Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1
Arun and Aparna are here.Arun and Aparna is here.Arun’s families is here.Arun’s family is here.Which of the above sentences are grammatically CORRECT?
(A)
(i) and (ii)
(B)
(i) and (iv)
(C)
(ii) and (iv)
(D)
(iii) and (iv)
Q.2
The mirror image of the above text about the x-axis is
Q.3
Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:
(A)
1/36
(B)
1/12
(C)
1/8
(D)
1/4
Q.4
⊕ and ⊙ are two operators on numbers p and q such that𝒑 ⊙ 𝒒 = 𝒑 − 𝒒, and p ⊕ q = p× qThen, (𝟗 ⊙ (𝟔 ⊕ 𝟕)) ⊙ (𝟕 ⊕ (𝟔 ⊙ 𝟓))=
(A)
40
(B)
-26
(C)
-33
(D)
-40
Q.5
Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
(A)
6
(B)
9
(C)
18
(D)
24
Q.6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3)
Q.6
On a planar field, you travelled 3 units East from a point O. Next you travelled 4 units South to arrive at point P. Then you travelled from P in the North-East direction such that you arrive at a point that is 6 units East of point O. Next, you travelled in the North-West direction, so that you arrive at point Q that is 8 units North of point P.The distance of point Q to point O, in the same units, should be____________
(A)
3
(B)
4
(C)
5
(D)
6
Q.7
The author said, “Musicians rehearse before their concerts. Actors rehearse their roles before the opening of a new play. On the other hand, I find it strange that many public speakers think they can just walk on to the stage and start speaking. In my opinion, it is no less important for public speakers to rehearse their talks.”Based on the above passage, which one of the following is TRUE?
(A)
The author is of the opinion that rehearsing is important for musicians, actors and public speakers.
(B)
The author is of the opinion that rehearsing is less important for public speakers than for musicians and actors.
(C)
The author is of the opinion that rehearsing is more important only for musicians than public speakers.
(D)
The author is of the opinion that rehearsal is more important for actors than musicians.
Q.8
Some football players play cricket.All cricket players play hockey.Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is:
(A)
No football player plays hockey.
(B)
Some football players play hockey.
(C)
All football players play hockey.
(D)
All hockey players play football.
Q.9
In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.The probability that any point picked randomly within the square falls in the shaded area is___________
(A)
4-π/2
(B)
1/2
(C)
π/2-1
(D)
𝜋/4
Q.10
In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.The minimum area of the triangle PQR possible, in cm2, is
(A)
18
(B)
24
(C)
48√3
(D)
144√3
Physics (PH)
– Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1
Choose the graph that best describes the variation of dielectric constant (𝝐𝒓) with temperature (𝑻) in a ferroelectric material.(𝑻𝑪is the Curie temperature)
Q.2
A matter wave is represented by the wave function 𝚿(𝒙, 𝒚, 𝒛, 𝒕) = 𝑨𝒆𝒊 (𝟒𝒙+𝟑𝒚+𝟓𝒛−𝟏𝟎𝝅𝒕)where 𝑨 is a constant. The unit vector representing the direction of propagation of this matter wave is
(A)
(B)
(C)
(D)
Q.3
As shown in the figure, X-ray diffraction pattern is obtained from a diatomic chain of atoms 𝑷 and 𝑸. The diffraction condition is given by 𝒂 𝐜𝐨𝐬 𝜽 = 𝒏𝝀, where 𝒏 is the order of the diffraction peak. Here, 𝒂 is the lattice constant and 𝝀 is the wavelength of the X-rays. Assume that atomic form factors and resolution of the instrument do not depend on 𝜽. Then, the intensity of the diffraction peaks is
(A)
lower for even values of 𝑛, when compared to odd values of 𝑛
(B)
lower for odd values of 𝑛, when compared to even values of 𝑛
(C)
zero for odd values of 𝑛
(D)
zero for even values of 𝑛
Q.4
As shown in the figure, two metal-semiconductor junctions are formed between an n-type semiconductor 𝑺 and metal 𝑴. The work functions of 𝑺 and 𝑴 are 𝝋𝑺 𝐚𝐧𝐝 𝝋𝑴, respectively with 𝝋𝑴 > 𝝋𝑺.
