ long, $2.4\ m$ wide and $3.2\ m$ high. It is moving with a velocity $v$. A particle moving horizontally with a speed $u$, perpendicular to the direction of $v$, enters through a hole at an upper corner $A$ and strikes the diagonally opposite corner $B$. Assume $g=10 m/{ s }^{ 2 }$. Which of the following values of $u$ and $v$ is(are) correct?

Options:
A. $$u = 2\ m/s $$
B. $$u=3.5\ m/s$$
C. u=2.9\ m/s
D. u=2.5\ m/s
E. $$u=3\ m/s$$
F. $$v=15 \ m/s$$
G. u=3.2\ m/s
H. u = 2.8\ m/s
I. u=2.7\ m/s
Answer: E
A rocket of initial mass $5000 kg$ ejects gets at a constant rate of $60kg/s$ with a relative speed of $2050 m/s$. Acceleration of the rocket 15 second after it is blasted off from the surface of earth will be ($g = 10m/{s}^{2}$?)

Options:
A. $$40m/{s}^{2}$$
B. $$30m/{s}^{2}$$
C. 35m/{s}^{2}
D. $$25m/{s}^{2}$$
E. 32m/{s}^{2}
F. 22m/{s}^{2}
G. $$20m/{s}^{2}$$
H. $$10m/{s}^{2}$$
I. 15m/{s}^{2}
Answer: B
The relation between $\lambda_3:$ wavelength of series limit of Lyman series, $\lambda_2:$ the wavelength of the series limit of Balmer series & $\lambda_1:$ the wavelength of first line of Lyman series is

Options:
A. \(\frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{2\lambda_3} \)
B. \frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{3\lambda_3}
C. \(\frac{1}{\lambda_1} + \frac{1}{\lambda_2} = \frac{1}{\lambda_3} \)
D. \frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{2\lambda_3}
E. $$\lambda_1=\lambda_2+\lambda_3$$
F. $$\lambda_2=\lambda_3-\lambda_1$$
G. $$\frac{1}{\lambda_1}-\frac{1}{\lambda_2}=\frac{1}{\lambda_3}$$
H. \(\frac{1}{\lambda_1} + \frac{1}{\lambda_2} = \frac{2}{\lambda_3}\)
I. $$\lambda_3=\lambda_1+\lambda_2$$
J. \(\frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{4\lambda_3} \)
Answer: G
A parallel-plate air capacitor has initial capacitance C. If plate separation is slowly increased from $d_1$ to $d_2$, then mark the correct statement(s). [Take potential of the capacitor to be constant, i.e., throughout the process it remains connected to battery.]

Options:
A. Work done by battery $=3$ times the change in electric potential energy stored in capacitor
B. Work done by battery $=\frac{7}{2}$ times the change in electric potential energy stored in capacitor
C. Work done by battery $=\frac{3}{2}$ times the change in electric potential energy stored in capacitor
D. Work done by electric force $
eq -$ve of work done by external agent
E. Work done by battery $=\frac{5}{2}$ times the change in electric potential energy stored in capacitor
F. Work done by external force $=-\displaystyle\int ec{F}\cdot ec{d}x$, where $ec{F}$ is the electric force of attraction between the plates at plate separation x
G. Work done by battery $=\frac{1}{2}$ times the change in electric potential energy stored in capacitor
H. Work done by battery $=1$ times the change in electric potential energy stored in capacitor
I. Work done by battery $=2$ times the change in electric potential energy stored in capacitor
Answer: I
For emission line of atomic hydrogen from $n_1 = 8$ to $n_1 = n$, the plot of $(\frac{1}{n^2})$ will be (The Rydberg constnat, $R_H$ is in wave  number unit)

Options:
A. linear with slope - 4R_H
B. linear with slope 2R_H
C. linear with intercept - 2R_H
D. non linear
E. linear with intercept - $R_H$
F. linear with slope $R_H$
G. linear with slope - $R_H$
H. linear with intercept - 3R_H
I. linear with slope - 3R_H
J. linear with slope - 2R_H
Answer: D
The only electron in the hydrogen atom resides under ordinary conditions on the first orbit. When energy is supplied, the electron moves to higher energy orbit depending on the amount of energy absorbed. When this electron returns to any of the lower orbits, it emits energy. Lyman series is formed when the electron returns to the lowest orbit while Balmer series is formed when the electron returns to second orbit. Similarly, Paschen, Brackett and Pfund series are formed when electron returns to the third, fourth and fifth orbits from higher energy orbits respectively. The maximum number of lines produced when an electron jumps from nth level to ground level is equal Maximum number of lines produced when an electron jumps from nth level to ground level is equal to. $\displaystyle \:\frac{1\left ( n-1 \right )}{2}.$ For example, in the case of $n= 4, $ number of lines produces is 6. $\displaystyle \:\left ( 4\rightarrow 3,4\rightarrow 2,4\rightarrow 1,3\rightarrow 2,3\rightarrow 1,2\rightarrow 1 \right ).$ When an electron returns from to state, the number of lines in the spectrum will be equal to $\displaystyle \:\frac{\left ( n_{2}-n_{1} \right )\left ( n_{2}-n_{1}+1 \right )}{2}$ If the electron comes back from energy level having energy to energy level having energy, then the difference may be expressed in the terms of energy of photon as $\displaystyle \:E_{2}-E_{1}\Delta E,\lambda = \frac{hc}{\Delta E}$  Since h and c are constants, $\Delta E$ corresponds to definite energy; thus each transition from one energy level to another will produce a light of definite wavelength. This is observed as a line in the spectrum of the hydrogen atom. Wavenumber of line is given by the formula $\displaystyle \:\bar{v}= \left ( \frac{1}{n_{1}^{2}} -\frac{1}{n_{2}^{2}}\right ).$ where R is a Rydberg's constant $\displaystyle \:\left ( R= 1.1\times 10^{7}m^{-1} \right )$The energy photon emitted corresponding to transition n = 3 to n=1 is:[$h=6\times10^{-34} J-sec]$

