Match the statements/expressions given in Column I with the values given in Column II.
Column I
(A) Root(s) of the equation $$2\sin^{2}\theta+\sin^{2}2\theta=2$$ (B) Points of discontinuity of the function $$f(x)=\left[\frac{6x}{\pi}\right]\cos\left[\frac{3x}{\pi}\right],$$ where $[y]$ denotes the largest integer less than or equal to $y$
(C) Volume of the parallelopiped with its edges represented by the vectors $$\hat{i}+\hat{j},\quad\hat{i}+2\hat{j}\text{ and }\hat{i}+\hat{j}+\pi\hat{k}$$ (D) Angle between vectors $\vec{a}$ and $\vec{b}$ where $\vec{a},\vec{b}$ and $\vec{c}$ are unit vectors satisfying $$\vec{a}+\vec{b}+\sqrt{3}\vec{c}=\overrightarrow{0}$$
Column II
(p) $\frac{\pi}{6}$
(q) $\frac{\pi}{4}$
(r) $\frac{\pi}{3}$
(s) $\frac{\pi}{2}$
(t) $\pi$

Match the statements/expressions given in Column I with the values given in Column II.
Column I
(A) The number of solutions of the equation $$xe^{\sin x}-\cos x=0$$ in the interval $\left(0,\frac{\pi}{2}\right)$
(B) Value(s) of $k$ for which the planes $kx+4y+z=0,4x+ky+2z=0$ and $2x+2y+z=0$ intersect in a straight line
(C) Value(s) of $k$ for which $$|x-1|+|x-2|+|x+1|+|x+2|=4k$$ has integer solution(s)
(D) If $$y^{\prime}=y+1\text{ and }y(0)=1$$ then value(s) of $y(\ln2)$
Column II
(p) 1
(q) 2
(r) 3
(s) 4
(t) 5

5. $\quad \mathrm { AgNO } _ { 3 }$ (aq.) was added to an aqueous KCl solution gradually and the conductivity of the solution was measured. The plot of conductance ( $\Lambda$ ) versus the volume of $\mathrm { AgNO } _ { 3 }$ is
[Figure]
volume
(P)
[Figure]
volume (Q)
[Figure]
volume
(R) [Figure] volume (S)
(A) (P)
(B) (Q)
(C) $( \mathrm { R } )$
(D) (S)
ANSWER: D

6. Among the following compounds, the most acidic is
(A) $p$-nitrophenol
(B) p-hydroxybenzoic acid
(C) o-hydroxybenzoic acid
(D) $p$-toluic acid
ANSWER: C

7. The major product of the following reaction is [Figure]
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER:A

SECTION - II (Total Marks : 16)

(Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE may be correct.

8. Extraction of metal from the ore cassiterite involves
(A) carbon reduction of an oxide ore
(B) self-reduction of a sulphide ore
(C) removal of copper impurity
(D) removal of iron impurity

ANSWER: AD

1. The correct statement(s) pertaining to the adsorption of a gas on a solid surface is (are)
   (A) Adsorption is always exothermic.
   (B) Physisorption may transform into chemisorption at high temperature.
   (C) Physisorption increases with increasing temperature but chemisorption decreases with increasing temperature.
   (D) Chemisorption is more exothermic than physisorption, however it is very slow due to higher energy of activation.

ANSWER: ABD

1. According to kinetic theory of gases
   (A) collisions are always elastic.
   (B) heavier molecules transfer more momentum to the wall of the container.
   (C) only a small number of molecules have very high velocity.
   (D) between collisions, the molecules move in straight lines with constant velocities.

ANSWER: ACD

1. Amongst the given options, the compound(s) in which all the atoms are in one plane in all the possible conformations (if any), is (are)

(A) [Figure]
(B) [Figure]
(C) $\mathrm { H } _ { 2 } \mathrm { C } = \mathrm { C } = \mathrm { O }$
(D) $\mathrm { H } _ { 2 } \mathrm { C } = \mathrm { C } = \mathrm { CH } _ { 2 }$
ANSWER: BC

SECTION - III (Total Marks : 15)

(Paragraph Type)

This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph 2 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question Nos. 12 to 14

When a metal rod $\mathbf { M }$ is dipped into an aqueous colourless concentrated solution of compound $\mathbf { N }$, the solution turns light blue. Addition of aqueous NaCl to the blue solution gives a white precipitate $\mathbf { O }$. Addition of aqueous $\mathrm { NH } _ { 3 }$ dissolves $\mathbf { O }$ and gives an intense blue solution.

