GENERAL SCIENCES
MODEL QUESTION PAPER
PART A
ANSWER ANY 15 QUESTONS
1. Profit of a firm grows at a rate of 15% per year for the first three consecutive years. For the next three years, the profit level remains stagnant. From the 6th year till the 9th year, it again grows at a rate of 15% per year. Which of the following graphs depicts these facts?
2. A pond is deepest at its centre and becomes shallow uniformly towards the edge. If the depth of water at the centre in May is half its value in August, the water contained in the pond
(1) in May is greater than half that in August
(2) in August is equal to twice that in May
(3) in May is less than half that in August
(4) in August is less than twice that in May
3. The series representing the sum of the areas of the shaded equilateral triangles in the figure below is
1. 3+2+1+1
2. 23411113333
3. 23411114444
4. 1111481632
4. Which of the following vitamins will not be synthesized in a person confined to a dark cell for a long time ?
1. A
2. B
3. C
4. D
5. Flowering is some plants is strongly influenced by the photo period. A farmer was growing two species of plants, A and B near a sea coast where a light house was located. He observed that species A flowered profusely while species B did not. Which of the following is correct ?
1. Species A requires long duration of day while species B needs a shorter day
2. Species B requires longer duration of day while species A needs a shorter duration
3. Both species require short duration of day
4. Both species require long duration of day
6. Pneumatophores are modified roots in some plants like Rhizophora growing in swampy areas that come out of the ground and grow vertically upwards. The main function of such roots is to
(1) help obtain oxygen for respiration
(2) provide support
(3) adsorb and conduct water and minerals
(4) store food
7. A cube of side 1 cm is painted by putting a lacquer of thickness δ, negligible compared to the side of the cube. The volume of the painted cube is approximately
(1) 1 + δ cm3
(2) 1 + δ3 cm3
(3) 1 + 3δ3 cm3
(4) 1 + 3δ cm3
8. A candle is burning inside a sealed glass jar. The pressure and temperature of the air within the jar are plotted as a function of time. Which of the following graphs represents this process correctly?
9. The result of taking 1’s complement of the sum of the binary numbers 110 and 101 will be
(1) 1011
(2) 0011
(3) 0100
(4) 0110
10. Which of the following straight lines passes through the point (1,1)?
(1) y = 2x + 3
(2) 2y = x−6
(3) x = 1
(4) x = y + 1
11. Which of the following 1 molar (aqueous) solution has the highest number density of ions?
(1) Glucose
(2) CaCl2
(3) NaNO3
(4) KCl
12. How many two-digit even numbers can be composed from nine digits 1, 2, 3 … 9?
(1) 50
(2) 81
(3) 45
(4) 36
13. Complete combustion of cyclohexane (C6H12) is represented by the equation
C6H12 + x O2 → y CO2 + z H2O
The values of x, y and z, respectively, are
(1) 9, 6, 6
(2) 10, 6, 4
(3) 6, 12, 10
(4) 4, 8, 12
14. How many distinct trichlorobenzenes (C6H3Cl3) should exist, given that benzene (C6H6) has a regular hexagonal geometry?
(1) 6
(2) 1
(3) 2
(4) 3
15. Mercury is closer to the Sun than Venus. Yet Venus is hotter because it has
(1) a dominant CO2 atmosphere
(2) a dominant methane atmosphere
(3) sulphuric acid clouds
(4) an atmosphere devoid of oxygen
16. In a simple pendulum experiment, a student records the following readings. If the true period of the pendulum is 10 s, then the percent error is the largest for the observation with serial number
Serial Number
Number of Oscillations
Time
1
2
3
4
10
20
50
100
100.2
200.3
500.5
1000.8
(1) 1
(2) 2
(3) 3
(4) 4
17. A container holding normal air (1 bar pressure, room temperature) is being evacuated. The normal composition of air is approximately 78% N2, 21% O2, 0.9% Ar and traces of CO2 (0.04%) and water vapour (0.02%). After the pressure in the container falls to about 10–3 mbar, the relative fractions of the components will be
1. N2 and O2 approximately equal and greater than H2O
2. N2, O2, Ar approximately equal and greater than H2O
3. N2, O2, Ar in the original proportion, but N2 less than H2O
4. N2, O2, Ar in the original proportion, and N2 greater than H2O
18. An endoscope is a device for observing internal organs, using a combination of a lamp and an optical fibre. The image seen is due to
(1) light reflected by the organ and transmitted by internal reflection through the fibre
(2) light refracted by the organ and transmitted by refraction through the fibre.