Options:
A. $1.76 \times 10^{-20}$ J
B. $1.76 \times 10^{-22}$ J
C. $1.98\times 10^{-18}$ J
D. $1.76 \times 10^{-19}$ J
E. None of these
F. 1.76 \times 10^{-16} J
G. $1.76 \times 10^{-18}$ J
H. 1.76 \times 10^{-21} J
I. $1.76 \times 10^{-17}$ J
Answer: G
Chloride samples are prepared for analysis by using $NaCl, KCl$ and $ NH_{4}Cl $ separately or as mixture. What minimum volume of $5$% by weight $AgNO_{3}$ solution (specific gravity $= 1.02\:g\:mL^{-1})$ must be added to a sample of $0.321$ g, in order to ensure complete precipitation of chloride in every possible case?

Options:
A. $40$ mL
B. 25 mL
C. $20$ mL
D. $50$ mL
E. 18 mL
F. 10 mL
G. 22 mL
H. $30$ mL
I. 15 mL
J. 21 mL
Answer: C
A plane polarized monochromatic EM wave is travelling a vacuum along $z$ direction such that at $t={t}_{1}$ it is found that the electric field is zero at a spatial point ${z}_{1}$. The next zero that occurs in its neighborhood. is at ${z}_{2}$. The frequency of the electromagnetic wave is:

Options:
A. \cfrac { 3.3 \times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| }
B. $$\cfrac { 1.5\times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| } $$
C. $$\cfrac { 1 }{ { t }_{ 1 }+\cfrac { \left| { z }_{ 2 }-{ z }_{ 1 } \right| }{ 3\times { 10 }^{ 8 } } } $$
D. \cfrac { 2.5 \times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| }
E. \cfrac { 4.5 \times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| }
F. \cfrac { 3.5 \times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| }
G. \cfrac { 2.8 \times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| }
H. $$\cfrac { 6\times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| } $$
I. $$\cfrac { 3\times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| } $$
J. \cfrac { 2.9 \times { 10 }^{ 8 } }{ \left| { z }_{ 2 }-{ z }_{ 1 } \right| }
Answer: I
The wave function of $1\ s$ orbital of $H-$atom is $\Psi =\dfrac {1}{\sqrt {\pi}} \left (\dfrac {1}{a_0}\right)^{3/2} e^{-r/a_0}$ where $a_0=$ Bohr radius. The probability finding the electrons at a distance $'r'$ from the nucleus is given by:

Options:
A. \(\displaystyle \int \Psi ^2 4\pi r^{1.5} \ dr\)
B. $$\Psi =\Psi ^{2}\ dr$$
C. \Psi ^2 4\pi r \ dr
D. \(\displaystyle \int \Psi ^2 4\pi r^3 \ dr\)
E. zero
F. $$\displaystyle \int \Psi ^2 4\pi r^2 \ dr$$
G. $$\Psi ^24\pi r^2 \ dr$$
H. \Psi ^2 4\pi r^3 \ dr
I. \Psi ^2 4\pi r^2
J. \displaystyle \int \Psi ^2 4\pi r \ dr
Answer: E
Water enters through end  $A$  with speed  $v_{1}$  and leaves through end  $B$  with speed  $v_{2}$  of a cylindrical tube $AB.$  The tube is always completely filled with water. In case $I$ tube is horizontal and in case $II$ it is vertical with end $A$ upwards and in case III it is vertical with end $B$ upwards. We have $v_{1} = v _{2}$ for

Options:
A. Case I and Case III only
B. Case $III$
C. Case I and Case II only
D. Case I only
E. Case $II$
F. Case $I$
G. Case II only
H. Case I and Case II and Case III
I. Each case
J. Case II and Case III only
Answer: I
A body of mass of 8 Kg lies on a rough horizontal surface. It is found that a horizontal force of P newton gives the body an acceleration of $-1 ms^{-2}$  and that a horizontal force of 2P N gives an acceleration of 16 $ ms^{-2}$ . If $g=10 ms^2 $ the coefficient of friction between the body and the table is:

Options:
A. 0.2
B. 0.6
C. 0.5
D. 0.7
E. 0.1
F. 0.15
G. 0.8
H. 0.4
I. 0.9
J. 0.3
Answer: A
A tube filled with water and close at both ends uniformly rotates in a horizontal plane about the $ OO' $ axis. The manometers $A$ and $B$ fixed in the tube distance $ { r }_{ 1 }$ and $ { r }_{ 2 }$ form rotational axis indicate pressure ${ p }_{ 1 }$ and $ { p }_{ 2 }$, respectively. Determine the angular velocity $ \omega $ of rotation of the tube.

