35. The phase space diagram for a ball thrown vertically up from ground is
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]

ANSWER: D

1. The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $\mathrm { E } _ { 1 }$ and $\mathrm { E } _ { 2 }$ are the total mechanical energies respectively. Then
   (A) $\quad E _ { 1 } = \sqrt { 2 } E _ { 2 }$
   (B) $E _ { 1 } = 2 E _ { 2 }$
   (C) $E _ { 1 } = 4 E _ { 2 }$
   (D) $E _ { 1 } = 16 E _ { 2 }$ [Figure]

ANSWER: C

1. Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]

ANSWER: B

Paragraph for Question Nos. 38 and 39

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let ' $N$ ' be the number density of free electrons, each of mass ' $m$ '. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency ' $\omega _ { \mathrm { p } }$ ', which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega _ { \mathrm { p } }$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

38. Taking the electronic charge as ' $e$ ' and the permittivity as ' $\varepsilon _ { 0 }$ ', use dimensional analysis to determine the correct expression for $\omega _ { p }$.
(A) $\sqrt { \frac { N e } { m \varepsilon _ { 0 } } }$
(B) $\sqrt { \frac { m \varepsilon _ { 0 } } { N e } }$
(C) $\sqrt { \frac { N e ^ { 2 } } { m \varepsilon _ { 0 } } }$
(D) $\sqrt { \frac { m \varepsilon _ { 0 } } { N e ^ { 2 } } }$

ANSWER: C

1. Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $\mathrm { N } \approx 4 \times 10 ^ { 27 } \mathrm {~m} ^ { - 3 }$. Take $\varepsilon _ { 0 } \approx 10 ^ { - 11 }$ and $\mathrm { m } \approx 10 ^ { - 30 }$, where these quantities are in proper SI units.
   (A) 800 nm
   (B) 600 nm
   (C) 300 nm
   (D) 200 nm

ANSWER: B

SECTION - IV (Total Marks : 28)

(Integer Answer Type)

This section contains $\mathbf { 7 }$ questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9 . The bubble corresponding to the correct answer is to be darkened in the ORS.

40. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of $0.3 \mathrm {~m} / \mathrm { s } ^ { 2 }$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is $( P / 10 )$. The value of $P$ is
[Figure]
ANSWER: 4

1. A block is moving on an inclined plane making an angle $45 ^ { \circ }$ with the horizontal and the coefficient of friction is $\mu$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define $\mathrm { N } = 10 \mu$, then N is

ANSWER: 5

42. Four point charges, each of $+ q$, are rigidly fixed at the four corners of a square planar soap film of side ' $a$ '. The surface tension of the soap film is $\gamma$. The system of charges and planar film are in equilibrium, and $a = k \left[ \frac { q ^ { 2 } } { \gamma } \right] ^ { 1 / N }$, where ' $k$ ' is a constant. Then $N$ is

ANSWER: 3

1. Steel wire of length ' L ' at $40 ^ { \circ } \mathrm { C }$ is suspended from the ceiling and then a mass ' m ' is hung from its free end. The wire is cooled down from $40 ^ { \circ } \mathrm { C }$ to $30 ^ { \circ } \mathrm { C }$ to regain its original length ' L '. The coefficient of linear thermal expansion of the steel is $10 ^ { - 5 } / { } ^ { \circ } \mathrm { C }$, Young's modulus of steel is $10 ^ { 11 } \mathrm {~N} / \mathrm { m } ^ { 2 }$ and radius of the wire is 1 mm . Assume that $L \gg$ diameter of the wire. Then the value of ' $m$ ' in kg is nearly

ANSWER:3

1. The activity of a freshly prepared radioactive sample is $10 ^ { 10 }$ disintegrations per second, whose mean life is $10 ^ { 9 } \mathrm {~s}$. The mass of an atom of this radioisotope is $10 ^ { - 25 } \mathrm {~kg}$. The mass (in mg ) of the radioactive sample is

ANSWER: 1

1. A long circular tube of length 10 m and radius 0.3 m carries a current $I$ along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $I = I _ { 0 } \cos ( 300 t )$ where $I _ { 0 }$ is constant. If the magnetic moment of the loop is $N \mu _ { 0 } I _ { 0 } \sin ( 300 t )$, then ' $N$ ' is

[Figure]
ANSWER: 6

1. Four solid spheres each of diameter $\sqrt { 5 } \mathrm {~cm}$ and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm . The moment of inertia of the system about the diagonal of the square is $\mathrm { N } \times 10 ^ { - 4 } \mathrm {~kg} - \mathrm { m } ^ { 2 }$, then N is

ANSWER: 9

PART III : MATHEMATICS

SECTION-1 (Total Marks : 21)

(Single Correct Answer Type)

This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

2. Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures $2 T$ and $3 T$ respectively. The temperature of the middle (i.e. second) plate under steady state condition is
(A) $\left( \frac { 65 } { 2 } \right) ^ { \frac { 1 } { 4 } } T$
(B) $\left( \frac { 97 } { 4 } \right) ^ { \frac { 1 } { 4 } } T$
(C) $\left( \frac { 97 } { 2 } \right) ^ { \frac { 1 } { 4 } } T$
(D) $( 97 ) ^ { \frac { 1 } { 4 } } T$

ANSWER : C

1. Consider a thin spherical shell of radius $R$ with its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $| \vec { E } ( r ) |$ and the electric potential $V ( r )$ with the distance $r$ from the centre, is best represented by which graph?