(3) light refracted by the organ and transmitted by internal reflection through the fibre.
(4) light emitted by the organ and transmitted by refraction through the fibre.
19. How many times in a day is the angle between the minute and hour hands of a clock equal to an angle θ, where 00 < θ < 1800
(1) 24
(2) 12
(3) 36
(4) 48
20. A typical enzyme catalyzed reaction is shown below
What do you think the component x might be?
1. Substrate concentration or temperature
2. Substrate concentration or enzyme concentration
3. Substrate concentration or pH
4. pH or temperature
PART B
21. The value of the integral
2
0
x xe dx
is equal to
1.
2
2.
3. 0
4.
1
2
22. If 1
Tij2 xi y j xj yi and ij i j j i S x y x y
2
1
are components of tensors T and
S(i,j = 1, 2, 3) respectively then
,
ij ij
i j
T S is
1. 0
2. 2 2
i i
i
x y
3. 2 2
i i
i
x y
4. 4 4
i i
i
x y
23. An unbiased coin is tossed n times. The probability that exactly m heads will
come up is
1.
m
n
2
2.
!( )!
!
2
1
m n m
n
n
3.
!( )!
!
2
1
m n m
n
m
4. n
m
2
24. Given the Legendre polynomials ( ) 1, ( ) , 0 1 P x P x x and
2
3 1
( )
2
2
x
P x , then
the polynomial (3x2 + x – 1) can be expressed as
1. P2(x) – P1(x)
2. 2P2(x) + P1(x)
3. P2(x) + P1(x)
4. 2P2(x) + P1(x)+ P0(x)
25. A circular ring rotates about an axis passing through its centre and perpendicular to its
plane. Each point on it moves with a speed c 2 , where c is the speed of light in vacuum.
The relative velocity between any two diametrically opposite points on the ring is
1. 2c 5
2. c 2
3. 4c 5
4. c
26. For what value of will transformation q Q = q cos2p and pP=q sin2p be
canonical?
1. = 1
2. = ½
3. = ‒1
4. = 2
27. Consider a spherical ball of radius R and constant density . Which of the following
graphs shows the gravitational potential V(r) as a function of the radial coordinate r ?
.
.
.
28. A quantum particle of mass m moves in two dimensions in an anisotropic harmonic
oscillator potential 1 2 2 2 2
2 V(x, y) m x 2m y . The energy eigenvalues are (n is a
positive integer or zero)
1. 2n1
2. n1
3. 2 n1
4. 3
2 n
29. The ground state energy of the Hydrogen atom is 13.6 eV. The energy of the second
excited state is
1. 27.2 eV
2. 6.8 eV
3. 1.5eV
4. 4.5 eV
30. Consider a one-dimensional infinite square well potential
otherwise
x L
V x
0
( )
If each of the lowest two energy levels are occupied by identical non-interacting
bosonic particles (one in each level), then the unnormalized wave function of the
combined system is
1.
L
2 x
sin
L
x
x ,x sin 1 2
1 2
2. 1 2 2 1
1 2
2 2
( , ) sin sin sin sin
x x x x
x x
L L L L
3. 1 2 1 2
1 2
2 2 2
( , ) sin sin sin sin
x x x x
x x
L L L L
4. 1 2 1 2
1 2
2 2
( , ) sin sin sin sin
x x x x
x x
L L L L
31. The total spin-angular momentum of a system of three free electrons is
1. Stot =
1
2
and Stot =
3
2
2. Stot =
1
2
only
3. Stot =
3
2
only
4. Stot = 0 and Stot = 1
32. A conducting spherical shell of inner radius a and outer radius b has a point
charge q located at the centre of the shell. The potential at a distance r from the
centre (a < r < b) is
1. 0
2.
0
1
4
q
a
3.
0
1
4
q
b
4.
0
1
4
q
r
33. Consider the electric and magnetic fields of an accelerating charge. How should the
fields vary with r (the retarded distance) for the Poynting vector to remain finite at
arbitrarily large distances?