Options:
A. $$ \omega =\dfrac { \sqrt { 2({ p }_{ 2 }-{ p }_{ 1 }) } }{ \rho { r }_{ 1 }{ r }_{ 2 } } $$
B. $$ \omega =\sqrt { \dfrac { 2\left( { p }_{ 2 }-{ p }_{ 1 } \right) }{ \left( \rho { r }_{ 2 }^{ 2 }-{ r }_{ 1 }^{ 2 } \right) } } $$
C. \omega =\sqrt { \dfrac { 2({ p }_{ 2 }-{ p }_{ 1 }) }{ \rho ( { r }_{ 2 }^{ 2 }-{ r }_{ 1 }^{ 2 } ) - 1 } }
D. $$ \omega =\sqrt { \dfrac { 2({ r }_{ 2 }^{ 2 }-{ r }_{ 1 }^{ 2 }) }{ { p }_{ 1 }-{ p }_{ 2 } } } $$
E. \omega =\sqrt { \dfrac { 2({ p }_{ 2 }-{ p }_{ 1 }) }{ \rho ( { r }_{ 2 }^{ 2 }-{ r }_{ 1 }^{ 2 } ) + 1 } }
F. \omega =\sqrt { \dfrac { 2({ p }_{ 2 }-{ p }_{ 1 }) }{ \rho ( { r }_{ 1 }^{ 2 }+{ r }_{ 2 }^{ 2 } ) } }
G. \omega =\sqrt { \dfrac { 2({ p }_{ 2 }-{ p }_{ 1 }) }{ \rho ( { r }_{ 1 }^{ 2 }+{ r }_{ 2 }^{ 2 } ) - 1 } }
H. \omega =\sqrt { \dfrac { 2({ p }_{ 2 }-{ p }_{ 1 }) }{\rho(  { r }_{ 2 }^{ 2 }+{ r }_{ 1 }^{ 2 } )} }
I. $$ \omega =\sqrt { \dfrac { 2({ p }_{ 2 }-{ p }_{ 1 }) }{\rho(  { r }_{ 1 }^{ 2 }-{ r }_{ 2 }^{ 2 } )} } $$
J. \omega =\sqrt { \dfrac { 2({ p }_{ 2 }-{ p }_{ 1 }) }{ \rho ( { r }_{ 2 }^{ 2 }-{ r }_{ 1 }^{ 2 } ) } }
Answer: I
A Young's Double slit experiment is performed using light of wavelength $\lambda=5000\  \mathring {A}$ which emerges in phase from two slits $3\ \times 10^{-7}\ m$ apart. A transparent sheet of thickness $t=1.5\ \times 10^{-7}\ m$ is placed over one of the slits. The refractive index of the material of this sheet is $\mu=1.7$. The central maximum of the interference pattern appears at

Options:
A. $$8.84^{o}$$
B. 5.08^{o}
C. $$8.48^{o}$$
D. $$4.88^{o}$$
E. 4.18^{o}
F. $$3.50^{o}$$
G. 3.88^{o}
H. 4.58^{o}
I. 4.28^{o}
J. 4.48^{o}
Answer: D
A particle is executing SHM with amplitude $A$, time period $T$, maximum acceleration ${a_{0}}$ and maximum velocity ${v_{0}}$. It starts from mean position at $t=0$ and at time $t$ it has the displacement $A/2$, acceleration $a$ and velocity $v$, then

Options:
A. a=\dfrac{a_{0}}{8}
B. a=\dfrac{a_{0}}{7}
C. $$t=\dfrac{T}{8}$$
D. a=\dfrac{a_{0}}{5}
E. a=\dfrac{a_{0}}{3}
F. $$t=\dfrac{T}{12}$$
G. a=\dfrac{a_{0}}{6}
H. $$v=\dfrac{v_{0}}{2}$$
I. a=\dfrac{a_{0}}{4}
J. $$a=\dfrac{a_{0}}{2}$$
Answer: J
For a reaction involving a single reactant and order other than one, which of the following statement(s) is/are true?
($C=$ concentration of reactant at time $t$)

Options:
A. If $f$ is the fraction of the reactant concentration consumed over time interval $t$, then $\log { \left( \cfrac { f }{ t } \right) } $ is linearly related to $\left( \log { C } \right) $, when $f\ll 1$
B. If ${ t }_{ 1/2 }$ and ${ t }_{ 3/4 }$ are the time intervals for $50$% and $75$% consumption of the reactant, then ${ t }_{ 3/4 }:{ t }_{ 1/2 }={ 2 }^{ n-1 } \times 2$
C. If ${ t }_{ 1/2 }$ and ${ t }_{ 3/4 }$ are the time intervals for $50$% and $75$% consumption of the reactant, then ${ t }_{ 3/4 }:{ t }_{ 1/2 }={ 2 }^{ n-1 } \times 4
D. A plot of ${C}^{(n-1)}$ vs $t$ is linear with slope proportional to $(n-1)$
E. If ${ t }_{ 1/2 }$ and ${ t }_{ 3/4 }$ are the time intervals for $50$% and $75$% consumption of the reactant, then ${ t }_{ 3/4 }:{ t }_{ 1/2 }={ 2 }^{ n-1 }-2$
F. If ${ t }_{ 1/2 }$ and ${ t }_{ 3/4 }$ are the time intervals for $50$% and $75$% consumption of the reactant, then ${ t }_{ 3/4 }:{ t }_{ 1/2 }={ 2 }^{ n-1 }-1$
G. If ${ t }_{ 1/2 }$ and ${ t }_{ 3/4 }$ are the time intervals for $50$% and $75$% consumption of the reactant, then ${ t }_{ 3/4 }:{ t }_{ 1/2 }={ 2 }^{ n-1 } \times 3$
H. If ${ t }_{ 1/2 }$ and ${ t }_{ 3/4 }$ are the time intervals for $50$% and $75$% consumption of the reactant, then ${ t }_{ 3/4 }:{ t }_{ 1/2 }={ 2 }^{ n-1 }+1$
I. If ${ t }_{ 1/2 }$ and ${ t }_{ 3/4 }$ are the time intervals for $50$% and $75$% consumption of the reactant, then ${ t }_{ 3/4 }:{ t }_{ 1/2 }={ 2 }^{ n-1 }+2
J. A plot of $\log { C } \quad vs\quad \log { t } $ is linear with slope $=1$
Answer: H
A solid cylinder of mass $3kg$ is placed on a rough inclined plane of inclination ${30}^{o}$. If $g=10m{ s }^{ -2 }$, then the minimum frictional force required for it to roll without slipping down the plane is

Options:
A. $$5N$$
B. 6N
C. $$2N$$
D. $$18N$$
E. 7N
F. 12N
G. 20N
H. 8N
I. 10N
J. $$15N$$
Answer: A
In the given circuit initially the charge on capacitor is $\displaystyle Q_{0} $ At t = 0 the switch S is closed Which of the following statement(s) is/are correct?