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER : D

PHYSICS

1. In the determination of Young's modulus $\left( Y = \frac { 4 M L \mathrm {~g} } { \pi l d ^ { 2 } } \right)$ by using Searle's method, a wire of length $L = 2 \mathrm {~m}$ and diameter $d = 0.5 \mathrm {~mm}$ is used. For a load $M = 2.5 \mathrm {~kg}$, an extension $l = 0.25 \mathrm {~mm}$ in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm . The number of divisions on their circular scale is 100 . The contributions to the maximum probable error of the $Y$ measurement
   (A) due to the errors in the measurements of $d$ and $l$ are the same.
   (B) due to the error in the measurement of $d$ is twice that due to the error in the measurement of $l$.
   (C) due to the error in the measurement of $l$ is twice that due to the error in the measurement of $d$.
   (D) due to the error in the measurement of $d$ is four times that due to the error in the measurement of $l$.

ANSWER : A

1. A small block is connected to one end of a massless spring of un-stretched length 4.9 m . The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at $t = 0$. It then executes simple harmonic motion with angular frequency $\omega = \frac { \pi } { 3 } \mathrm { rad } / \mathrm { s }$. Simultaneously at $t = 0$, a small pebble is projected with speed $v$ from point $P$ at an angle of $45 ^ { \circ }$ as shown in the figure. Point $P$ is at a horizontal distance of 10 m from $O$. If the pebble hits the block at $t = 1 \mathrm {~s}$, the value of $v$ is (take $\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ ) [Figure]
   (A) $\sqrt { 50 } \mathrm {~m} / \mathrm { s }$
   (B) $\sqrt { 51 } \mathrm {~m} / \mathrm { s }$
   (C) $\sqrt { 52 } \mathrm {~m} / \mathrm { s }$
   (D) $\sqrt { 53 } \mathrm {~m} / \mathrm { s }$

ANSWER : A

1. Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are $\beta _ { G } , \beta _ { R }$ and $\beta _ { B }$, respectively. Then,
   (A) $\beta _ { G } > \beta _ { B } > \beta _ { R }$
   (B) $\beta _ { B } > \beta _ { G } > \beta _ { R }$
   (C) $\beta _ { R } > \beta _ { B } > \beta _ { G }$
   (D) $\beta _ { R } > \beta _ { G } > \beta _ { B }$

ANSWER: D

PHYSICS

1. A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x - y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by $\vec { L } _ { O }$ and $\vec { L } _ { P }$ respectively, then [Figure]
   (A) $\vec { L } _ { O }$ and $\vec { L } _ { P }$ do not vary with time.
   (B) $\vec { L } _ { O }$ varies with time while $\vec { L } _ { P }$ remains constant.
   (C) $\vec { L } _ { O }$ remains constant while $\vec { L } _ { P }$ varies with time.
   (D) $\vec { L } _ { O }$ and $\vec { L } _ { P }$ both vary with time.

ANSWER : C

8. A mixture of 2 moles of helium gas (atomic mass $= 4 \mathrm { amu }$ ) and 1 mole of argon gas (atomic mass $= 40 \mathrm { amu }$ ) is kept at 300 K in a container. The ratio of the rms speeds $\left( \frac { v _ { r m s } ( \text { helium } ) } { v _ { r m s } ( \text { argon } ) } \right)$ is
(A) 0.32
(B) 0.45
(C) 2.24
(D) 3.16

ANSWER : D

1. Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage source of potential difference $X$. A proton is released at rest midway between the two plates. It is found to move at $45 ^ { \circ }$ to the vertical JUST after release. Then $X$ is nearly
   (A) $1 \times 10 ^ { - 5 } \mathrm {~V}$
   (B) $1 \times 10 ^ { - 7 } \mathrm {~V}$
   (C) $1 \times 10 ^ { - 9 } \mathrm {~V}$
   (D) $1 \times 10 ^ { - 10 } \mathrm {~V}$

ANSWER : C

1. A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index $n$ of the first lens is 1.5 and that of the second lens is 1.2. Both the curved surfaces are of the same radius of curvature $R = 14 \mathrm {~cm}$. For this bi-convex lens, for an object distance of 40 cm , the image distance will be [Figure]
   (A) - 280.0 cm .
   (B) 40.0 cm .
   (C) 21.5 cm .
   (D) 13.3 cm .