The 𝑰 − 𝑽 characteristics (on linear scale) of the junctions is best represented by
(A)
(B)
(C)
(D)
Q.5
Consider a tiny current loop driven by a sinusoidal alternating current. If the surface integral of its time-averaged Poynting vector is constant, then the magnitude of the time-averaged magnetic field intensity, at any arbitrary position, ⃗𝒓→, is proportional to
(A)
1/r3
(B)
1/r2
(C)
1/𝑟
(D)
r
Q.6
Consider a solenoid of length 𝑳 and radius 𝑹, where 𝑹 ≪ 𝑳. A steady-current flows through the solenoid. The magnetic field is uniform inside the solenoid and zero outside.
Among the given options, choose the one that best represents the variation in the magnitude of the vector potential, (𝟎, 𝑨𝝋, 𝟎) at 𝒛 = 𝑳/𝟐, as a function of the radial distance (𝒓) in cylindrical coordinates.Useful information: The curl of a vector ⃗𝑭→, in cylindrical coordinates is
(A)
(B)
(C)
(D)
Q.7
Assume that 13N (𝒁 = 𝟕) undergoes first forbidden β+decay from its ground state with spin-parity 𝑱i𝝅, to a final state 𝑱𝒇𝝅. The possible values for 𝑱i𝝅and𝑱f𝝅, respectively, are
(A)
(B)
(C)
(D)
Q.8
In an experiment, it is seen that an electric-dipole (𝑬𝟏) transition can connect an initial nuclear state of spin-parity 𝑱i𝝅 = 𝟐+to a final state 𝑱𝒇𝝅. Allpossible values of 𝑱f𝝅are
(A)
1+, 2+
(B)
1+, 2+, 3+
(C)
1−, 2−
(D)
1−, 2−, 3−
Q.9
Choose the correct statement from the following.
(A)
Silicon is a direct band gap semiconductor.
(B)
Conductivity of metals decreases with increase in temperature.
(C)
Conductivity of semiconductors decreases with increase in temperature.
(D)
Gallium Arsenide is an indirect band gap semiconductor.
Q.10 – Q.16 Multiple Select Question (MSQ), carry ONE mark each (no negative marks).
Q.10
A two-dimensional square lattice has lattice constant 𝒂.𝒌 represents the wavevector in reciprocal space. The
coordinates (𝒌𝒙, 𝒌𝒚)of reciprocal space where band gap(s) can occur, are
Q.11
As shown in the figure, an electromagnetic wave with intensity 𝑰𝑰is incident at the interface of two media having refractive indices 𝒏𝟏 = 𝟏 and 𝒏𝟐 = √𝟑. The wave is reflected with intensity 𝑰𝑹and transmitted with intensity 𝑰𝑻. Permeability of each medium is the same. (Reflection coefficient 𝑹 = 𝑰𝑹/𝑰𝑰and Transmission coefficient 𝑻 = 𝑰𝑻/𝑰𝑰).
Choose the correct statement(s).
(A)
𝑅 = 0 if 𝜃𝐼 = 0o and polarization of incident light is parallel to the plane of incidence.
(B)
𝑇 = 1 if 𝜃𝐼 = 60o and polarization of incident light is parallel to the plane of incidence.
(C)
𝑅 = 0 if 𝜃𝐼 = 60o and polarization of incident light is perpendicular to the plane of incidence.
(D)
𝑇 = 1 if 𝜃𝐼 = 60o and polarization of incident light is perpendicular to the plane of incidence.
Q.12
A material is placed in a magnetic field intensity 𝑯. As a result, bound current density 𝑱𝒃is induced and magnetization of the material is 𝑴. The magnetic flux density is 𝑩. Choose the correct option(s) valid at the surface of the material.