Options:
A. \(\displaystyle \frac{Q_{0}^{2}}{3C}\)
B. Maximum magnetic energy can be stored in the inductor is \(\displaystyle \frac{Q_{0}^{2}}{L}\)
C. \(\displaystyle \frac{Q_{0}^{2}}{3L}\)
D. Maximum magnetic energy can be stored in the inductor is $\displaystyle \frac{Q_{0}^{2}}{2L}$
E. Maximum magnetic energy can be stored in the inductor is \(\displaystyle \frac{Q_{0}^{2}}{4C}\)
F. \(\displaystyle \frac{Q_{0}^{2}}{4L}\)
G. Maximum magnetic energy can be stored in the inductor is $\displaystyle \frac{Q_{0}^{2}}{2C}$
Answer: G
During dischargning of lead storage cell concentration ${ H }_{ 2 }{ SO }_{ 4 }$ reduces from 40% w/w to 30% w/w. Find the total charged produced. Given volume of solution 4.9 litre and density = 1.2 gm/ml [Assuming volume and density remain constant]

Options:
A. 18 faraday
B. 15 faraday
C. 3 faraday
D. 10 faraday
E. 9 faraday
F. 5 faraday
G. 12 faraday
H. 6 faraday
I. 7 faraday
J. 8 faraday
Answer: H
Calculate energy of aromatisation of cyclohexane according to the following reaction-
Given, bond energies: $C-C = 348, C-H = 415, C = C = 600, H-H = 436\ kJ\ mol^{-1}$ respectively, sublimation energy of $C(gr)$ is $717\ kJ/mol$, Resonance energy of $C_{6}H_{6}(l) = -152\ kJ\ mol^{-1}, \triangle H_{V}^{\circ}$ of benzene $= 30.8$ and cyclohexane is $33\ kJ/ mol$.

Options:
A. 276.0\ k
B. $$2.762\ k$$
C. 275.4\ k
D. 275.2\ k
E. $$326.5\ k$$
F. $$276.2\ k$$
G. 175.4\ k
H. 267.2\ k
I. 276.5\ k
Answer: F
A mixture of ethane $(C_{2}H_{6})$ and ethene $(C_{2}H_{4})$ occupies $40 L$ at $1.00$ atm and $400 K$. The mixture reacts completely with $130 g$ of $O_{2}$ to produce $CO_{2}$ and $H_{2}O$. Assuming ideal gas behaviour, mole fractions of $C_{2}H_{4}$ and $C_{2}H_{6}$ in the mixture will be :

Options:
A. x_{C_{2}H_{6}}=0.45, x_{C_{2}H_{4}}=0.55
B. x_{C_{2}H_{6}}=0.55, x_{C_{2}H_{4}}=0.45
C. $x_{C_{2}H_{6}}=0.6$, $x_{C_{2}H_{4}}=0.4$
D. $x_{C_{2}H_{6}}=0.3707$, $x_{C_{2}H_{4}}=0.6293$
E. $x_{C_{2}H_{6}}=0.5$, $x_{C_{2}H_{4}}=0.5$,
F. $x_{C_{2}H_{6}}=0.6707$, $x_{C_{2}H_{4}}=0.3293$,
G. none of these
H. $x_{C_{2}H_{6}}=0.3293$, $x_{C_{2}H_{4}}=0.6707$,
I. $x_{C_{2}H_{6}}=0.4$, $x_{C_{2}H_{4}}=0.6$
J. $x_{C_{2}H_{6}}=0.3293$, $x_{C_{2}H_{4}}=0.6707$
Answer: F
The cathode of a photo electric cell is changed such that the work function changes from $W_{1} \  to \ \ W_{2}(W_{1} >W_{2})$. If the currents before and after change are $I_{1}$ and $I_{2}$, all other conditions unchanged, then :(assuming $h_{ v} >W_{2})$

Options:
A. $$I_{1} \neq I_{2}$$
B. I_{1} \leq I_{2}
C. $$I_{1} > I_{2}$$
D. $$I_{1} \geq I_{2}$$
E. $$I_{1} = I_{2}$$
F. $$I_{1} < I_{2}$$
Answer: E
The vapour pressure of a certain liquid is given by the equation:$log_{10}P=3.54595+\dfrac {313.7}{T}+1.40655 log_{10}T$, where, P is the vapour pressure in mm and T is temperature in K. The molar latent heat of vaporisation as a function of temperature and  its value in cal at 80 K is :

Options:
A. 1024.56
B. 822.84
C. $$922.84$$
D. $$1056.24$$
E. 922.48
F. $$1194$$
G. $$597$$
H. $$778.56$$
Answer: F
A geostationary satellite orbits around the earth in a circular orbit of radius $36000 km$. Then, the time period of a spy satellite orbiting a few hundred kilometers above the earth's surface will approximately be $($ Given: $R_{Earth}=6400$ km $)$

Options:
A. 3.5 h
B. 1.5 h
C. $$2 h$$
D. 2.5 h
E. $$1 h$$
F. $$\dfrac{1}{2} h$$
G. $$4 h$$
H. 2.2 h
I. 3 h
J. 6 h
Answer: C
The velocity of a particle moving along a straight line increases according to the linear law $v = v_{0} + kx$ , where $k$ is a constant. Then

Options:
A. velocity varies linearly with displacement with slope of velocity displacement curve equal to $k$.
B. velocity varies linearly with displacement with slope of velocity displacement curve equal to \( k + 1 \)
C. velocity varies linearly with displacement with slope of velocity displacement curve equal to \( -k \).
D. velocity varies linearly with displacement with slope of velocity displacement curve equal to \( 3k \)
E. the acceleration of the particle is \( k(v_{0} - kx) \)
F. velocity varies linearly with displacement with slope of velocity displacement curve equal to \( 2k \)
G. velocity varies linearly with displacement with slope of velocity displacement curve equal to \( -2k \)
H. the acceleration of the particle is $k (v_{0}+kx)$.
I. the acceleration of the particle is zero.
Answer: A
What is the ratio of the enthalpy yield on combustion of hydrogen atoms to steam to the yield on combustion of an equal mass of hydrogen molecules to steam? Given : $\displaystyle \:H_{2}(g)+\dfrac{1}{2}O_{2}(g)\rightarrow H_{2}O(g), \ \ \Delta H= -242$ kJ,Bond energy $(H-H)= 436$ kJ.