ANSWER : B

SECTION II : Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

9. For light incident from air on a meta-material, the appropriate ray diagram is
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]

1. Choose the correct statement.
   (A) The speed of light in the meta-material is $v = c | n |$
   (B) The speed of light in the meta-material is $v = \frac { c } { | n | }$
   (C) The speed of light in the meta-material is $v = \mathrm { c }$.
   (D) The wavelength of the light in the meta-material $\left( \lambda _ { m } \right)$ is given by $\lambda _ { m } = \lambda _ { \text {air } } | n |$, where $\lambda _ { \text {air } }$ is the wavelength of the light in air.

ANSWER : B

Paragraph for Questions 11 and 12

The $\beta$-decay process, discovered around 1900, is basically the decay of a neutron ( $n$ ). In the laboratory, a proton ( $p$ ) and an electron ( $e ^ { - }$) are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has a continuous spectrum. Considering a three-body decay process, i.e. $n \rightarrow p + e ^ { - } + \bar { v } _ { e }$, around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino $\left( \bar { v } _ { e } \right)$ to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is $0.8 \times 10 ^ { 6 } \mathrm { eV }$. The kinetic energy carried by the proton is only the recoil energy.

11. A cubical region of side $a$ has its centre at the origin. It encloses three fixed point charges, $- q$ at $( 0 , - a / 4,0 ) , + 3 q$ at $( 0,0,0 )$ and $- q$ at $( 0 , + a / 4,0 )$. Choose the correct option(s). [Figure]
(A) The net electric flux crossing the plane $x = + a / 2$ is equal to the net electric flux crossing the plane $x = - a / 2$.
(B) The net electric flux crossing the plane $y = + a / 2$ is more than the net electric flux crossing the plane $y = - a / 2$.
(C) The net electric flux crossing the entire region is $\frac { q } { \varepsilon _ { 0 } }$.
(D) The net electric flux crossing the plane $z = + a / 2$ is equal to the net electric flux crossing the plane $x = + a / 2$.
ANSWER : ACD

11. What is the maximum energy of the anti-neutrino?
(A) Zero.
(B) Much less than $0.8 \times 10 ^ { 6 } \mathrm { eV }$.
(C) Nearly $0.8 \times 10 ^ { 6 } \mathrm { eV }$.
(D) Much larger than $0.8 \times 10 ^ { 6 } \mathrm { eV }$.

ANSWER : C

1. If the anti-neutrino had a mass of $3 \mathrm { eV } / \mathrm { c } ^ { 2 }$ (where c is the speed of light) instead of zero mass, what should be the range of the kinetic energy, $K$, of the electron?
   (A) $0 \leq K \leq 0.8 \times 10 ^ { 6 } \mathrm { eV }$
   (B) $3.0 \mathrm { eV } \leq K \leq 0.8 \times 10 ^ { 6 } \mathrm { eV }$
   (C) $3.0 \mathrm { eV } \leq K < 0.8 \times 10 ^ { 6 } \mathrm { eV }$
   (D) $0 \leq K < 0.8 \times 10 ^ { 6 } \mathrm { eV }$

ANSWER : D

Paragraph for Questions 13 and 14

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed $\omega$, the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the $z$-axis, and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points $P$ and $Q$ ). Both these motions have the same angular speed $\omega$ in this case. [Figure]
Now consider two similar systems as shown in the figure: Case (a) the disc with its face vertical and parallel to $x - z$ plane; Case (b) the disc with its face making an angle of $45 ^ { \circ }$ with $x - y$ plane and its horizontal diameter parallel to $x$-axis. In both the cases, the disc is welded at point $P$, and the systems are rotated with constant angular speed $\omega$ about the $z$-axis.
[Figure]
r
Case (a)
[Figure]
Case (b)

1. Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?
   (A) It is vertical for both the cases (a) and (b).
   (B) It is vertical for case (a); and is at $45 ^ { \circ }$ to the $x - z$ plane and lies in the plane of the disc for case (b).
   (C) It is horizontal for case (a); and is at $45 ^ { \circ }$ to the $x$ - $z$ plane and is normal to the plane of the disc for case (b).
   (D) It is vertical for case (a); and is at $45 ^ { \circ }$ to the $x$ - $z$ plane and is normal to the plane of the disc for case (b).

ANSWER:A

1. Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?
   (A) It is $\sqrt { 2 } \omega$ for both the cases.
   (B) It is $\omega$ for case (a); and $\frac { \omega } { \sqrt { 2 } }$ for case (b).
   (C) It is $\omega$ for case (a); and $\sqrt { 2 } \omega$ for case (b).
   (D) It is $\omega$ for both the cases.