(A)
𝛁. 𝑴 = 0
(B)
𝛁. 𝑩 = 0
(C)
𝛁. 𝑯 = 0
(D)
𝛁. 𝑱𝑏 = 0
Q.13
For a finite system of Fermions where the density of states increases with energy, the chemical potential
(A)
decreases with temperature
(B)
increases with temperature
(C)
does not vary with temperature
(D)
corresponds to the energy where the occupation probability is 0.5
Q.14
Among the term symbols𝟒𝑺𝟏, 𝟐𝑫𝟕/𝟐, 𝟑𝑺𝟏and 𝟐𝑫𝟓/𝟐choose the option(s) possible in the 𝑳𝑺 coupling notation.
(A)
4S1
(B)
2𝐷7/2
(C)
3𝑆1
(D)
2𝐷5/2
Q.15
To sustain lasing action in a three-level laser as shown in the figure, necessary condition(s) is(are)
(A)
lifetime of the energy level 1 should be greater than that of energy level 2
(B)
population of the particles in level 1 should be greater than that of level 0
(C)
lifetime of the energy level 2 should be greater than that of energy level 0
(D)
population of the particles in level 2 should be greater than that of level 1
Q.16
If 𝒚𝒏(𝒙) is a solution of the differential equation𝒚′′ − 𝟐𝒙𝒚′ + 𝟐𝒏𝒚 = 𝟎where 𝒏 is an integer and the prime ( ′) denotes differentiation with respect to 𝒙, then acceptable plot(s) of 𝝍𝒏(𝒙) = 𝒆−𝒙⁄𝟐 𝒚𝒏(𝒙), is(are)
(A)
(B)
(C)
(D)
Q.17 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).
Q.17
The donor concentration in a sample of n-type silicon is increased by a factor of 100. Assuming the sample to be non-degenerate, the shift in the Fermi level (in meV) at 300 K (rounded off to the nearest integer) is ________.(Given: 𝒌𝐁𝑻= 25 meV at 300 K)
Q.18
Two observers O and Oʹ observe two events P and Q. The observers have a constant relative speed of 0.5c. In the units, where the speed of light, c, is taken as unity, the observer O obtained the following coordinates:Event P: 𝒙 = 5, 𝒚 = 3, 𝒛 = 5, 𝒕 = 3 Event Q: 𝒙 = 5,𝒚 = 1, 𝒛 = 3, 𝒕 = 5The length of the space-time interval between these two events, as measured by Oʹ, is 𝑳. The value of |𝑳| (in integer) is ________________.
Q.19
A light source having its intensity peak at the wavelength 289.8 nm is calibrated as 10,000 K which is the temperature of an equivalent black body radiation. Considering the same calibration, the temperature of light source (in K) having its intensity peak at the wavelength 579.6 nm (rounded off to the nearest integer) is ______.
Q.20
A hoop of mass 𝑴 and radius 𝑹 rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass 𝒎 slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is_______.
Q.21
Three non-interacting bosonic particles of mass 𝒎 each, are in a one- dimensional infinite potential well of width 𝒂. The energy of the third excitedstate of the system is 𝒙 ×. The value of 𝒙 (in integer) is _________.
Q.22
The spacing between two consecutive S-branch lines of the rotational Raman spectra of hydrogen gas is 𝟐𝟒𝟑. 𝟐 𝐜𝐦−𝟏. After excitation with a laser of wavelength 𝟓𝟏𝟒. 𝟓 𝐧𝐦, the Stoke’s line appeared at 𝟏𝟕𝟔𝟏𝟏. 𝟒 𝐜𝐦−𝟏for a particular energy level. The wavenumber (rounded off to the nearest integer), in 𝐜𝐦−𝟏, at which Stoke’s line will appear for the next higher energy level is______________.
Q.23
Thetransition line, asshownin the figure, arises between𝟐𝑫𝟑/𝟐and 𝟐𝑷𝟏/𝟐states without any external magnetic field. The number of lines that will appear in the presence of a weak magnetic field (in integer) is______.
