“He often _____ the numbers. False claims are not going to help. Honesty _____ trust”, said the manager. Choose the option with the correct order of words to fill the blanks.
a.
exaggerates; engenders
Correct Answer
b.
excels; encourages
c.
aggravates; alleviates
d.
diminishes; eliminates
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Q2mcq
In the sequence of tiles shown below, the missing tile indicated by the question mark should be
a.
b.
Correct Answer
c.
d.
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Q3mcq
A school has 100 students distributed among 1 st to 10 th standards. Based on this, which one of the following statements is always correct?
a.
There are at least 10 students who belong to the same standard.
Correct Answer
b.
There is at least one student in each standard.
c.
There are at most 10 students in 10 th standard.
d.
The total number of students from 1 st to 5 th standards is at least 50.
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Q4mcq
How many 3-digit numbers can be formed using three distinct single digit prime numbers?
a.
64
b.
24
Correct Answer
c.
12
d.
4
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Q5mcq
In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____
a.
18
b.
20
c.
24
Correct Answer
d.
32
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Q6mcq
Charity: P:: Retaliation: Q Choose the appropriate pair of words P and Q that fit the analogy.
a.
P = Parsimonious; Q = Vengeful
b.
P = Altruistic; Q = Amicable
c.
P = Resentful; Q = Spiteful
d.
P = Magnanimous; Q = Vindictive
Correct Answer
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Q7mcq
A paper shown in Panel I is folded along the dashed lines (- - -) to construct a cube. The shaded regions shown in Panel I appear on the outer surface of the cube. Referring to cubes shown in Panel II, which one of the options is correct?
a.
Only (i) can correspond to the unfolded cube in Panel I.
Correct Answer
b.
Only (ii) can correspond to the unfolded cube in Panel I.
c.
Both (i) and (ii) can correspond to the unfolded cube in Panel I.
d.
Neither (i) nor (ii) can correspond to the unfolded cube in Panel I.
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Q8mcq
Consider the cube shown below with its 8 corners labelled a, b, c, d, e, f, g, and h. The figure is representative. All corners are to be colored such that any two corners that are connected by an edge must be of different colors. The minimum number of colors required to achieve this is ________
a.
8
b.
4
c.
3
d.
2
Correct Answer
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Q9mcq
Four hills H1, H2, H3, and H4 are present in an area. The following observations are made about them: i. Neither H2 nor H3 is the easternmost hill. ii. Neither H2 nor H3 is the westernmost hill. iii. Neither the easternmost hill nor the westernmost hill is the southernmost iv. Two hills are located to the west of H2. v. The southernmost hill has at least two hills to its east.
hill. The southernmost hill is ________.
a.
H1
b.
H2
c.
H3
Correct Answer
d.
H4
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Q10mcq
As shown in the figure, circle $C$ 1 with center $O$ 1 and radius $r$ 1 touches the square $VWXY$ at points $P$ and $Q$ while circle $C$ 2 with center $O$ 2 and radius $r$ 2 touches the square $VWXY$ at points $R$ and $S.$ The two circles touch each other at $T.$
̅̅̅̅̅ = 4 cm, $r$ = _____ cm. Given $r$ = 1 cm and ̅̅̅̅ $VY$ = $VW$
2 1
a.
4 - $3\sqrt{2}$
b.
1 + $2\sqrt{2}$
c.
7 - $4\sqrt{2}$
Correct Answer
d.