ANSWER : D

SECTION III : Multiple Correct Answer(s) Type

This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

12. For the resistance network shown in the figure, choose the correct option(s). [Figure]
(A) The current through $P Q$ is zero.
(B) $I _ { 1 } = 3 \mathrm {~A}$.
(C) The potential at $S$ is less than that at $Q$.
(D) $I _ { 2 } = 2 \mathrm {~A}$.

ANSWER : ABCD

1. A small block of mass of 0.1 kg lies on a fixed inclined plane $P Q$ which makes an angle $\theta$ with the horizontal. A horizontal force of 1 N acts on the block through its center of mass as shown in the figure. The block remains stationary if (take $\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ ) [Figure]
   (A) $\theta = 45 ^ { \circ }$
   (B) $\theta > 45 ^ { \circ }$ and a frictional force acts on the block towards $P$.
   (C) $\theta > 45 ^ { \circ }$ and a frictional force acts on the block towards $Q$.
   (D) $\theta < 45 ^ { \circ }$ and a frictional force acts on the block towards $Q$.

ANSWER : AC

14. Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec { E } = E _ { 0 } \hat { j }$ and $\vec { B } = B _ { 0 } \hat { j }$. At time $t = 0$, this charge has velocity $\vec { v }$ in the $x - y$ plane, making an angle $\theta$ with the $x$-axis. Which of the following option(s) is(are) correct for time $t > 0$ ?
(A) If $\theta = 0 ^ { \circ }$, the charge moves in a circular path in the $x - z$ plane.
(B) If $\theta = 0 ^ { \circ }$, the charge undergoes helical motion with constant pitch along the $y$-axis.
(C) If $\theta = 10 ^ { \circ }$, the charge undergoes helical motion with its pitch increasing with time, along the $y$-axis.
(D) If $\theta = 90 ^ { \circ }$, the charge undergoes linear but accelerated motion along the $y$-axis.

ANSWER : CD

1. A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe,
   (A) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is open.
   (B) a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is open.
   (C) a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed.
   (D) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed.

ANSWER : BD

SECTION III: Integer Answer Type

This section contains $\mathbf { 5 }$ questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).

15. In the given circuit, the AC source has $\omega = 100 \mathrm { rad } / \mathrm { s }$. Considering the inductor and capacitor to be ideal, the correct choice(s) is(are) [Figure]
(A) The current through the circuit, $I$ is 0.3 A .
(B) The current through the circuit, $I$ is $0.3 \sqrt { 2 } \mathrm {~A}$.
(C) The voltage across $100 \Omega$ resistor $= 10 \sqrt { 2 } \mathrm {~V}$.
(D) The voltage across $50 \Omega$ resistor $= 10 \mathrm {~V}$.

ANSWER : C or AC

1. A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement(s) is(are)
   (A) The emf induced in the loop is zero if the current is constant.
   (B) The emf induced in the loop is finite if the current is constant.
   (C) The emf induced in the loop is zero if the current decreases at a steady rate.
   (D) The emf induced in the loop is finite if the current decreases at a steady rate.

ANSWER : AC

1. Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Given that $K = \frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { q } { L ^ { 2 } }$, which of the following statement(s) is(are) correct? [Figure]
   (A) The electric field at $O$ is 6 K along $O D$.
   (B) The potential at $O$ is zero.
   (C) The potential at all points on the line $P R$ is same.
   (D) The potential at all points on the line $S T$ is same.

ANSWER : ABC

1. Two solid cylinders $P$ and $Q$ of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder $P$ has most of its mass concentrated near its surface, while $Q$ has most of its mass concentrated near the axis. Which statement(s) is(are) correct ?
   (A) Both cylinders $P$ and $Q$ reach the ground at the same time.
   (B) Cylinder $P$ has larger linear acceleration than cylinder $Q$.
   (C) Both cylinders reach the ground with same translational kinetic energy.
   (D) Cylinder $Q$ reaches the ground with larger angular speed.