1. 1 r
2. 2 1 r
3. r
4. 2 r
34. According to Kirchoff‟s laws for circuits, the sum of the currents at any junction is equal
to zero. Which of the following equations for the current density j
describes this
situation?
1. j 0
2. .j 0
3. 2 j 0
4. j/t0
35. Two equal and opposite charges are placed at a distance r from each other. A
large metallic sheet is placed at a distance d from them as shown in the figure.
Due to the presence of the sheet, the attractive force between the charges along
the direction joining them
1. decreases
2. increases
3. remains unchanged
4. decreases if r > d/2 and increases if r < d/2.
36. A non-interacting system has two energy levels, 0 and . The lower level is
doubly degenerate while that of energy is non-degenerate. If the system is in
thermal equilibrium at temperature T, the single-particle partition function is
1. Z e kBT / 2
2. Z e kBT / 1 2
3. Z e kBT / 2
4. Z e kBT / 2 1
37. A system of weakly interacting two-dimensional harmonic oscillators is in
thermal contact with a heat bath of absolute temperature T. The average kinetic
energy of an oscillator is
1. kBT
2. 2kBT
3. 3 kBT/2
4. kBT/2
38. Which of the following expresses the Second law of thermodynamics? (Symbols have
their usual meanings.)
1. FW
2. S Q T
3. U QW
4. S 0
39. For the op-amp circuit shown in the figure below, which is the correct output
waveform?
1.
2.
3.
4.
40. Which of the following circuits has an appropriate voltage divider for biasing the
transistor?
1. 2.
3. 4.
PART C
COMPULSORY QUESTIONS
41. The value of the contour integral
1
2i
dz f (z)
C
where 2
1 2
( )
2 1
z z
f z
z z
and the contour C is a circle of radius 2 centered at the origin, traversed in the
counter-clockwise direction is
1. 2
2. 1/2
3. 1
4. 3
42. The matrix A, defined by
1 0 0
0
0
A a b
b a
is orthogonal if
1. a = 1, b = –1
2. a = 1/ 2 , b = 1/ 2
3. a = 1/ 2 , b = i/ 2
4. a = 1, b = 1
43. Given the Lagrangian L =
2
1
dt
dx
where x is a co-ordinate and t is the time, the Hamiltonian is
1.
dx
dt
2
/ 1
dx
dt
2
2.
2
1
dt
dx
3.
2
1
1
dt
dx
k
4.
2
dt
dx
44. In a nonmagnetic dielectric medium with dielectric constant 4 r ,the electric
field of a propagating plane wave with = 10 8 rad/s is given by
E i j j t k r
ˆ 3ˆ exp .
The propagation vector k
(in units of m-1) is given by
1. k i ˆj
3
ˆ 1
3
1
2. k zˆ
3
1
3. k i ˆj
6
ˆ 1
2 3
1
4. k i ˆj
3
ˆ 1
3
1
45. Two small circular loops, each of area 1cm2 carry currents 1A and 2A
respectively. They are placed in a plane at a distance of 5 meters from each other.
If one of the loops is lifted 5 meters in the vertical direction while maintaining it
flat, the electrical work done in the process is ( 0 is the magnetic permeability of
vacuum)
1. 2.75 10 9 2
0 J
2. 1.46 10 10 2
0 J
3. 1.88 10 10 2
0 J
4. 4.74 10 9 2
0 J
46. Consider a three dimensional harmonic oscillator with Hamiltonian
H= 2 2 2 2
2 2 2
2
1
2 2 2
m x y z
m
p
m
p
m
p x y z .
A. The number of distinct eigenstates with energy eigenvalue
2
5
is
1. 5
2. 3
3. 2
4. 1
B. The eigenvalue of L 2 (where L
is the angular momentum operator) in the
ground state is
1. 2
2. 2 2
3. 0
4. 2 6
47. The states |m>, m = -1, 0, 1 are the eigenstates of S z , the z component of the spin
angular momentum of a particle with S =1. Sz m m m.