Q.24
Consider the atomic system as shown in the figure, where the Einstein 𝑨 coefficients for spontaneous emission for the levels are 𝑨𝟐→𝟏 = 𝟐 × 𝟏𝟎𝟕s–1 and 𝑨𝟏→𝟎 = 𝟏𝟎𝟖s–1. If 1014 atoms/cm3 are excited from level 0 to level 2 and a steady state population in level 2 is achieved, then the steady state population at level 1 will be 𝒙 × 𝟏𝟎𝟏𝟑 cm–3. The value of 𝒙 (in integer) is_______________.
Q.25
If 𝒂⃗→ and ⃗𝒃→ are constant vectors, 𝒓⃗→ and ⃗𝒑→ are generalized positions and conjugate momenta, respectively, then for the transformation 𝑸 = 𝒂⃗ ∙ 𝒓⃗ and 𝑷 = ⃗𝒃 ∙ 𝒓⃗ to be canonical, the value of ⃗𝒂→ ∙ 𝒃→ (in integer) is _______________.
Q.26 – Q.41 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).
Q.26
The above combination of logic gates represents the operation
(A)
OR
(B)
NAND
(C)
AND
(D)
NOR
Q.27
In a semiconductor, the ratio of the effective mass of hole to electron is 2:11 and the ratio of average relaxation time for hole to electron is 1:2. The ratio of the mobility of the hole to electron is
(A)
4:9
(B)
4:11
(C)
9:4
(D)
11:4
Q.28
Consider a spin 𝑺 = ℏ⁄𝟐 particle in the state |𝝓⟩ =. The probability that a measurement finds the state with 𝑺𝒙 = +ℏ⁄𝟐 is
(A)
5/18
(B)
11/18
(C)
15/18
(D)
17/18
Q.29
An electromagnetic wave having electric field 𝑬 = 𝟖 𝐜𝐨(𝒌𝒛 − ω𝒕) 𝒚̂ 𝐕 𝐜𝐦−𝟏is incident at 𝟗𝟎°(normal incidence) on a square slab from vacuum (with refractive index 𝒏𝟎 = 𝟏. 𝟎) as shown in the figure. The slab is composed of two different materials with refractive indices 𝒏𝟏and 𝒏𝟐. Assume that the permeability of each medium is the same. After passing through the slab for the first time, the electric field amplitude, in 𝐕 𝐜𝐦−𝟏, of the electromagnetic
wave, which emerges from the slab in region 2, is closest to
(A)
11/1.6
(B)
11/3.2
(C)
11/13.8
(D)
11/25.6
Q.30
Consider a point charge +𝑸 of mass 𝒎 suspended by a massless, inextensible string of length 𝒍 in free space (permittivity 𝗌𝟎) as shown in the figure. It is placed at a height 𝒅 (𝒅 > 𝒍) over an infinitely large, grounded conducting plane. The gravitational potential energy is assumed to be zero at the position of the conducting plane and is positive above the plane.
If 𝜽 represents the angular position and 𝒑𝜽its corresponding canonical momentum, then the correct Hamiltonian of the system is
(A)
(𝑝𝜃2/2𝑚𝑙2 ) _ [𝑄2/16𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)] − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(B)
(P𝜃2/2𝑚𝑙2 ) – [𝑄2+ /8𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)]-𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(C)
(𝑝𝜃2/2𝑚𝑙2) -[𝑄2/8𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)] − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(D)
(𝑝𝜃2/2𝑚𝑙2) -[𝑄2/16𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)] + 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
Q.31
Consider two concentric conducting spherical shells as shown in the figure. The inner shell has a radius a and carries a charge +𝑸. The outer shell has a radius b and carries a charge −𝑸. The empty space between them is half- filled by a hemispherical shell of a dielectric having permittivity ε𝟏. The remaining space between the shells is filled with air having the permittivity ε𝟎.