ANSWER : D

1. Two spherical planets $P$ and $Q$ have the same uniform density $\rho$, masses $M _ { P }$ and $M _ { Q ^ { \prime } }$, and surface areas $A$ and 4A, respectively. A spherical planet $R$ also has uniform density $\rho$ and its mass is $\left( M _ { P } + M _ { Q } \right)$. The escape velocities from the planets $P , Q$ and $R$, are $V _ { P } , V _ { Q }$ and $V _ { R }$, respectively. Then
   (A) $V _ { Q } > V _ { R } > V _ { P }$
   (B) $V _ { R } > V _ { Q } > V _ { P }$
   (C) $V _ { R } / V _ { P } = 3$
   (D) $V _ { P } / V _ { Q } = \frac { 1 } { 2 }$
2. The figure shows a system consisting of (i) a ring of outer radius $3 R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius $2 R$ rotating anti-clockwise with angular speed $\omega / 2$. The ring and disc are separated by frictionless ball bearings. The system is in the $x - z$ plane. The point $P$ on the inner disc is at a distance $R$ from the origin, where $O P$ makes an angle of $30 ^ { \circ }$ with the horizontal. Then with respect to the horizontal surface, [Figure]
   (A) the point $O$ has a linear velocity $3 R \omega \hat { i }$.
   (B) the point $P$ has a linear velocity $\frac { 11 } { 4 } R \omega \hat { i } + \frac { \sqrt { 3 } } { 4 } R \omega \hat { k }$.
   (C) the point $P$ has a linear velocity $\frac { 13 } { 4 } R \omega \hat { i } - \frac { \sqrt { 3 } } { 4 } R \omega \hat { k }$.
   (D) the point $P$ has a linear velocity $\left( 3 - \frac { \sqrt { 3 } } { 4 } \right) R \omega \hat { i } + \frac { 1 } { 4 } R \omega \hat { k }$.

ANSWER : AB

PART II: CHEMISTRY

SECTION 1 : Single Correct Answer Type

This section contains $\mathbf { 8 }$ multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

16. An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of radius $R / 2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a distance $2 R$ from the axis of the cylinder, is given by the expression $\frac { 23 \rho R } { 16 k \varepsilon _ { 0 } }$. The value of $k$ is [Figure]

ANSWER : 6

1. A cylindrical cavity of diameter $a$ exists inside a cylinder of diameter $2 a$ as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density $J$ flows along the length. If the magnitude of the magnetic field at the point $P$ is given by $\frac { N } { 12 } \mu _ { 0 } a J$, then the value of $N$ is

[Figure]
ANSWER : 5

1. A lamina is made by removing a small disc of diameter $2 R$ from a bigger disc of uniform mass density and radius $2 R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I _ { O }$ and $I _ { P }$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac { I _ { P } } { I _ { O } }$ to the nearest integer is [Figure]

ANSWER : 3

1. A circular wire loop of radius $R$ is placed in the $x - y$ plane centered at the origin $O$. A square loop of side $a ( a \ll R )$ having two turns is placed with its center at $z = \sqrt { 3 } R$ along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of $45 ^ { \circ }$ with respect to the $z$-axis. If the mutual inductance between the loops is given by $\frac { \mu _ { 0 } a ^ { 2 } } { 2 ^ { p / 2 } R }$, then the value of $p$ is

[Figure]
ANSWER : 7

1. A proton is fired from very far away towards a nucleus with charge $Q = 120 e$, where $e$ is the electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm ) of the proton at its start is: (take the proton mass, $m _ { p } = ( 5 / 3 ) \times 10 ^ { - 27 } \mathrm {~kg}$; $\mathrm { h } / e = 4.2 \times 10 ^ { - 15 } \mathrm {~J} . \mathrm { s } / \mathrm { C } ; \frac { 1 } { 4 \pi \varepsilon _ { 0 } } = 9 \times 10 ^ { 9 } \mathrm {~m} / \mathrm { F } ; 1 \mathrm { fm } = 10 ^ { - 15 } \mathrm {~m}$ )

PART II : CHEMISTRY

SECTION 1 : Single Correct Answer Type

This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

21. In allene $\left( \mathrm { C } _ { 3 } \mathrm { H } _ { 4 } \right)$, the type(s) of hybridisation of the carbon atoms is (are)
(A) $s p$ and $s p ^ { 3 }$
(B) $s p$ and $s p ^ { 2 }$
(C) only $s p ^ { 2 }$
(D) $s p ^ { 2 }$ and $s p ^ { 3 }$

ANSWER : B

1. For one mole of a van der Waals gas when $b = 0$ and $T = 300 \mathrm {~K}$, the $P V$ vs. $1 / \mathrm { V }$ plot is shown below. The value of the van der Waals constant $a \left( \mathrm {~atm} . \mathrm { liter } ^ { 2 } \mathrm {~mol} ^ { - 2 } \right)$ is [Figure]
   (A) 1.0
   (B) 4.5
   (C) 1.5
   (D) 3.0

ANSWER: C

1. The number of optically active products obtained from the complete ozonolysis of the given compound is [Figure]
   (A) 0
   (B) 1
   (C) 2
   (D) 4

ANSWER : A

1. A compound $\mathrm { M } _ { \mathrm { p } } \mathrm { X } _ { \mathrm { q } }$ has cubic close packing (ccp) arrangement of X . Its unit cell structure is shown below. The empirical formula of the compound is [Figure]
   (A) MX
   (B) $\mathrm { MX } _ { 2 }$
   (C) $M _ { 2 } X$
   (D) $M _ { 5 } X _ { 14 }$