Options:
A. 2.60 : 1
B. 1 : 0.80
C. 2.40 : 1
D. 3.00 : 1
E. 0.80 :1
F. 1.80 : 1
G. 2.50 : 1
H. 2.20 : 1
I. 2.70 : 1
J. 2.80 : 1
Answer: J
A circuit connected to an ac source of emf $e = e_0 \sin (100 t)$ with t in seconds, gives a phase difference of $\dfrac{\pi}{4}$ between the emf e and current i. Which of the following circuits will exhibit this ?

Options:
A. RC circuit with \( R = 10 k \Omega \) and \( C = 10 \mu F \)
B. RC circuit with $R = 10 k \Omega$ and $C = 10 \mu F$
C. RC circuit with $R = 10 k \Omega$ and $C = 1 \mu F$
D. RL circuit with $R = 1 k \Omega $ and $L = 1 m H$
E. RL circuit with $R = 1 k \Omega $ and $L = 10 mH$
F. RL circuit with \( R = 10 k \Omega \) and \( L = 1 mH \)
G. RC circuit with \( R = 1 k \Omega \) and \( C = 1 \mu F \)
H. RL circuit with \( R = 1 k \Omega \) and \( L = 100 mH \)
I. RC circuit with $R = 1 k \Omega $ and $C = 10 m F$
J. RC circuit with $R = 1 k \Omega$ and $C = 1 \mu F$
Answer: I
A cart is moving horizontally along a straight line with a constant speed of $30$m/s.A projectile is to be fired from the moving cart in such away that it will return to the cart (at the same point on cart) must be projectile be fired?$(Take = 10m/{s^2})$

Options:
A. $$\dfrac{{80}}{3}m/s$$
B. \dfrac{{70}}{3}m/s
C. \dfrac{{60}}{3}m/s
D. \dfrac{{30}}{3}m/s
E. $$10m/s$$
F. $$\dfrac{{20}}{3}m/s$$
G. \dfrac{{50}}{3}m/s
H. \dfrac{{35}}{3}m/s
I. \dfrac{{55}}{3}m/s
J. $$\dfrac{{40}}{3}m/s$$
Answer: J
In an interference arrangement similar to Young's double-slit experiment, the slits $S_{1}$ and $S_{2}$ are illuminated with coherent microwave sources, each of frequency $10^{6}Hz$. The source are synchronized to have zero phase difference. The slits are separated by a distance $d = 150.0\ m$. The intensity $I(\theta)$ is measured as a function of $\theta$, where $\theta$ is defined as shown. If $I_{0}$ is the maximum intensity, then $I(\theta)$ for $0\leq \theta \ 90^{\circ}$ is given by

Options:
A. $I(\theta)$ is constant for all values of $\theta$
B. $I(\theta) = I_{0}$ for $\theta = 30^{\circ}$
C. $I(\theta) = I_{0}$ for $\theta = 60^{\circ}$
D. $I(\theta) = I_{0}$ for $\theta = 0^{\circ}$
E. $I(\theta) = \dfrac{I_{0}}{2}$ for $\theta = 30^{\circ}$
F. I(θ) = I_{0} for θ = 15^{\circ}
G. $I(\theta) = \dfrac{I_{0}}{4}$ for $\theta = 90^{\circ}$
H. $I(\theta) = \dfrac{I_{0}}{2}$ for $\theta = 45^{\circ}$
I. $I(\theta) = I_{0}$ for $\theta = 45^{\circ}$
J. $I(\theta) = \dfrac{I_{0}}{3}$ for $\theta = 0^{\circ}$
Answer: D
Angular part of wave function for an orbital is =$\left(\frac{15}{4 \pi}\right)^{1/2} sin\theta cos\theta sin\Phi, \, where \, \theta\, $ = angle from z-axis, Angular node(s) are _

Options:
A. XY and XZ plane only
B. XY and YZ plane only
C. XY plane only
D. YZ and ZX plane only
E. XY and ZX plane only
F. XY, YZ, and ZX plane
G. YZ plane only
H. YZ and XZ plane only
Answer: A
The displacement of a particle executing periodic motion is given by $y = 4{\cos ^2}\frac{t}{2}\left( {1000t} \right)$.
The number of independent simple harmonic motions is

Options:
A. $$1$$
B. 6
C. $$2$$
D. $$5$$
E. 3
F. $$4$$
Answer: C
A particle of mass $m$ is released from a height $H$ on a smooth curved surface which ends into a vertical loop of radius $R$, as shown. Choose the correct alternative(s) if $H = \ 2R$.