A. The expectation value of S z in the state 1
2
1
0
2
1
1
2
1
is
1. / 4
2. - / 2
3. / 2
4. - / 4
B. Consider the states 1
3
1
0
2
1
1
6
1
1 and
1
3
1
1 0
6
1
2 a . The value of a for which these states are
orthogonal is
1. 1/ 3
2. 1/2
3. 1/ 2
4. 1/ 2
48. The free energy of a system having magnetization m is given by
F = - 2
2
m
J
k T B
2
1
ln
2
1
2
1
ln
2
1 m m m m
,
where J is the exchange energy, T is the Temperature, and B k is the Boltzmann
constant.
A. The equilibrium value of magnetization is determined by
1. m = sinh
k T
Jm
B
2. m = cosh
k T
Jm
B
3. m = tanh
k T
Jm
B
4. m =
2
3
1
1
k T
Jm
k T
Jm
B B
B. If m is determined through the relationship m =
2
3
1
1
k T
Jm
k T
Jm
B B
,
then the system undergoes
1. no phase-transition
2. a second order phase transition at T C B J / k
3. a first order phase transition at T C B J / k
4. a phase transition that cannot be classified as either first or second
order.
49. A piece of metal of heat capacity 500 JK–1assumed to be independent of
temperature is at 500 K. The metal piece is cooled to 300K in two steps: it is first
plunged into a liquid bath at 400 K. After cooling it is plunged into a colder
liquid bath at 300 K. The metal piece is then heated to 500 K in two steps: it is
plunged into a liquid bath at 400 K first and then into a liquid bath at 500 K.
During the cooling-heating process, the metal piece and the liquid baths gain or
lose entropy. The total change in entropy of the system (the metal piece and the
liquid baths) is
1. 1500 ln (5/3)JK-1
2. Zero
3. –200 JK–1
4. +200JK–1
50. Shown in the figure is a circuit to measure light intensity and convert it to a digital
signal. The photodiode P has a responsivity of 0.1A per watt of incident light
intensity. The op-amp converts the induced photocurrent to a voltage which is
digitized by the 10-bit A/D converter with a reference voltage of 4V.
A. For a light intensity of 25 μW incident on the photodiode, the voltage
output of the op-amp is
1. 0.25V
2. 1.0V
3. 4.0V
4. 2.5V
B. The range of light intensity which can be measured by this set up is
1. 100nW to 100μW
2. 100nW to 100μW
3. 40nW to 40μW
4. 40nW to 40μW
ANSWER ANY 10 QUESTIONS OUT OF THE REMAINING 15
51. Under a small rotation about the x – axis, the component of a second
rank tensor Γyz transforms as
1. Γyz = (1 + ) Γyz
2. Γyz = Γyz + (Γyy – Tzz)
3. Γyz = Γyz + (Γyx – Γzx)
4. Γyz = Γyz + Γxx
52. The partial differential equation
0
u u
x y
x y
.
Upon change of variables to z = x + iy z* =x–iy becomes
1.
u
z
u
z* 0
2.
z
u
z* z* u
z
0
3.
z
u
z
z* u
z* 0
4.
z
u
z
z* u
z* 0
53. A large slab consists of two materials with dielectric constants 1 and 3 divided by
a plane interface as shown in the figure. An electric field exists in this slab, which
uniformly makes an angle of 30 with the normal to the interface on the left side
of the (=1). The angle the field makes on the other side of the interface is
1. tan 1 (3)
2. sin
2 3
1 1
3. 60
4. tan 1 (1/3)
54. Given the vector A y,x,0
, the line integral .
c
Adl
, where C is a circle of
radius 5 units with its centre at the origin, (correct to the first decimal place) is
1. 172.8
2. 157.1
3. -146.3
4. 62.8
55. An electron is placed in an uniform magnetic field H
that points in the
xˆ direction. The Hamiltonian of the system is H= KS H
where k >0 is a
constant and S
is the electron spin operator.
A. The energy of the ground state of the Hamiltonian is
1. KS
2. KH
2
1
3. KH
2
1
4. 0
B. The expectation value of z S in the ground state of this Hamiltonian is
1. / 2
2. 0
3. / 2
4.
56. Positronium is an atom formed by an electron and a positron. The mass of a
positron is the same as that of an electron and its charge is equal in magnitude but
opposite in sign to that of an electron. The positronium atom is thus similar to the
hydrogen atom with the positron replacing the proton.