5 + $3\sqrt{2}$
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In free space, an electromagnetic wave is travelling whose wavevector is $k⃗$ = $10(x̂$ + $\sqrt{3y}̂)$ m -1. The electric field component of this electromagnetic wave is given by $E⃗ (r$, $t)$ = $ẑ$ 600 $cos(k⃗$ ⋅ $r$ - $ωt)$ V.m -1. The speed of light in free space is
⃗ $(r$, $t)$ is $c$ = 3.0 $\times$ 10 8 m.s -1. The corresponding magnetic field $B$
An infinitely large non-conducting thin sheet in the $xy$ plane $(z=0)$ carries a uniform surface charge density $σ$ = 17.70 $\times$ 10 -12 C.m -2. The electric field in the region $z$ < 0 is $E⃗$ 2 = $x̂$ + $2ŷ$ + $3ẑ$. Then, the electric field $E⃗$ 1 in the region $z$ > 0 will be $(ε$ 0 = 8.85 $\times$ 10 -12 C 2. N -1. m -2)
a.
$E⃗$ = $x̂$ + $2ŷ$ + $5ẑ$ 1
b.
$E⃗$ = $x̂$ + $2ŷ$ + $4ẑ$ 1
c.
$E⃗$ = $x̂$ + $2ŷ$ + $3ẑ$ 1
d.
$E⃗$ = $x̂$ + $4ŷ$ + $ẑ$ 1
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On a horizontal plane, a projectile of mass $m$ is launched from the ground with speed $v$ 0 at an angle $θ$ 0 with the horizontal. In addition to the gravitational force $(mg),$ it also experiences a drag force $F drag$ = $-γv$, where $v$ is its velocity and $γ$ is a constant. It hits the ground at a distance $R$ from the point of launch with its velocity making an angle $θ$ with the horizontal, as shown schematically in the figure. Then which of the following options is correct? $R= θ$ < $θ$ 0 $v$ 02 $sin2θ$
$g$
a.
b.
$R<$ 0 $θ$ < $θ$ 0
$g$
c.
$R<$ 0 $θ$ > $θ$ 0
$g$
d.
$R= θ$ > $θ$ 0
$g$
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Consider the Otto cycle for an ideal gas engine consisting of two quasistatic adiabatic and two quasistatic isochoric processes. The correct temperature-entropy (T-S) phase diagram for the cycle is
a.
b.
c.
d.
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The formula for energy $E$ of a photon gas at temperature $T$ in a two-dimensional box at equilibrium with $g 2d (ν)$ denoting the density of states of photons is given below where symbols $ν,$ ℎ and $k B$ have their standard meaning. The specific heat $(C V$) of this photon gas obeys
$\infty hν$
$E$ = $\int dν g 2d (ν)$
0 exp ()-1 $k T$
$hν$
$B$
a.
$C V \propto T$
b.
$C V \propto T$ 2
c.
$C V \propto T$ 3
d.
$C V \propto T$ 4
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For the electric field of an electromagnetic wave given below, which of the following statements is correct?
$π$
$E⃗$ = $x̂ E cos(ωt)$ + $ŷ 2E cos (ωt$ +)
0 0 2
a.
The electric field is linearly polarised with slope 2.
b.
The electric field is circularly polarised with radius $E$ 0.
c.
The electric field is elliptically polarised with a ratio of major to minor axis being 2.
d.
The electric field is unpolarised with the two components being phase shifted by $π/2.$
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A gas of non-interacting 4 He atoms (of mass $m)$ is in a three-dimensional trap whose energy levels can be approximated by those of a harmonic oscillator potential
1 $V(x, y, z)$ = $mω$ 2 $(x$ 2 + $y$ 2 + $z$ 2). The chemical potential of the gas at $T=0$ K is
2
a.
0
b.
2
c.
3 $ħω$ 2
d.
$3ħω$
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⃗ is shown below. Here, length and arrow head Sketch of a two-dimensional vector field $V$ of the arrows denote magnitude and direction of the vector field, respectively. Which of ⃗ ? the following statements is correct for ⃗ ∇ $\times V$ O
a.
It is zero everywhere in the two-dimensional space.
b.
Its magnitude is non-zero and its direction is out of the two-dimensional plane.
c.
Its magnitude is non-zero and its direction is into the two-dimensional plane.
d.
It points in opposite directions above and below the x-axis.