Options:
A. The particle breaks off at a height $h$ from the base of the loop such that $ R < h < 3R $
B. The particle breaks off at a height $h$ from the base of the loop such that $ 0 < h < 3R $
C. The particle breaks off at a height $h$ from the base of the loop such that $ 0 < h < 2R $
D. The particle breaks off at a height $h$ from the base of the loop such that $ R < h < 2R $
E. The particle breaks of at a height $h = R$ from the base of the loop
F. The particle reaches the top of the loop with zero velocity.
G. The particle breaks off at a height $h$ from the base of the loop such that $ 0 < h < R $
H. The particle cannot reach the top of the loop.
I. The particle breaks off at a height $h$ from the base of the loop such that $ 2R < h < 3R $
J. The particle breaks off at a height $h$ from the base of the loop such that $ 1.5R < h < 2.5R $
Answer: D
The emission series of hydrogen atom is given by $\dfrac {1}{\lambda} = R\left (\dfrac {1}{n_{1}^{2}} - \dfrac {1}{n_{2}^{2}}\right )$ where $R$ is the Rydberg constant. For a transition from $n_{2}$ to $n_{1}$, the relative change $\triangle \lambda / \lambda$ in the emission wavelength if hydrogen is replaced by deuterium (assume that the mass of proton and neutron are the same and approximately $2000$ times larger than that of electrons) is

Options:
A. 0.010%
B. 0.0125%
C. $0.05$%
D. $0.025$%
E. 0.0175%
F. $0.005$%
G. 0.030%
H. 0.0075%
I. $0.0025$%
J. 0.015%
Answer: D
It has been estimated that is all the $O_3$ in the atmosphere were brought to sea level at $STP$, the gas would form a layer $0.3\space cm$ thick. Estimate the number of $O_3$ molecules in the earth's atmosphere.
(radius of the earth is 4000 mile, 1 km = 0.6214 mile)

Options:
A. 4.3 \times 10^{37} \text{ ozone molecules}
B. 4.1 \times 10^{37} \text{ ozone molecules}
C. 4.4 \times 10^{37} \text{ ozone molecules}
D. 4.0 \times 10^{37} \text{ ozone molecules}
E. $2.48\times10^{37}$ ozone molecules
F. $4.2\times10^{37}$ ozone molecules
G. $4\times10^{35}$ ozone molecules
H. 4.5 \times 10^{37} \text{ ozone molecules}
I. $4\times10^{37}$ ozone molecules
Answer: I
Let us identify the system as the spacecraft and the earth but not the portion of the fuel in the spacecraft that we use to change the orbit . In a given orbit, the mechanical energy of the spacecraft - earth system is given by $E = -\dfrac {GMm}{2r} .$
This energy includes the kinetic energy of the spacecraft and the potential energy associated with the gravitational force between the spacecraft and the earth. If the rocket engines are fired, the trust force moves the spacecraft through a displacement. As a result, the mechanical energy of the spacecraft-earth system increases.
The spacecraft has a new higher energy but is constrained to be in an orbit that includes the original starting point. It can not be in a higher energy circular orbit having a larger radius because this orbit would not contain the starting point. The only possibility is that the orbit is elliptical as shown in the Fig. $E = -\dfrac {GMm}{2a}.$
Above equation gives the energy of the spacecraft-earth system for an elliptical orbit where 'a' is semimajor axis. Thus if we know the new energy of the orbit, we can find the semi-major axis of the elliptical orbit. Conversely , if we know the semi-major axis of an elliptical orbit we would like to achieve , we can calculate how much additional energy is required from the rocket engines. A spacecraft is moving in a circular orbit  around the earth (radius 6400 km), at a heightb 300 km from the surface. To place the spacecraft in an elliptical orbit, the magnitude of the mechanical energy of the spacecraft-earth system is decreased by $10.0\%$.
Maximum height of the spacecraft above the surface of the earth will be :

Options:
A. 1374 km
B. 970 km
C. 300 km
D. 1404 km
E. 1774 km
F. 1474 km
G. 1574 km
H. 1874 km
I. 1274 km
J. 1674 km
Answer: F
A rectangular frame $ABCD$ made of a uniform metal wire has a straight connection between $E$ and $F$ made of the same wire as shown in figure. $AEFD$ is a square of side $1m$ and $EB=FC=0.5m$. The entire circuit is placed in a steadily increasing, uniform magnetic field directed into the plane of paper and normal to it. The rate of change of the magnetic field is $1$ $T{s}^{-1}$. The resistance per unit length of the wire is $1\Omega$ ${m}^{-1}$. Find the magnitude and direction of the current in the segment $AE$, $BE$ and $EF$.

Options:
A. $ \dfrac{1}{22}$A, E to A; $\dfrac{6}{22}$A, E to B; $ \dfrac{7}{22}$A, F to E
B. $\dfrac{7}{22}$A, A to E; $\dfrac{6}{22}$A, E to B; $\dfrac{1}{22}$A, E to F
C. $\dfrac{7}{22} $A, E to A; $\dfrac{6}{22}$A, B to E; $\dfrac{1}{22}$A, F to E
D. $\dfrac{7}{22} $A, E to A; $\dfrac{6}{22}$A, E to B; $\dfrac{1}{22}$A, E to F
E. $\dfrac{6}{22}$A, E to A; $\dfrac{7}{22}$A, B to E; $\dfrac{1}{22}$A, E to F
F. $\dfrac{7}{22}$A, A to E; $\dfrac{6}{22}$A, E to B; $\dfrac{1}{22}$A, F to E
G. zero, zero, zero
H. $\dfrac{6}{22}$A, A to E; $\dfrac{7}{22}$A, E to B; $\dfrac{1}{22}$A, E to F
I. $ \dfrac{6}{22}$A, E to A; $\dfrac{7}{22}$A, B to E; $\dfrac{1}{22}$A, F to E
Answer: C
A small pond of depth 0.5 m deep is exposed to a cold winter with outside temperature of 263 K. Thermal conductivity of ice is K = 2.2 W $m^{-1}$ $K^{-1}$, latent heat L = 3.4 $\times$ $10^5$ J$kg^{-1}$ and density  $\rho$ = 0.9 $\times$ $10 ^3$ kg$m^{-3}$. take the temperature of the pond to be 273 K. The time taken for the whole pond to freeze is about.