The electric field at a radial distance 𝒓 from the center and between the shells (𝒂 < 𝒓 < 𝒃) is
(A)
(B)
(C)
(D)
Q.32
For the given sets of energy levels of nuclei X and Y whose mass numbers are odd and even, respectively, choose the best suited interpretation.
(A)
Set I: Rotational band of X Set II: Vibrational band of Y
(B)
Set I: Rotational band of Y Set II: Vibrational band of X
(C)
Set I: Vibrational band of X Set II: Rotational band of Y
(D)
Set I: Vibrational band of Y Set II: Rotational band of X
Q.33
Consider a system of three distinguishable particles, each having spin 𝑺=𝟏/𝟐 such that 𝑺𝒛 = ±𝟏/𝟐 with corresponding magnetic moments 𝝁𝒛 = ±𝝁. When the system is placed in an external magnetic field 𝑯 pointing along the𝒛-axis, the total energy of the system is 𝝁𝑯. Let 𝒙 be the state where the first spin has 𝑺𝒛 = 𝟏/𝟐. The probability of having the state 𝒙 and the mean magnetic moment (in the +𝒛 direction) of the system in state 𝒙 are
(A)
(B)
(C)
(D)
Q.34
Consider a particle in a one-dimensional infinite potential well with its walls at 𝒙 = 𝟎 and 𝒙 = 𝑳. The system is perturbed as shown in the figure.
The first order correction to the energy eigenvalue is
(A)
𝑉0 /4
(B)
𝑉0 /3
(C)
𝑉0 /2
(D)
𝑉0 /5
Q.35
Consider a state described by 𝝍(𝒙, 𝒕) = 𝝍𝟐(𝒙, 𝒕) + 𝝍𝟒(𝒙, 𝒕), where 𝝍𝟐(𝒙, 𝒕) and 𝝍𝟒(𝒙, 𝒕) are respectively the second and fourth normalized harmonic oscillator wave functions and 𝑚 is the angular frequency of the harmonic oscillator. The wave function 𝝍(𝒙, 𝒕 = 𝟎) will be orthogonal to 𝝍(𝒙, 𝒕) at time 𝒕 equal to
(A)
𝜋/2𝜔
(B)
𝜋/𝜔
(C)
𝜋/4𝜔
(D)
𝜋/6𝜔
Q.36
(A)
ħ𝜔/2+ 𝛽−1 ln[1 − exp(𝛽ħ𝜔)]
(B)
ħ𝜔/2+ 𝛽−1 ln[1 − exp(−𝛽ħ𝜔)]
(C)
ħ𝜔/2+ 𝛽−1 ln[1 + exp(−𝛽ħ𝜔)]
(D)
𝛽−1 ln[1 − exp(−𝛽ħ𝜔)]
Q.37
A system of two atoms can be in three quantum states having energies 0, 𝜖 and 2𝜖. The system is in equilibrium at temperature 𝑇 = (𝑘𝐵𝛽)−1. Match the following Statistics with the Partition function.
(A)
CD:Z1, CI:Z2, FD:Z3, BE:Z4
(B)
CD:Z2, CI:Z3, FD:Z4, BE:Z1
(C)
CD:Z3, CI:Z4, FD:Z1, BE:Z2
(D)
CD:Z4, CI:Z1, FD:Z2, BE:Z3
Q.38
The free energy of a ferromagnet is given by 𝑭 = 𝑭𝟎 + 𝒂𝟎(𝑻 − 𝑻𝑪)𝑴𝟐 +𝒃𝑴𝟒, where 𝑭𝟎, 𝒂𝟎, and 𝒃 are positive constants, 𝑴 is the magnetization, 𝑻 is the temperature, and 𝑻𝑪is the Curie temperature. The relation between𝑴𝟐and 𝑻 is best depicted by
(A)
(B)
(C)
(D)
Q.39
Consider a spherical galaxy of total mass 𝑴 and radius 𝑹, having a uniform matter distribution. In this idealized situation, the orbital speed 𝒗 of a star of mass 𝒎 (𝒎 ≪ 𝑴) as a function of the distance 𝒓 from the galactic centre is best described by(𝑮 is the universal gravitational constant)
(A)
(B)
(C)
(D)
Q.40
Consider the potential 𝑼(𝒓) defined aswhere 𝑎 and 𝑼𝟎are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with 𝑼(𝒓) for a (wave-vector) momentum transfer 𝒒 and 𝑎 → 𝟎, is proportional to(Useful integral:
(A)
q−2
(B)
q−1
(C)
𝑞
(D)
q2
Q.41
As shown in the figure, inverse magnetic susceptibility (𝟏/𝝌) is plotted as a function of temperature (𝑻) for three different materials in paramagnetic states.