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Schematic variation of the specific heat $C p$ of an ideal gas of diatomic molecules with temperature $T$ is shown in the figure below. For rotational energy $E R$ and vibrational energy $E v$ of the molecule, which of the following options is/are correct? Here $k B$ is the Boltzmann constant.
a.
$E R$ ≅ $k B T$ 1
b.
$E R$ ≅ $k B T$ 2
c.
$E v$ ≅ $k B T$ 1
d.
$E v$ ≅ $k B T$ 2
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If the perturbation $V$ = $λx$ 3 is added to the Hamiltonian of a one-dimensional harmonic oscillator, the matrix element $⟨m|V|0⟩$ is/are non-zero for which of the following states? Here, the eigenstates of the harmonic oscillator are denoted by $|n⟩.$
a.
$|m$ = 3⟩
b.
$|m$ = 1⟩
c.
$|m$ = 2⟩
d.
$|m$ = 5⟩
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A projectile of mass $m$ is launched from the ground with the initial speed $v$ 0 at an angle 30° from the horizontal. Take the ground to be horizontal. Ignoring the drag, the magnitude of Hamilton’s action $\int Ldt$ for the particle from the beginning till it hits the
$mv$ 3 ground is $f \times$ (0). The value of $f$ (rounded off to two decimal places) is _____
$g$
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Consider an electron in the energy eigenstate $ψ$ 211 $(r$) of the hydrogen atom. Given that the radial probability distribution of the electron in such a state takes its maximum value at $r$ = $n$ 0 $a,$ where $a$ is the Bohr radius, and $n$ 0 is an integer. The value of $n$ 0 (in integer) is _____ The radial part of the wavefunction $ψ$ 211 $(r$) is given by $R$ 21 $(r)$ =
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1 $re -r/2a$. $\sqrt{24a}$ 5 A dielectric sphere carries a uniform polarization $P$ = 26 μC. cm -2. The magnitude of the electric field at the center of the sphere is $E \times$ 10 9 N. C -1. The value of $E$ (rounded off to one decimal place) is _____ $(ε$ 0 = 8.85 $\times$ 10 -12 C 2. N -1. m -2)
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Consider a metal-superconductor junction connected to a dc voltage $V.$ At $T$ < $T c$, where $T$ is the superconductor’s transition temperature, the current $I$ versus $V$ behavior of this $c$ junction is shown schematically in the figure below. If the superconducting energy gap is $D$ meV. The value of $D$ (rounded off to one decimal place) is _____ For the energy dispersion of an electron in a one-dimensional solid $E(k)$ = $E$ 0 - $2γ cos(ka),$ the ratio of the effective mass of the electron in the solid to the free electron mass $(m e$) at $k$ = 0 is $R$ 0. Taking $γ$ = 0.5 eV and $a$ = 0.5 nm, the value of $R$ 0 (rounded off to two decimal place) is _____ (ℏ = 1.054 $\times$ 10 -34 J. s, $m e$ = 9.1 $\times$ 10 -31 kg, electron charge = 1.6 $\times$ 10 -19 C)
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The specific heat $C p (T)$ of one mole of a material as a function of temperature $T$ is given as $C p (T)$ = $AT$ + $BT$ 3, where $A$ = 0.695 mJ. mol -1. K -2 and $B$ = 0.045 mJ. mol -1. K -4. When $T$ is changed from 1 K to 10 K at constant pressure, then the change in entropy $ΔS$ in mJ. mol -1. K -1 (rounded off to one decimal place) is _____ Raman spectrum of a molecule was recorded using a source of wavelength 5000 Å. The first Stokes line is observed at 5100 Å. The first anti-Stokes line will appear at a wavelength $L$ (in Å). The value of $L$ (rounded off to nearest integer) is _____ A rocket of length 18.0 m is moving at speed 0.9c (where c is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is _____
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The Hamiltonian for a quantum particle of mass $m$ is given below, where $ω$ < Ω. The Schrö dinger equation for this system can be solved exactly using the orthogonal
$x$ - $x x$ + $x$ transformations: $x$ = 1 2 and $y$ = 1 2.