Options:
A. 32 days
B. 20 days
C. 27 days
D. 25 days
E. 22 days
F. 28 days
G. 30 days
H. 37 days
I. 35 days
J. 33 days
Answer: G
An electric component manufactured by 'RASU Electronics' is tested for its defectiveness by a sophisticated testing device. Let $A$ denote the event "the device is defective" and $B$ the event "the testing device reveals the component to be defective". Suppose $P(A)=lpha$ and $P(B|A)=P(B'|A')=1-lpha$, where $0 < lpha < 1$, then

Options:
A. $$P(B)=2lpha(1-lpha)$$
B. $$P(A|B')=1/2$$
C. $$P(B')=(1-lpha)^2$$
D. P(B')=2\alpha(1-\alpha)
E. P(B')=2\alpha(1-\alpha)^2
F. P(B')=2(1-lpha)^2
G. P(B')=1-2(1-\alpha)^2
H. $$P(A'|B')=[lpha /(1-lpha)]^2$$
I. P(B')=2(1-lpha)^2 - lpha
J. P(B')=1-2\alpha(1-\alpha)
Answer: C
If s= $ {e  }^{  t} $ (sint - cost) , then the acceleration is

Options:
A. $ {e }^{ t} $ (Cost - 2sint)
B. $ 2{e }^{ t} $ (sint + cost)
C. $ 2{e }^{ t} $ (cost + sint)
D. $ {e }^{ t} $ (cost - sint)
E. $ {e }^{ t} $ (2cost - sint)
F. $ 2{e }^{ t} $ (sint - cost)
G. $ 2{e }^{ t} $ (Cost - sint)
H. $ 2{e }^{ t} $ (sint - 2cost)
I. $ {e }^{ t} $ (cost + sint)
J. $ 2{e }^{ t} $ (Cost + sint)
Answer: G
The resistance value of an unknown resistor is calculated using the formula $R = d \frac{V}{I}$ where $V$ and $I$ be the readings of the voltmeter and the ammeter respectively. Consider the circuits below. The internal resistance of the voltmeter and the ammeter ($R_V$ and $R_G$ respectively) are finite and non zero. The relation between $R_B$ and the actual value of R is

Options:
A. R = R_B + \frac{R_V}{R_G}
B. dependent upon E and r
C. $$R < R_B$$
D. R = R_B + \frac{R_V}{2R_G}
E. R = R_B + \frac{R_G}{R_V}
F. $$R = R_B$$
G. R = R_B - \frac{R_G}{R_V}
H. R = R_B - \frac{R_V}{R_G}
I. R > R_B
Answer: C
A coil of radius R carries a current I. Another concentric coil of radius $\displaystyle r\left( r<<R \right) $ caries current $\displaystyle \frac { I }{ 2 } $. Initially planes of the two coils are mutually perpendicular and both the coils are free to rotate about common diameter. They are released from rest from this position. The masses of the coils are M and m respectively $\displaystyle \left( m<M \right) $. During the subsequent motion let $\displaystyle { K }_{ 1 }$ and $\displaystyle { K }_{ 2 }$ be the maximum kinetic energies of the two coils respectively and let U be the magnitude of maximum potential energy of magnetic interaction of the system of the coils. Choose the correct options.

Options:
A. \frac{ \mu_0 \pi I^2 r^2 }{ 4R^3 }
B. U = \frac{ \mu_0 \pi I^2 r^2 }{ 2R }
C. U = \frac{ \mu_0 \pi I^2 r^2 }{ 3R }
D. $$\displaystyle U=\frac { { \mu }_{ 0 }\pi { I }^{ 2 }{ r }^{ 2 } }{ 4R } $$
E. $$\displaystyle \frac { { K }_{ 1 } }{ { K }_{ 2 } } =\frac { M }{ m } { \left( \frac { R }{ r } \right) }^{ 2 }$$
F. \frac{ \mu_0 \pi I^2 r^2 }{ 4R^2 }
G. U = \frac{ \mu_0 \pi I^2 r^2 }{ 8R }
Answer: D
Two sources of equal emf are connected to a resistance R. The internal resistance of these sources are $r_1$ and $r_2 (r_1 > r_2)$. If the potential difference across the source having internal resistance $r_2$ is zero, then

Options:
A. R = r_1 + r_2
B. R = \left(\dfrac{r_1 r_2}{r_2 - r_1} \right)
C. R = r_1 - r_2
D. R = \left(\dfrac{r_1 r_2}{r_1 - r_2} \right)
E. R = \left(\dfrac{r_1 r_2}{r_2 + r_1} \right)
F. $$R = r_2 - r_1$$
G. R = \left(\dfrac{r_1 r_2}{r_1 + r_2} \right)
H. $$R = r_2 \left(\dfrac{r_1 + r_2}{r_2 - r_1} \right )$$
I. $$R = \dfrac{r_1 r_2}{r_2 - r_1}$$
Answer: F
Which of the following combinations of quantum numbers can refer to an electron in a ground-state cobalt atom (z=27)?

Options:
A. n=4,l=2,$m_{l}=-2$
B. n=3,l=0,m_{l}=1
C. n=3,l=0,$m_{l}=2$
D. n=3,l=0,m_{l}=-2
E. n=3,l=0,m_{l}=4
F. n=2,l=2,$m_{l}=0$
G. n=3,l=1,$m_{l}=0$
H. n=3,l=0,m_{l}=0
I. n=3,l=0,m_{l}=3
Answer: C
Two particle of mass m and 2m moving in opposite direction collide head on. They have same de-Broglie wavelength before collision after the collision.

