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SECTION 1 (Maximum Marks:12)

• This section contains FOUR (04) questions.
• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
• For each question, choose the option corresponding to the correct answer.
• Answer to each question will be evaluated according to the following marking scheme:
  
  
  • Full Marks: +3 If ONLY the correct option is chosen;
  • Zero Marks: 0 If none of the options is chosen (i.e. the question is unanswered);
  • Negative Marks: -1 In all other cases.

1. The center of a disk of radius $𝑟$ and mass $𝑚$ is attached to a spring of spring constant $𝑘$, inside a ring of radius $𝑅 > 𝑟$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $𝑇 =\frac{2𝜋}{𝜔}$. The correct expression for $𝜔$ is ($𝑔$ is the acceleration due to gravity):
   
   
   
   
   
    
   1. $\sqrt{\frac{2}{3}\left(\frac{g}{R-r}+\frac{k}{m}\right)}$
   2. $\sqrt{\frac{2g}{3(R-r)}+\frac{k}{m}}$
   3. $\sqrt{\frac{1}{6}\left(\frac{g}{R-r}+\frac{k}{m}\right)}$
   4. $\sqrt{\frac{1}{4}\left(\frac{g}{R-r}+\frac{k}{m}\right)}$
2. In a scattering experiment, a particle of mass $2𝑚$ collides with another particle of mass $𝑚$, which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation $𝜃$ of the heavier particle, as shown in the figure, in radians is:
   
   
   
   
   
    
   1. $\pi$
   2. $\tan^{-1}\left(\frac{1}{2}\right)$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{6}$
3. A conducting square loop initially lies in the $𝑋𝑍$ plane with its lower edge hinged along the $𝑋-$axis. Only in the region $𝑦 ≥ 0$, there is a time dependent magnetic field pointing along the $𝑍-$direction, $\vec{B} (𝑡) $= $𝐵_0 (cos 𝜔𝑡)\hat{k}$, where $𝐵_0$ is a constant. The magnetic field is zero everywhere else. At time $𝑡 = 0$, the loop starts rotating with constant angular speed $𝜔$ about the $𝑋$ axis in the clockwise direction as viewed from the $+𝑋$ axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. $(𝑉)$ in the loop as a function of time:
   
   
   
   
   
    
   1. 
      
   2. 
      
   3. 
      
   4. 
      
4. Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $𝐷$ of a tube. The measured value of $𝐷$ is:
   
   
   
   
   
    
   1. $0.12 cm$
   2. $0.11cm$
   3. $0.13cm$
   4. $0.14cm$

SECTION 2 (Maximum Marks:12)

• This section contains THREE (03) questions.
• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s).
• For each question, choose the option(s) corresponding to (all) the correct answer(s).
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 ONLY if (all) the correct option(s) is(are) chosen;
  • Partial Marks: +3 If all the four options are correct but ONLY three options are chosen;
  • Partial Marks: +2 If three or more options are correct but ONLY two options are chosen, both of which are correct;
  • Partial Marks: +1 If two or more options are correct but ONLY one option is chosen and it is a correct option;
  • Zero Marks: 0 If none of the options is chosen (i.e. the question is unanswered);
  • Negative Marks: -2 In all other cases.
• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then
  choosing ONLY (A), (B) and (D) will get +4 marks;
  choosing ONLY (A) and (B) will get +2 marks;
  choosing ONLY (A) and (D) will get +2 marks;
  choosing ONLY (B) and (D) will get +2 marks;
  choosing ONLY (A) will get +1 mark;
  choosing ONLY (B) will get +1 mark;
  choosing ONLY (D) will get +1 mark;
  choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get -2 marks.

5. A conducting square loop of side $𝐿$, mass $𝑀$ and resistance $𝑅$ is moving in the $𝑋𝑌$ plane with its edges parallel to the $𝑋$ and $𝑌$ axes. The region $𝑦 ≥ 0$ has a uniform magnetic field, $\vec{B}$= $𝐵_0 \hat{k}$. The magnetic field is zero everywhere else. At time $𝑡 = 0$, the loop starts to enter the magnetic field with an initial velocity $𝑣_0 \hat{j} m/s$, as shown in the figure. Considering the quantity $𝐾 =\frac{𝐵_0^2𝐿^2}{𝑅𝑀}$in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:
   
   
   
   
   
    
   1. If $𝑣_0 = 1.5𝐾𝐿$, the loop will stop before it enters completely inside the region of magnetic field.
   2. When the complete loop is inside the region of magnetic field, the net force acting on the loopis zero.
   3. If $v_0$=$\frac{KL}{10}$,the loop comes to rest at $t$=$\left(\frac{1}{k}\right)ln\left(\frac{5}{2}\right)$.
   4. If $v_0$=$3Kl$, the complete loop enters inside the region of magnetic field at time $t$=$\left(\frac{1}{k}\right)ln\left(\frac{3}{2}\right)$.
6. Length, breadth and thickness of a strip having a uniform cross section are measured to be $10.5 cm$, $0.05 mm$, and $6.0 𝜇m$, respectively. Which of the following option(s) give(s) the volume of the strip in $cm^3$ with correct significant figures:
    