ANSWER : B

1. The number of aldol reaction(s) that occurs in the given transformation is [Figure]
   (A) 1
   (B) 2
   (C) 3
   (D) 4
2. The colour of light absorbed by an aqueous solution of $\mathrm { CuSO } _ { 4 }$ is
   (A) orange-red
   (B) blue-green
   (C) yellow
   (D) violet

ANSWER : A

1. The carboxyl functional group $( - \mathrm { COOH } )$ is present in
   (A) picric acid
   (B) barbituric acid
   (C) ascorbic acid
   (D) aspirin

ANSWER : D

1. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [ $a _ { 0 }$ is Bohr radius]
   (A) $\frac { \mathrm { h } ^ { 2 } } { 4 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$
   (B) $\frac { \mathrm { h } ^ { 2 } } { 16 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$
   (C) $\frac { \mathrm { h } ^ { 2 } } { 32 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$
   (D) $\frac { \mathrm { h } ^ { 2 } } { 64 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$

ANSWER: C

1. Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?
   (A) $\mathrm { HNO } _ { 3 } , \mathrm { NO } , \mathrm { NH } _ { 4 } \mathrm { Cl } , \mathrm { N } _ { 2 }$
   (B) $\mathrm { HNO } _ { 3 } , \mathrm { NO } , \mathrm { N } _ { 2 } , \mathrm { NH } _ { 4 } \mathrm { Cl }$
   (C) $\mathrm { HNO } _ { 3 } , \mathrm { NH } _ { 4 } \mathrm { Cl } , \mathrm { NO } , \mathrm { N } _ { 2 }$
   (D) $\mathrm { NO } , \mathrm { HNO } _ { 3 } , \mathrm { NH } _ { 4 } \mathrm { Cl } , \mathrm { N } _ { 2 }$

ANSWER : B

1. As per IUPAC nomenclature, the name of the complex $\left[ \mathrm { Co } \left( \mathrm { H } _ { 2 } \mathrm { O } \right) _ { 4 } \left( \mathrm { NH } _ { 3 } \right) _ { 2 } \right] \mathrm { Cl } _ { 3 }$ is
   (A) Tetraaquadiaminecobalt (III) chloride
   (B) Tetraaquadiamminecobalt (III) chloride
   (C) Diaminetetraaquacobalt (III) chloride
   (D) Diamminetetraaquacobalt (III) chloride

ANSWER: D

SECTION II : Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

21. $\mathrm { NiCl } _ { 2 } \left\{ \mathrm { P } \left( \mathrm { C } _ { 2 } \mathrm { H } _ { 5 } \right) _ { 2 } \left( \mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \right) \right\} _ { 2 }$ exhibits temperature dependent magnetic behaviour (paramagnetic/ diamagnetic). The coordination geometries of $\mathrm { Ni } ^ { 2 + }$ in the paramagnetic and diamagnetic states are respectively
(A) tetrahedral and tetrahedral
(B) square planar and square planar
(C) tetrahedral and square planar
(D) square planar and tetrahedral

ANSWER : C

1. In the cyanide extraction process of silver from argentite ore, the oxidizing and reducing agents used are
   (A) $\mathrm { O } _ { 2 }$ and CO respectively.
   (B) $\mathrm { O } _ { 2 }$ and Zn dust respectively.
   (C) $\mathrm { HNO } _ { 3 }$ and Zn dust respectively.
   (D) $\mathrm { HNO } _ { 3 }$ and CO respectively.

ANSWER : B

1. The reaction of white phosphorus with aqueous NaOH gives phosphine along with another phosphorus containing compound. The reaction type; the oxidation states of phosphorus in phosphine and the other product are respectively
   (A) redox reaction; - 3 and - 5
   (B) redox reaction; + 3 and + 5
   (C) disproportionation reaction; - 3 and + 5
   (D) disproportionation reaction; - 3 and + 3

Zero Marks to all

24. The shape of $\mathrm { XeO } _ { 2 } \mathrm {~F} _ { 2 }$ molecule is
(A) trigonal bipyramidal
(B) square planar
(C) tetrahedral
(D) see-saw

ANSWER : D

1. For a dilute solution containing 2.5 g of a non-volatile non-electrolyte solute in 100 g of water, the elevation in boiling point at 1 atm pressure is $2 ^ { \circ } \mathrm { C }$. Assuming concentration of solute is much lower than the concentration of solvent, the vapour pressure ( mm of Hg ) of the solution is (take $K _ { \mathrm { b } } = 0.76 \mathrm {~K} \mathrm {~kg} \mathrm {~mol} ^ { - 1 }$ )
   (A) 724
   (B) 740
   (C) 736
   (D) 718