A. The binding energy of a positronium atom is
1. 13.6 eV
2. 6.8 eV
3. 27.2 eV
4. 3.4 eV
B. If a positronium atom makes a transition from the state with n 3 to a
state with n 2, the energy of the photon that is emitted in this transition
is closest to
1. 1.88 eV
2. 0.94 eV
3. 1.13 eV
4. 2.27 eV
57. A particle of mass m and charge q is constrained to move along a straight line
joining two other equal charges q fixed at x = a. The time period of small
oscillations is
1. am
q
a
T 0
2
2. r am
q
a
T 0
4
3. r am
q
a
T 0
4. am
q
a
T 0
2
58. A Carnot engine operates between a heat source at 500 K and a heat sink at 300
K. The temperature of the source is increased by 20K. In order that the
efficiency of the engine remain unchanged, the temperature of the sink should be
changed by
1. +20K
2. +12K
3. –12K
4. –20K
59. The following table shows the relationship between an independent quantity x, a
two measured quantities y and z.
A. In order to get a straight line graph relating y to x we should plot
1. y vs log x
2. y2 vs x
3. y vs x2
4. y2 vs log x
B Similarly, in order to get a straight line graph relating z to x, we should
plot
1. z2 vs x3
2. (z + 4)2 vs x3
3. (z – 4)2 vs x3
4. log z vs log x
60. The cross-section for the excitation of a certain atomic level A under electron
impact is 20 2 1.4 10 cm A
. The level A has lifetime s 8 2 10 and decays
10% of the time to level B and 90% of the time to level C, as shown in figure.
x 0 1 2 3 4 5
y –0.1 2.1 8.1 17.9 32.2 49.7
z –4 –3 2 2 4
3 3 4 4 5 5 4
When an electron beam of 5mA/cm 2 is passed through a vapour of these atoms at
a pressure of 66.65 dynes/cm 2 at a temperature of 27 C, the equilibrium
population per cm3 in level A is
1. 1.4 10 4 cm 3
2. 1.55 10 5 cm 3
3. 4.66 10 5cm 3
4. 3.1 1016cm 3
61. Consider the CO molecule as a diatomic rigid rotor with a bond length of 1.12Å.
The reduced mass of the system is obtained from the atomic masses of C and O.
The rotational energies are defined in terms of B (the rotational constant) and J
(the rotational quantum number). If 1 and 2 denote the frequency of the first
rotational resonance lines for the molecules 12 C16 O and 13 C18 O respectively,
their ratio 1 / 2 is approximately
1. 1.5
2. 1.1
3. 0.9
4. 1.01
62. A metal at temperature T is placed in a static uniform electric field E
. An electron
in it experiences a collision, and then, after a time t, a second collision. In the
Drude model, energy mean speed of an electron emerging from a collision does not
depend on the energy that the electron acquired from the field since the time of the
last collision. The average energy lost by the electron in the second of the two
collisions mentioned previously (note that the average is taken over all directions in
which the electron may emerge after the first collision) is equal to
1.
2
2
eEt
m
2.
2
eEt
m
3.
2
1
4 2
eEt
m
4.
2
1
4
eEt
m
63. The Fermi energy of a two-dimensional electron gas with number density n is
equal to
1.
m
n
2
2.
m
n
2
2
2
2 2/ 3
3.
m
n
2
3
2
2 2/ 3
4.
m
n
2
2
64. The binding energy of a nucleus with atomic number Z and atomic mass number
A is given by the semi-empirical mass formula
B ( Z,A) = aA - bA s 2/3
A
A Z 2 2
- d 1/ 3 1/ 2
2
A A
Z
where a = 15.8 MeV, b = 18.3 MeV, s = 23.2 MeV, d = 0.7 MeV and
= (+11.2, 0, -11.2) MeV for nuclei which are (odd-odd, odd-even, even-even).
If we consider all possible isobars of mass number 216, the most stable nuclide is
1. Pb 216
82
2. Fr 216
87
3. At 216
85
4. Rn 216
86
65. The activity of a radioactive sample is defined as N where is the decay
constant and N is the number of atoms. In an experiment, the activity of a sample
of Cr 55
24 was found to change as
After(min) 0 5 10 15 20
N(m curie) 19.2 7.13 2.65 0.99 0.37
The half life of Cr 55
24 is
1. 5.08 min
2. 3.52 min
3. 3.57 min
4. 5.16 min
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