(Curie temperature of ferromagnetic material =𝑻𝐂Néel temperature of antiferromagnetic material= 𝑻𝐍)Choose the correct statement from the following
(A)
Material 1 is antiferromagnetic (𝑇 < 𝑇N), 2 is paramagnetic, and 3 is ferromagnetic (𝑇 < 𝑇C).
(B)
Material 1 is paramagnetic, 2 is antiferromagnetic (𝑇 < 𝑇N), and 3 is ferromagnetic (𝑇 < 𝑇C).
(C)
Material 1 ferromagnetic (𝑇 < 𝑇C), 2 is antiferromagnetic (𝑇 < 𝑇N), and 3 is paramagnetic.
(D)
Material 1 is ferromagnetic (𝑇 < 𝑇C), 2 is paramagnetic, and 3 is antiferromagnetic (𝑇 < 𝑇N).
Q.42 – Q.46 Multiple Select Question (MSQ), carry TWO mark each (no negative marks).
Q.42
A function (𝒕) is defined only for 𝒕 ≥ 𝟎. The Laplace transform of 𝒇(𝒕) is 𝐿(𝒇; 𝒔) = ∫∞𝟎 𝒆−𝒔𝒕 𝒇(𝒕) 𝒅𝒕 whereas the Fourier transform of 𝒇(𝒕) is 𝒇̃(𝑚) = ∫∞𝟎 𝒇(𝒕) 𝒆−𝒊𝑚𝒕 𝒅𝒕 . The correct statement(s) is(are)
(A)
The variable 𝑠 is always real.
(B)
The variable 𝑠 can be complex.
(C)
(𝑓; 𝑠) and 𝑓̃(𝜔) can never be made connected.
(D)
(𝑓; 𝑠) and 𝑓̃(𝜔) can be made connected.
Q.43
𝑷 and 𝑸 are two Hermitian matrices and there exists a matrix 𝑹, which diagonalizes both of them, such that 𝑹𝑷𝑹−𝟏 = 𝑺𝟏and 𝑹𝑸𝑹−𝟏 = 𝑺𝟐, where 𝑺𝟏and 𝑺𝟐are diagonal matrices. The correct statement(s) is(are)
(A)
All the elements of both matrices 𝑆1 and 𝑆2 are real.
(B)
The matrix 𝑃𝑄 can have complex eigenvalues.
(C)
The matrix 𝑄𝑃 can have complex eigenvalues.
(D)
The matrices 𝑃 and 𝑄 commute.
Q.44
A uniform block of mass 𝑴 slides on a smooth horizontal bar. Another mass 𝒎 is connected to it by an inextensible string of length l of negligible mass, and is constrained to oscillate in the X-Y plane only. Neglect the sizes of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as x and θ, as shown in the figure.