Options:
A. If e = 1, de-broglie wavelength of 2m is greater than that of m
B. If e = 1, de-broglie wavelength of 2m is less than that of m
C. If e = 1, de-broglie wavelength of 2m is half that of m
D. If e = 1, de-broglie wavelength of m is equal to that of 2 m
E. If e = 1, de-broglie wavelength of m is less than that of 2 m
F. If e = 1, de-broglie wavelength of 2m is equal to that of m
G. If e = 1, de-broglie wavelength of m is greater than that of 2 m
H. The de-Broglie wavelength of 2m remains same if e = 1
I. If e = 1, de-Broglie wavelength of 2 m is greater than that of m
J. The de-Broglie wavelength of m increases if e = 1
Answer: G
A particle of mass $10\ gm$ is placed in a potential field given by $V = (50x^{2} + 100)J$. The frequency of oscillation in cycle/sec is

Options:
A. \dfrac{15}{\pi}
B. $$\dfrac {50}{\pi}$$
C. \dfrac{20}{\pi}
D. $$\dfrac {10}{\pi}$$
E. \dfrac {30}{\pi}
F. $$\dfrac {5}{\pi}$$
G. \dfrac {25}{\pi}
H. $$\dfrac {100}{\pi}$$
Answer: F
In the spectrum of singly ionized helium, the wavelength of a line observed is almost the same as the first line of Balmer series of hydrogen. It is due to transition of electron

Options:
A. From $n_1= 6 $ to $n_2 = 5$
B. From $n_1= 6 $ to $n_2 = 2$
C. From $n_1= 5 $ to $n_2 = 2$
D. From $n_1= 3$ to $n_2 = 2$
E. From $n_1= 6 $ to $n_2 = 4$
F. From $n_1= 4 $ to $n_2 = 3$
G. From $n_1= 5$ to $n_2 = 3$
H. From $n_1= 5$ to $n_2 = 4$
I. From $n_1= 4$ to $n_2 = 2$
J. From $n_1= 6 $ to $n_2 = 3$
Answer: E
A cooling object was emitting radiations of time varying wavelength $\lambda = 3000 + 40t$, where $\lambda$ is in $\mathring{A}$ and $t$ is in second is incident on a metal sheet (of work function $2eV$) such that the power incident on sheet is constant at $100$ watt. This signal is switched on and off for time intervals of $2$ minutes and $1$ minute respectively. Each time the signal is switched on, $\lambda$ again starts from fresh value of $3000 \mathring A$. If the metal plate is grounded so that it always remains neutral and electron clouding is negligible then find the maximum photo-current (mA). The photo-emission efficiency is $0.01\%$ and remains constant. (Take he = $12400 cV-\mathring A$)

Options:
A. 5.1A
B. $$4.5A$$
C. $$5A$$
D. $$6A$$
E. 5.5A
F. $$3A$$
G. 5.2A
H. 5.3A
I. 4A
Answer: C
A double conical pendulum consists of two masses $m$ and $M$, connected by a massless string passing over a frictionless, massless pulley. The entire apparatus rotates freely at constant angular speed $\omega (rad/s)$ about the vertical axis (dashed line) passing through centre of pulley as shown. After the system comes in steady state, the length of string on either sides of pulley are $l$  and $L$. Pick up the correct option(s).

Options:
A. \cos\phi=\dfrac{g}{2\omega^{2}(l+L)}
B. \(\cos\phi=\dfrac{g}{3\omega^{2}l}\)
C. \(\cos\phi=\dfrac{g}{\omega^{2}(l+L)/2}\)
D. $$\cos\phi=\dfrac{g}{\omega^{2}l}$$
E. \(\cos\phi=\dfrac{g}{2\omega^{2}l}\)
F. \(\cos\phi=\dfrac{g}{\omega^{2}(l+L)}\)
G. \(\cos\phi=\dfrac{g}{2\omega^{2}L}\)
Answer: D
Each edge of a cubic unit cell is 400 pm long. If atomic mass of the element is 120 and its density is $ 6.25 \mathrm{g} / \mathrm{cm}^{3} $ the crystal lattice is (use $ N_{A}=6 \times 10^{23} $ )

Options:
A. face centered (with a lattice parameter of 400 pm)
B. end centered
C. primitive (with a lattice parameter of 300 pm)
D. face centered
E. body centered (with a lattice parameter of 400 pm)
F. body centered
G. primitive
H. face centered (with a lattice parameter of 300 pm)
I. primitive (with a lattice parameter of 400 pm)
J. body centered (with a lattice parameter of 300 pm)
Answer: G
Heat treatment of muscular pain involves radiation of wavelength of about 900 nm. Which spectral line of H-atom is suitable for this purpose ?
$$\left[ { R }_{ H }=1\times { 10 }^{ 5 }{ cm }^{ -1 },h=6.6\times { 10 }^{ -34 }Js,c=3\times { 10 }^{ 8 }{ ms }^{ -1 } \right] $$

Options:
A. Paschen, $7 \rightarrow 3$
B. Paschen, $4 \rightarrow 3$
C. Paschen, $5 \rightarrow 2 $
D. Lyman, $\infty \rightarrow 1$
E. Paschen, $\infty \rightarrow 3$
F. Paschen, $5 \rightarrow 3$
G. Paschen, $6 \rightarrow 3 $
H. Paschen, $6 \rightarrow 2 $
I. Balmer, $\infty \rightarrow 2$
Answer: F
Two soap bubbles combine under isothermal conditions to form a single soap bubble. If in the process, the change in volume is $V$ and change in area is $S$, then $(P$ at atmospheric pressure).

Options:
A. 3PV + 2TS = 0
B. 3PV + 7TS = 0
C. $$3PV + 4TS = 0$$
D. $$3PV + TS = 0$$
E. 3PV + 6TS = 0
F. $$4PV + 3TS = 0$$
G. 3PV + 5TS = 0
H. 2PV + 4TS = 0
I. $$PV + TS = 0$$
J. 2PV + 3TS = 0
Answer: C