   1. $3.2 × 10^{−5}$
   2. $32.0 × 10^{−6}$
   3. $3.0 × 10^{−5}$
   4. $3 × 10^{−5}$
7. Consider a system of three connected strings, $𝑆_1$, $𝑆_2$ and $𝑆_3$ with uniform linear mass densities $𝜇 kg/m$, $4𝜇 kg/m$ and $16𝜇 kg/m$, respectively, as shown in the figure. $𝑆_1$ and $𝑆_2$ are connected at the point $𝑃$, whereas $𝑆_2$ and $𝑆_3$ are connected at the point $𝑄$, and the other end of $𝑆_3$ is connected to a wall. A wave generator $O$ is connected to the free end of $𝑆_1$. The wave from the generator is represented by $𝑦 = 𝑦_0 cos(𝜔𝑡 − 𝑘𝑥) cm$, where $𝑦_0$, $𝜔$ and $𝑘$ are constants of appropriate dimensions. Which of the following statements is/are correct:
   
   
   
   
   
    
   1. When the wave reflects from $𝑃$ for the first time, the reflected wave is represented by$𝑦$ = $𝛼_1y_0 cos(𝜔𝑡 + 𝑘𝑥 + 𝜋) cm$, where $𝛼_1$ is a positive constant.
   2. When the wave transmits through $𝑃$ for the first time, the transmitted wave is represented by $𝑦$ = $𝛼_2y_0 cos(𝜔𝑡 − 𝑘𝑥) cm$, where $𝛼_2$ is a positive constant.
   3. When the wave reflects from $𝑄$ for the first time, the reflected wave is represented by $𝑦$ = $𝛼_3y_0 cos(𝜔𝑡 − 𝑘𝑥 + 𝜋) cm$, where $𝛼_3$ is a positive constant.
   4. When the wave transmits through $𝑄$ for the first time, the transmitted wave is represented by $𝑦$ = $𝛼_4y_0 cos(𝜔𝑡 − 4𝑘𝑥) cm$, where $𝛼_4$ is a positive constant.

SECTION 3 (Maximum Marks:24)

• This section contains SIX (06) questions.
• The answer to each question is a NON-NEGATIVE INTEGER.
• For each question, enter the correct integer corresponding to the answer using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If ONLY the correct integer is entered;
  • Zero Marks: 0 In all other cases.

8. A person sitting inside an elevator performs a weighing experiment with an object of mass $50 kg$. Suppose that the variation of the height $𝑦 (in m)$ of the elevator, from the ground, with time $𝑡 (in s)$is given by $𝑦$ = $8 [1 + sin \left(\frac{2𝜋𝑡}{𝑇}\right)]$, where $𝑇 = 40𝜋 s$. Taking acceleration due to gravity, $𝑔 = 10 m/s^2$, the maximum variation of the object’s weight (in $N$) as observed in the experiment is ____
9. A cube of unit volume contains $35 × 10^7$ photons of frequency $10^{15} Hz$. If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is $𝛼 × 10^{−9} T$. Taking permeability of free space $𝜇_0 = 4𝜋 × 10^{−7} Tm/A$, Planck’s constant $ℎ = 6 × 10^{−34}Js$ and $𝜋 =\frac{22}{7}$, the value of $𝛼$ is____
10. Two identical plates $P$ and $Q$, radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures $T_P$ and $T_Q$, respectively, with $T_Q$ < $T_P$, as shown in Fig. 1. The radiated power transferred per unit area from $P$ to $Q$ is $𝑊_0$. Subsequently, two more plates, identical to $P$ and $Q$, are introduced between $P$ and $Q$, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from $P$ to $Q$ (Fig. 2) in the steady state is $𝑊_𝑆$, then the ratio $\frac{𝑊_0}{𝑊_𝑆}$is ___
    
    
    
11. A solid glass sphere of refractive index $𝑛 = \sqrt{3}$ and radius $𝑅$ contains a spherical air cavity of radius $\frac{𝑅}{2}$, as shown in the figure. A very thin glass layer is present at the point $O$ so that the air cavity (refractive index $𝑛 = 1$) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source $𝑆$ emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point $O$ and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is $𝜃$. The value of $\sin 𝜃$ is ____
    
    
    
    
    