If 𝒑𝒙and 𝒑𝜽are the generalized momenta corresponding to x and θ, respectively, then the correct option(s) is(are)
(A)
𝑝𝑥 = (𝑚 + 𝑀)𝑥̇ + 𝑚𝑙 cos 𝜃 𝜃̇
(B)
𝑝𝜃 = 𝑚𝑙2𝜃̇ − 𝑚𝑙 cos 𝜃 𝑥̇
(C)
𝑝𝑥 is conserved
(D)
𝑝𝜃 is conserved
Q.45
The Gell-Mann – Okuba mass formula defines the mass of baryons as 𝑴 = 𝑴 + 𝒂𝒀 + 𝒃 [𝑰(𝑰 + 𝟏) − 1/4 𝒀𝟐], where 𝑴 , 𝒂 and 𝒃 are constants, 𝑰 represents the isospin and 𝒀 represents the hypercharge. If the mass of 𝚺 hyperons is same as that of 𝚲 hyperons, then the correct option(s) is(are)
(A)
𝑀 ∝ 𝐼(𝐼 + 1)
(B)
𝑀 ∝ 𝑌
(C)
𝑀 does not depend on 𝐼
(D)
𝑀 does not depend on 𝑌
Q.46
The time derivative of a differentiable function 𝒈(𝒒𝒊, 𝒕) is added to a Lagrangian 𝑳(𝒒𝒊, 𝒒̇𝒊, 𝒕) such that,where 𝒒𝒊, 𝒒̇ 𝒊, 𝒕 are the generalized coordinates, generalized velocities and time, respectively. Let 𝒑𝒊be the generalized momentum and 𝑯 the Hamiltonian associated with (𝒒𝒊, 𝒒̇ 𝒊, 𝒕). If 𝒑′and 𝑯′are those associated with 𝑳′, then the correct option(s) is(are)
(A)
Both 𝐿 and 𝐿′ satisfy the Euler-Lagrange’s equations of motion
(B)
(C)
If 𝑝𝑖 is conserved, then 𝑝′ is necessarily conserved
(D)
Q.47 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).
Q.47
A linear charged particle accelerator is driven by an alternating voltage source operating at 10 MHz. Assume that it is used to accelerate electrons. After a few drift-tubes, the electrons attain a velocity 𝟐. 𝟗 × 𝟏𝟎𝟖m s–1. The minimum length of each drift-tube, in m, to accelerate the electrons further (rounded off to one decimal place) is _______________.
Q.48
The Coulomb energy component in the binding energy of a nucleus is 18.432 MeV. If the radius of the uniform and spherical charge distribution in the nucleus is 3 fm, the corresponding atomic number (rounded off to the nearest integer) is _____________. (Given:e2/𝟒𝝅E𝟎= 𝟏. 𝟒𝟒 MeV fm)
Q.49
For a two-nucleon system in spin singlet state, the spin is represented through the Pauli matrices 𝝈𝟏, 𝝈𝟐for particles 1 and 2, respectively. The value of (𝝈𝟏 ⋅ 𝝈𝟐) (in integer) is ________.
Q.50
A contour integral is defined as where 𝒏 is a positive integer and 𝑪 is the closed contour, as shown in the figure, consisting of the line from −𝟏𝟎𝟎 to 𝟏𝟎𝟎 and the semicircle traversed in the counter-clockwise sense.
The value of 𝑰𝒏 (in integer) is _________.
Q.51
The normalized radial wave function of the second excited state of hydrogen atom iswhere 𝒂 is the Bohr radius and 𝒓 is the distance from the center of the atom.The distance at which the electron is most likely to be found is 𝒚 × 𝒂. The value of 𝒚 (in integer) is ____________.