ANSWER:A

1. The compound that undergoes decarboxylation most readily under mild condition is

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER : B

27. Using the data provided, calculate the multiple bond energy ( $\mathrm { kJ } \mathrm { mol } ^ { - 1 }$ ) of a $\mathrm { C } \equiv \mathrm { C }$ bond in $\mathrm { C } _ { 2 } \mathrm { H } _ { 2 }$. That energy is (take the bond energy of a $\mathrm { C } - \mathrm { H }$ bond as $350 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 }$.) $2 \mathrm { C } ( \mathrm { s } ) + \mathrm { H } _ { 2 } ( \mathrm {~g} ) \longrightarrow \mathrm { C } _ { 2 } \mathrm { H } _ { 2 } ( \mathrm {~g} )$
$$\begin{aligned} & \Delta H = 225 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \\ & \Delta H = 1410 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \\ & \Delta H = 330 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \end{aligned}$$
(A) 1165
(B) 837
(C) 865
(D) 815

ANSWER : D

1. The major product $\mathbf { H }$ of the given reaction sequence is [Figure]

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER : A

SECTION II : Paragraph Type

This section contains $\mathbf { 6 }$ multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Questions 29 and 30

Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.

29. Bleaching powder contains a salt of an oxoacid as one of its components. The anhydride of that oxoacid is
(A) $\mathrm { Cl } _ { 2 } \mathrm { O }$
(B) $\mathrm { Cl } _ { 2 } \mathrm { O } _ { 7 }$
(C) $\mathrm { ClO } _ { 2 }$
(D) $\mathrm { Cl } _ { 2 } \mathrm { O } _ { 6 }$

ANSWER : A

1. 25 mL of household bleach solution was mixed with 30 mL of 0.50 M KI and 10 mL of 4 N acetic acid. In the titration of the liberated iodine, 48 mL of $0.25 \mathrm {~N} \mathrm { Na } _ { 2 } \mathrm {~S} _ { 2 } \mathrm { O } _ { 3 }$ was used to reach the end point. The molarity of the household bleach solution is
   (A) 0.48 M
   (B) 0.96 M
   (C) 0.24 M
   (D) 0.024 M

ANSWER : C

Paragraph for Questions 31 and 32

The electrochemical cell shown below is a concentration cell. $\mathrm { M } \mid \mathrm { M } ^ { 2 + }$ (saturated solution of a sparingly soluble salt, $\left. \mathrm { MX } _ { 2 } \right) \left| \left| \mathrm { M } ^ { 2 + } \left( 0.001 \mathrm {~mol} \mathrm { dm } ^ { - 3 } \right) \right| \mathrm { M } \right.$ The emf of the cell depends on the difference in concentrations of $\mathrm { M } ^ { 2 + }$ ions at the two electrodes. The emf of the cell at 298 K is 0.059 V .

31. Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction, as shown in the given scheme. [Figure]
(A) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { OH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { COOH }$
(B) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { COOH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { OH }$
(C) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { OH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { OH }$
(D) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { OH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { COOH }$

ANSWER : BD

1. Choose the correct reason(s) for the stability of the lyophobic colloidal particles.
   (A) Preferential adsorption of ions on their surface from the solution
   (B) Preferential adsorption of solvent on their surface from the solution
   (C) Attraction between different particles having opposite charges on their surface
   (D) Potential difference between the fixed layer and the diffused layer of opposite charges around the colloidal particles

ANSWER : AD

1. Which of the following molecules, in pure form, is (are) unstable at room temperature ?

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER : BC

31. The solubility product $\left( K _ { s p } ; \mathrm { mol } ^ { 3 } \mathrm { dm } ^ { - 9 } \right)$ of $\mathrm { MX } _ { 2 }$ at 298 K based on the information available for the given concentration cell is (take $2.303 \times \mathrm { R } \times 298 / \mathrm { F } = 0.059 \mathrm {~V}$ )
(A) $1 \times 10 ^ { - 15 }$
(B) $4 \times 10 ^ { - 15 }$
(C) $1 \times 10 ^ { - 12 }$
(D) $4 \times 10 ^ { - 12 }$

ANSWER : B

1. The value of $\Delta G \left( \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \right)$ for the given cell is (take $\left. 1 \mathrm {~F} = 96500 \mathrm { C } \mathrm { mol } ^ { - 1 } \right)$
   (A) - 5.7
   (B) 5.7
   (C) 11.4
   (D) - 11.4

ANSWER : D

Paragraph for Questions 33 and 34

In the following reaction sequence, the compound $\mathbf { J }$ is an intermediate. [Figure]
J $\left( \mathrm { C } _ { 9 } \mathrm { H } _ { 8 } \mathrm { O } _ { 2 } \right)$ gives effervescence on treatment with $\mathrm { NaHCO } _ { 3 }$ and a positive Baeyer's test.