12. A single slit diffraction experiment is performed to determine the slit width using the equation,$\frac{𝑏𝑑}{𝐷}$=$𝑚λ$, where $𝑏$ is the slit width, $𝐷$ the shortest distance between the slit and the screen, $𝑑$ the distance between the $𝑚^{th}$ diffraction maximum and the central maximum, and $λ$ is the wavelength. $𝐷$ and $𝑑$are measured with scales of least count of $1 cm$ and $1 mm$, respectively. The values of $λ$ and $𝑚$ areknown precisely to be $600 nm$ and 3, respectively. The absolute error (in $𝜇m$) in the value of $𝑏$estimated using the diffraction maximum that occurs for $𝑚 = 3$ with $𝑑 = 5 mm$ and $𝐷 = 1 m$ is ___
13. Consider an electron in the $𝑛 = 3$ orbit of a hydrogen-like atom with atomic number $𝑍$. At absolute temperature $𝑇$, a neutron having thermal energy $𝑘_B𝑇$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $𝑇$ =$\frac{𝑍^2ℎ^2}{𝛼𝜋^2𝑎_0^2𝑚_N𝑘_B}$, (where $ℎ$ is the Planck’s constant, $𝑘_𝐵$is the Boltzmann constant, $𝑚_N$ is the mass of the neutron and $𝑎_0$ is the first Bohr radius of hydrogen atom) then the value of $𝛼$ is ___

SECTION 4 (Maximum Marks:12)

• This section contains FOUR (04) Matching List Sets.
• Each set has ONE Multiple Choice Question.
• Each set has TWO lists: List-I and List-II.
• List-I has Four entries (P), (Q), (R) and (S) and List-II has Five entries (1), (2), (3), (4) and (5).
• FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +3 ONLY if the option corresponding to the correct combination is chosen;
  • Zero Marks: 0   If none of the options is chosen (i.e. the question is unanswered);
  • Negative Marks: -1 In all other cases.

14. List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude $𝑝$, oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance $2𝑟$ apart along the $𝑥$ direction. The midpoint of the line joining the two dipoles is $𝑋$. The possible resultant electric fields $\vec{E}$ at $𝑋$are given in List-II.
    Choose the option that describes the correct match between the entries in List-I to those in List-II.                                                                                                     

    	List - I		List - II
    (P)		(1)	$\vec{E}$=0
    (Q)		(2)	$\vec{E}$=$-\frac{p}{2\pi \epsilon_0 r^3} \hat{j}$
    (R)		(3)	$\vec{E}$=$-\frac{p}{4\pi \epsilon_0 r^3} (\hat{i}-\hat{j})$
    (S)		(4)	$\vec{E}$=$\frac{p}{4\pi \epsilon_0 r^3} (2\hat{i}-\hat{j})$
    		(5)	$ \vec{E}$=$\frac{p}{\pi \epsilon_0 r^3} \hat{i}$

    
    The correct option is:
    
    1. P→3, Q→1, R→2, S→4
    2. P→4, Q→5, R→3, S→1
    3. P→2, Q→1, R→4, S→5
    4. P→2, Q→1, R→3, S→5
15. A circuit with an electrical load having impedance $𝑍$ is connected with an $AC$ source as shown in the diagram. The source voltage varies in time as $𝑉(𝑡)$ = $300 sin(400𝑡) V$, where $𝑡$ is time in $s$. List-I shows various options for the load. The possible currents $𝑖(𝑡)$ in the circuit as a function of time are given in List-II.
    
    
    
    Choose the option that describes the correct match between the entries in List-I to those in List-II.                                                                                                     

    	List - I		List - II
    (P)		(1)
    (Q)		(2)
    (R)		(3)
    (S)		(4)
    		(5)

    
    The correct option is:
    
    1. P→3, Q→5, R→2, S→1
    2. P→1, Q→5, R→2, S→3
    3. P→3, Q→4, R→2, S→1
    4. P→1, Q→4, R→2, S→5
16. List-I shows various functional dependencies of energy $(𝐸)$ on the atomic number $(𝑍)$. Energies associated with certain phenomena are given in List-II.
    Choose the option that describes the correct match between the entries in List-I to those in List-II.                                                                                                     

    	List - I		List - II
    (P)	$𝐸 ∝ 𝑍^2$	(1)	energy of characteristic $x-$rays
    (Q)	$𝐸 ∝ (𝑍 − 1)^2$	(2)	electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170
    (R)	$𝐸 ∝ 𝑍(𝑍 − 1)$	(3)	energy of continuous $x-$rays
    (S)	$𝐸$ is practically independent of $𝑍$	(4)	average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170
    		(5)	energy of radiation due to electronic transitions from hydrogen-like atoms

    
    The correct option is:
    
    1. P→4, Q→3, R→1, S→2
    2. P→5, Q→2, R→1, S→4
    3. P→5, Q→1, R→2, S→4
    4. P→3, Q→2, R→1, S→5

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