2 2 $√ √$
$H=$ - [ + ] + $mΩ$ 2 $(x$ 2 + $y$ 2) + $mω$ 2 $xy$
ℏ 2 $∂$ 2 $∂$ 2 1
2 2 $2m ∂x ∂y$ 2 The ground state energy of this system is
a.
$[\sqrt{Ω}$ 2 - $ω$ 2 + $\sqrt{Ω}$ 2 + $ω$ 2 ] 2
b.
ℏ $[\sqrt{Ω}$ 2 - $ω$ 2 + $\sqrt{Ω}$ 2 + $ω$ 2 ]
c.
$[\sqrt{Ω}$ 2 + $Ωω$ + $\sqrt{Ω}$ 2 - $Ωω$ ] 2
d.
ℏ $[\sqrt{Ω}$ 2 + $Ωω$ - $\sqrt{Ω}$ 2 - $Ωω$ ]
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The energy $E$ and degeneracy $d$ of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency $ω$ are $\sqrt{5}$ 2
, $c$ = 1
1 $\sqrt{5}$
1 $\sqrt{2}$
Two identical particles with a fixed total energy $E$ = $2ħω$ are in thermal equilibrium in a
1 one-dimensional harmonic oscillator potential $mω$ 2 $x$ 2. Let the entropy of the particles
2 be denoted by $S F$ if they are fermions with spin $(s z$ = $\pm$) and $S B$ if they are
1 ℏ
2 2 bosons with spin 0. Then, which of the following options is correct? $(k B$ is the Boltzmann constant)
a.
$S F$ = $k B$ ln 2, $S B$ = $k B$ ln 2
b.
$S F$ = $2k B$ ln 2, $S B$ = 0
c.
$S F$ = $4k B$ ln 2, $S B$ = 0
d.
$S F$ = $2k B$ ln 2, $S B$ = $k B$ ln 2
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A gas of $N$ classical particles that can occupy energy levels, $ε$ 1 and $ε$ 2 = $ε$ 1 + Δ is in equilibrium with a reservoir at temperature $T.$ From the schematics shown below, choose the correct dependence of the internal energy $U$ on $T$.
a.
b.
c.
d.
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The Lagrangian
1 1
$L$ 0 = $mq̇$ 2 - $mω$ 2 $q$ 2
2 2 with the generalized coordinate $q$ is transformed to $L$ = $L$ 0 + $α df(q)$. Consider the following statements:
$dt$ (i) Expression for the canonical momentum does not change. (ii) The equation of the motion does not change. Which of the following options is correct for the above statements?
a.
Both (i) and (ii) are correct.
b.
Both (i) and (ii) are not correct.
c.
(i) is correct and (ii) is not correct.
d.
(i) is not correct and (ii) is correct.
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An infinitely large thin sheet in the $xy-plane$ carries uniform positive charge density and is moving with constant velocity $v$ in the $+x$ direction (see figure below). The direction of the corresponding Poynting vector is
a.
$+x$ for both $z$ < 0 and $z$ > 0
b.
$+x$ for $z$ < 0 and $-x$ for $z$ > 0
c.
$-x$ for $z$ < 0 and $+x$ for $z$ > 0
d.
$-x$ for both $z$ < 0 and $z$ > 0
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A positive point charge is fixed at the origin. At some distance from it on the $x$ axis, a point dipole is kept pointing in the $+y$ direction. The force on the dipole is
a.