33. The compound I is
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]

ANSWER : A

1. The compound $\mathbf { K }$ is
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]

ANSWER : C

SECTION III : Multiple Correct Answer(s) Type

This section contains $\mathbf { 6 }$ multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

34. Which of the following hydrogen halides react(s) with $\mathrm { AgNO } _ { 3 } ( \mathrm { aq } )$ to give a precipitate that dissolves in $\mathrm { Na } _ { 2 } \mathrm {~S} _ { 2 } \mathrm { O } _ { 3 } ( \mathrm { aq } )$ ?
(A) HCl
(B) HF
(C) HBr
(D) HI

ANSWER : ACD

1. For an ideal gas, consider only $P$ - $V$ work in going from an initial state $X$ to the final state $Z$. The final state $\boldsymbol { Z }$ can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct ? [take $\Delta S$ as change in entropy and $w$ as work done] [Figure]
   (A) $\Delta S _ { \mathrm { x } \rightarrow \mathrm { z } } = \Delta S _ { \mathrm { x } \rightarrow \mathrm { y } } + \Delta S _ { \mathrm { y } \rightarrow \mathrm { z } }$
   (B) $w _ { \mathrm { x } \rightarrow \mathrm { z } } = w _ { \mathrm { x } \rightarrow \mathrm { y } } + w _ { \mathrm { y } \rightarrow \mathrm { z } }$
   (C) $w _ { x \rightarrow y \rightarrow z } = w _ { x \rightarrow y }$
   (D) $\Delta S _ { \mathrm { x } \rightarrow \mathrm { y } \rightarrow \mathrm { z } } = \Delta S _ { \mathrm { x } \rightarrow \mathrm { y } }$

ANSWER : AC

SECTION III : Integer Answer Type

This section contains $\mathbf { 5 }$ questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).

35. The reversible expansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the following statement(s) is (are) correct? [Figure]
(A) $T _ { 1 } = T _ { 2 }$
(B) $T _ { 3 } > T _ { 1 }$
(C) $w _ { \text {isothermal } } > w _ { \text {adiabatic } }$
(D) $\Delta U _ { \text {isothermal } } > \Delta U _ { \text {adiabatic } }$
ANSWER : ACD or AD

36. The substituents $\mathbf { R } _ { \mathbf { 1 } }$ and $\mathbf { R } _ { \mathbf { 2 } }$ for nine peptides are listed in the table given below. How many of these peptides are positively charged at $\mathrm { pH } = 7.0$ ? [Figure]

Peptide	$\mathbf { R } _ { \mathbf { 1 } }$	$\mathbf { R } _ { \mathbf { 2 } }$
1	H	H
II	H	$\mathrm { CH } _ { 3 }$
III	$\mathrm { CH } _ { 2 } \mathrm { COOH }$	H
IV	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$
V	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$
VI	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$
VII	$\mathrm { CH } _ { 2 } \mathrm { COOH }$	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$
VIII	$\mathrm { CH } _ { 2 } \mathrm { OH }$	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$
IX	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$	$\mathrm { CH } _ { 3 }$

ANSWER : 4

36. The given graphs / data I, II, III and IV represent general trends observed for different physisorption and chemisorption processes under mild conditions of temperature and pressure. Which of the following choice(s) about I, II, III and IV is (are) correct? [Figure]
(A) I is physisorption and II is chemisorption
(B) I is physisorption and III is chemisorption
(C) IV is chemisorption and II is chemisorption
(D) IV is chemisorption and III is chemisorption

ANSWER : AC

1. For the given aqueous reactions, which of the statement(s) is (are) true ? [Figure]
   (A) The first reaction is a recux reaction.
   (B) White precipitate is $\mathrm { Zn } _ { 3 } \left[ \mathrm { Fe } ( \mathrm { CN } ) _ { 6 } \right] _ { 2 }$.
   (C) Addition of filtrate to starch solution gives blue colour.
   (D) White precipitate is soluble in NaOH solution.

ANSWER : ACD

1. With respect to graphite and diamond, which of the statement(s) given below is (are) correct ?
   (A) Graphite is harder than diamond.
   (B) Graphite has higher electrical conductivity than diamond.
   (C) Graphite has higher thermal conductivity than diamond.
   (D) Graphite has higher C-C bond order than diamond.

ANSWER : BD

1. With reference to the scheme given, which of the given statment(s) about $\mathbf { T } , \mathbf { U } , \mathbf { V }$ and $\mathbf { W }$ is (are) correct ? [Figure]
   (A) T is soluble in hot aqueous NaOH
   (B) $\mathbf { U }$ is optically active
   (C) Molecular formula of $\mathbf { W }$ is $\mathrm { C } _ { 10 } \mathrm { H } _ { 18 } \mathrm { O } _ { 4 }$
   (D) $\mathbf { V }$ gives effervescence on treatment with aqueous $\mathrm { NaHCO } _ { 3 }$

ANSWER : ACD