Q.52
Consider an atomic gas with number density 𝒏 = 𝟏𝟎𝟐𝟎 𝐦−𝟑, in the ground state at 300 K. The valence electronic configuration of atoms is 𝒇𝟕. The paramagnetic susceptibility of the gas 𝝌 = 𝒎 × 𝟏𝟎−𝟏𝟏. The value of 𝒎 (rounded off to two decimal places) is ________.(Given: 𝐌𝐚𝐠𝐧𝐞𝐭𝐢𝐜 𝐩𝐞𝐫𝐦𝐞𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐨𝐟 𝐟𝐫𝐞𝐞 𝐬𝐩𝐚𝐜𝐞 𝝁𝟎 = 𝟒𝝅 × 𝟏𝟎−𝟕 𝐇 𝐦−𝟏𝐁𝐨𝐡𝐫 𝐦𝐚𝐠𝐧𝐞𝐭𝐨𝐧 𝝁𝐁 = 𝟗. 𝟐𝟕𝟒 × 𝟏𝟎−𝟐𝟒𝐀 𝐦𝟐𝐁𝐨𝐥𝐭𝐳𝐦𝐚𝐧𝐧 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝒌𝐁 = 𝟏. 𝟑𝟖𝟎𝟕 × 𝟏𝟎−𝟐𝟑 𝐉 𝐊−𝟏)
Q.53
Consider a cross-section of an electromagnet having an air-gap of 𝟓 cm as shown in the figure. It consists of a magnetic material (𝝁 = 𝟐𝟎𝟎𝟎𝟎𝝁𝟎) and is driven by a coil having 𝑵𝑰 = 𝟏𝟎𝟒A, where 𝑵 is the number of turns and 𝑰 is the current in Ampere.
Ignoring the fringe fields, the magnitude of the magnetic field ⃗𝑩⃗→ (in Tesla, rounded off to two decimal places) in the air-gap between the magnetic poles is ____________ .
Q.54
The spin ⃗𝑺→ and orbital angular momentum ⃗𝑳→ of an atom precess about 𝑱→, the total angular momentum. 𝑱→ precesses about an axis fixed by a magnetic field⃗𝑩𝟏→ = 𝟐𝑩𝟎𝒛̂ , where 𝑩𝟎is a constant. Now the magnetic field is changed to 𝑩⃗𝟐→ = 𝑩𝟎(𝒙̂ + √𝟐𝒚̂ + 𝒛̂). Given the orbital angular momentum quantum number 𝒍 = 𝟐 and spin quantum number 𝒔 = 𝟏⁄𝟐, 𝜽 is the angle between𝑩⃗𝟏and j→ for the largest possible values of total angular quantum number 𝒋 and its z-component 𝒋𝒛. The value of 𝜽 (in degree, rounded off to the nearest integer) is ____.
Q.55
The spin-orbit effect splits the 𝟐𝑷→𝟐𝑺 transition (wavelength, 𝝀 =𝟔𝟓𝟐𝟏 Å) in Lithium into two lines with separation of ∆𝝀 = 𝟎. 𝟏𝟒 Å. The corresponding positive value of energy difference between the above two lines, in eV, is 𝒎 × 𝟏𝟎−𝟓. The value of 𝒎 (rounded off to the nearest integer) is _________ .(Given: Planck’s constant, 𝐡 = 𝟒. 𝟏𝟐𝟓 × 𝟏𝟎−𝟏𝟓𝐞𝐕 𝐬Speed of light, 𝐜 = 𝟑 × 𝟏𝟎𝟖 𝐦 𝐬−𝟏)
Q.1
. The value of 𝒙 (in integer) is _________.
. The probability that a measurement finds the state with 𝑺𝒙 = +ℏ⁄𝟐 is
where 𝑎 and 𝑼𝟎 are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with 𝑼(𝒓) for a (wave-vector) momentum transfer 𝒒 and 𝑎 → 𝟎, is proportional to(Useful integral: 
where 𝒒𝒊, 𝒒̇ 𝒊, 𝒕 are the generalized coordinates, generalized velocities and time, respectively. Let 𝒑𝒊 be the generalized momentum and 𝑯 the Hamiltonian associated with (𝒒𝒊, 𝒒̇ 𝒊, 𝒕). If 𝒑′ and 𝑯′ are those associated with 𝑳′, then the correct option(s) is(are)
where 𝒏 is a positive integer and 𝑪 is the closed contour, as shown in the figure, consisting of the line from −𝟏𝟎𝟎 to 𝟏𝟎𝟎 and the semicircle traversed in the counter-clockwise sense.
where 𝒂 is the Bohr radius and 𝒓 is the distance from the center of the atom.The distance at which the electron is most likely to be found is 𝒚 × 𝒂. The value of 𝒚 (in integer) is ____________.