0
b.
in the $+y$ direction
c.
in the $-y$ direction
d.
in the $+x$ direction
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Two frames $S$ (solid lines) and $S′$ (dashed lines) with common origin are shown in the figure below. Frame $S$ is inertial while $S′$ is rotating about the common $z-axis.$ There is a point mass fixed at $P$ on the $x-axis$ of the $S$ frame. The magnitude of the centrifugal force and the Coriolis force experienced by the mass in the $S′$ frame is $F cen$ and $F cor$, respectively. Which of the following options is correct for these forces?
a.
$F cen$ = 0 and $F cor$ = 0
b.
$F cen ≠$ 0 and $F cor ≠$ 0 and $F cen$ =
c.
$F cen ≠$ 0 and $F cor ≠$ 0 and $F cen$ = $2F cor$
d.
$F cen ≠$ 0 and $F cor ≠$ 0 and $F cen$ = $F cor$
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Consider two operators $Â$ and $B̂$ which are related as $Â$ = $exp(iθB̂).$ If $θ$ is a non-zero real number, which of the following statements is/are true?
a.
If $B̂$ is Hermitian, then $Â$ is unitary.
b.
If $B̂$ is anti-Hermitian, then $Â$ is unitary.
c.
If $B̂$ is Hermitian, then $|Det(Â)|$ = 1.
d.
If $B̂$ is anti-Hermitian, then $Â$ is Hermitian.
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Consider operators $Â$, $B̂$, and $Ĉ$ for three observables of a quantum system satisfying $[Â, B̂]$ = 0, $[B̂, Ĉ$ ] = 0, and $[Â, Ĉ$ ] $≠$ 0, with uncertainties $ΔA, ΔB, ΔC,$ respectively. From the options given below, which is/are implied by the commutation relations among $Â$, $B̂$, and $Ĉ$ ?
a.
$ΔA ΔB$ > 0
b.
$ΔA ΔC$ > 0
c.
$Â, B̂$ can be simultaneously diagonalized.
d.
$Â, B̂, Ĉ$ can be simultaneously diagonalized.
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Consider the distribution of outcomes generated by $N (N$ ≫ 1) independent throws of (i) a coin or (ii) a six-sided dice. A coin (dice) is unbiased if both (all) its sides have equal probability to show up in a throw; it is biased otherwise. For the cases (i) and (ii) above, which of the following statements is/are true?
a.
The entropy of an unbiased coin is smaller than that of an unbiased dice.
b.
The entropy of an unbiased coin is greater than that of an unbiased dice.
c.
The entropy of a biased dice is smaller than that of an unbiased dice.
d.
The entropy of a biased coin is greater than that of an unbiased coin.
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The dispersion $(E(k))$ of the conduction band (CB) and valence band (VB) for a semiconductor are shown schematically in the figure. Considering the possibility of an electron making a transition from the bottom of the CB to the top of the VB, which of the following options is/are correct?
a.
The transition is forbidden.
b.
A photon can be emitted with an energy exactly equal to $E g$.
c.
A photon can be emitted with an energy less than $E g$.
d.
A phonon can be created with a crystal momentum $ħq.$
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A symmetric rigid body has moment of inertia $I$ 1, $I$ 2, $I$ 3 about its principal axes 1, 2, and 3, respectively, with $I$ 1 = $I$ 3 = $I$ ⊥ and $I$ 2 $≠ I$ ⊥. It is rotating in space with no torque on it so that its angular momentum $L⃗$ is constant. Let $ω$ 1, $ω$ 2, $ω$ 3 be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?
a.
$ω$ 1 + $ω$ 3
b.
$ω$ 12 + $ω$ 32
c.
Angle between axis 2 and $L⃗$
d.
$ω$ 2
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An $α$ particle moves towards a fixed nucleus carrying charge $Ze,$ with initial speed $v$ 0 and impact parameter $b.$ Starting from a large distance from the nucleus, its distance of closest approach is $r m$ and its speed there is $v m$. Then which of the following options is/are correct?
1
$Ze$ 2
$r$ =
$k$
$4πε$
0
$mv$ 02
0
a.
$v$ 0 $b$ = $v m r m$
b.
$v$ 0 $b$ = $2v m r$ 0
c.
For ≪ 1, $r m$ = $4r$ 0 + ignoring higher order corrections in $r 2r r$
d.
For ≪ 1, $r m$ = $4r$ 0 + ignoring higher order corrections in $r 8r r$ 0 0 0
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Consider a particle of mass $m$ = 9.0 $\times$ 10 -5 kg and charge $q$ = 3.0 $\times$ 10 -4 C in a
⃗ = 3 $ẑ$ V. m -2. s. The particle is uniform electromagnetic field $E⃗$ = 2 $x̂$ V. m -1, $B$ released from the coordinates (0, 5 m, 0) at time $t$ = 0. Starting from initial speed zero, it comes back to the $y-axis$ for the first time at time $t.$ The value of $t$ in seconds (rounded off to two decimal places) is _____ An atom has two energy levels with energy difference 2.2 eV between them. A gas of these atoms is with 8 $\times$ 10 20 atoms in the upper state and 5 $\times$ 10 20 atoms in the lower state. Ignoring spontaneous emission, the maximum possible energy released by this gas of atoms by stimulated emission is $E$ Joules. The value of $E$ (rounded off to one decimal place) is _____ $(e$ = 1.6 $\times$ 10 -19 C)
$A$
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The Hamiltonian of two interacting spin-1/2 particles is $H$ = 2 $S$ 1 ⋅ $S$ 2, where $S$ 1 and $S$ 2
ℏ are the spin angular momenta of particles 1 and 2, respectively. Here, $A$ = 10.56 eV. The energy in eV required to induce an excitation from the ground state to the excited state (rounded off to two decimal places) is _____
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The output signal (current $I d$) of a reversed biased (with a voltage $V b$) photodiode, on which light is incident, is fed to an amplifier (see figure). The output voltage is digitized by a 10 bit Analogue to Digital convertor (ADC) which has a reference voltage of 5 V. The smallest current $I d$ which can be measured by the circuit in nano-Amperes (rounded off to one decimal place) is _____
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A capacitor is made of two circular metal plates of radius 1 m separated by a distance of $d$ = 1 mm. The space between them is filled by a dielectric with permittivity $ε r$ = 5. The capacitor is connected to a voltage $V$ = 10 $sin(2π \times$ 10 6 $\times t)$ volts, where $t$ is in seconds. The maximum value of the magnetic field in between the plates at a radial distance $r$ = 0.5 m from the centre of the capacitor is $B \times$ 10 -6 T. The value of $B$ (rounded off to two decimal places) is _____ (Speed of light in vacuum $c$ = 3 $\times$ 10 8 m. s -1)
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Rotational spectrum of a diatomic molecule consists of lines of equal spacing with an interval of 20.0 cm -1. Its moment of inertia is found to be $I$ 0 $\times$ 10 -47 kg. m 2, the value of $I$ 0 (rounded off to one decimal place) is _____ (ℎ = 6.6 $\times$ 10 -34 J. s, speed of light in vacuum c = 3 $\times$ 10 8 m. s -1)
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Copper has an electron number density of 8.3 $\times$ 10 28 m -3. Its Fermi energy in eV (rounded off to one decimal place) is _____ (ℏ = 1.06 $\times$ 10 -34 J. s, mass of electron m e = 9.10 $\times$ 10 -31 kg, charge of electron = 1.60 $\times$ 10 -19 C)
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Two 1 kg blocks are connected to two massless springs of spring constants 8 N. m -1 and 4 N. m -1. The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the
$ω x-axis$ with the normal mode frequencies $ω$ and $ω (ω$ > $ω$). The ratio $H$ (rounded
$H L H L ω$
$L$ off to two decimal places) is _____
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A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one’s index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed as it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle $θ.$ The value of $θ$ in degrees (rounded off to one decimal place) is _____ (g = 9.8 m.s -2)
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