12. The metal rod $\mathbf { M }$ is
(A) Fe
(B) Cu
(C) Ni
(D) Co

ANSWER:B

1. The compound $\mathbf { N }$ is
   (A) $\mathrm { AgNO } _ { 3 }$
   (B) $\mathrm { Zn } \left( \mathrm { NO } _ { 3 } \right) _ { 2 }$
   (C) $\mathrm { Al } \left( \mathrm { NO } _ { 3 } \right) _ { 3 }$
   (D) $\mathrm { Pb } \left( \mathrm { NO } _ { 3 } \right) _ { 2 }$

ANSWER:A

14. The final solution contains
(A) $\left[ \mathrm { Pb } \left( \mathrm { NH } _ { 3 } \right) _ { 4 } \right] ^ { 2 + }$ and $\left[ \mathrm { CoCl } _ { 4 } \right] ^ { 2 - }$
(B) $\left[ \mathrm { Al } \left( \mathrm { NH } _ { 3 } \right) _ { 4 } \right] ^ { 3 + }$ and $\left[ \mathrm { Cu } \left( \mathrm { NH } _ { 3 } \right) _ { 4 } \right] ^ { 2 + }$
(C) $\left[ \mathrm { Ag } \left( \mathrm { NH } _ { 3 } \right) _ { 2 } \right] ^ { + }$and $\left[ \mathrm { Cu } \left( \mathrm { NH } _ { 3 } \right) _ { 4 } \right] ^ { 2 + }$
(D) $\left[ \mathrm { Ag } \left( \mathrm { NH } _ { 3 } \right) _ { 2 } \right] ^ { + }$and $\left[ \mathrm { Ni } \left( \mathrm { NH } _ { 3 } \right) _ { 6 } \right] ^ { 2 + }$

ANSWER: C

Paragraph for Question Nos. 15 and 16

An acyclic hydrocarbon $\mathbf { P }$, having molecular formula $\mathrm { C } _ { 6 } \mathrm { H } _ { 10 }$, gave acetone as the only organic product through the following sequence of reactions, in which $\mathbf { Q }$ is an intermediate organic compound.
(i) conc. $\mathrm { H } _ { 2 } \mathrm { SO } _ { 4 }$ (catalytic amount) $\left( \mathrm { C } _ { 6 } \mathrm { H } _ { 10 } \right)$ [Figure] [Figure] [Figure]
(ii) $\mathrm { NaBH } _ { 4 }$ /ethanol
(iii) dil. acid
(ii) $\mathrm { O } _ { 3 }$
(iii) $\mathrm { Zn } / \mathrm { H } _ { 2 } \mathrm { O }$

15. The structure of compound $\mathbf { P }$ is
(A) $\mathrm { CH } _ { 3 } \mathrm { CH } _ { 2 } \mathrm { CH } _ { 2 } \mathrm { CH } _ { 2 } - \mathrm { C } \equiv \mathrm { C } - \mathrm { H }$
(B)
(C) [Figure]
(D) [Figure]
ANSWER: D

16. The structure of the compound $\mathbf { Q }$ is
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
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.

17. The difference in the oxidation numbers of the two types of sulphur atoms in $\mathrm { Na } _ { 2 } \mathrm {~S} _ { 4 } \mathrm { O } _ { 6 }$ is
ANSWER : 5

18. Reaction of $\mathrm { Br } _ { 2 }$ with $\mathrm { Na } _ { 2 } \mathrm { CO } _ { 3 }$ in aqueous solution gives sodium bromide and sodium bromate with evolution of $\mathrm { CO } _ { 2 }$ gas. The number of sodium bromide molecules involved in the balanced chemical equation is

ANSWER : 5

1. The maximum number of electrons that can have principal quantum number, $n = 3$, and spin quantum number, $m _ { \mathrm { s } } = - 1 / 2$, is

ANSWER: 9

20. The work function ( $\phi$ ) of some metals is listed below. The number of metals which will show photoelectric effect when light of 300 nm wavelength falls on the metal is

Metal	Li	Na	K	Mg	Cu	Ag	Fe	Pt	W
$\phi ( \mathrm { eV } )$	2.4	2.3	2.2	3.7	4.8	4.3	4.7	6.3	4.75

ANSWER: 4

1. To an evacuated vessel with movable piston under external pressure of $1 \mathrm {~atm} . , 0.1 \mathrm {~mol}$ of He and 1.0 mol of an unknown compound (vapour pressure 0.68 atm . at $0 ^ { \circ } \mathrm { C }$ ) are introduced. Considering the ideal gas behaviour, the total volume (in litre) of the gases at $0 ^ { \circ } \mathrm { C }$ is close to

ANSWER:7

1. The total number of alkenes possible by dehydrobromination of 3-bromo-3-cyclopentylhexane using alcoholic KOH is

ANSWER: 5

1. A decapeptide (Mol. Wt. 796) on complete hydrolysis gives glycine (Mol. Wt. 75), alanine and phenylalanine. Glycine contributes $47.0 \%$ to the total weight of the hydrolysed products. The number of glycine units present in the decapeptide is

ANSWER: 6

PART II: PHYSICS

SECTION-I (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.

24. A police car with a siren of frequency 8 kHz is moving with uniform velocity $36 \mathrm {~km} / \mathrm { hr }$ towards a tall building which reflects the sound waves. The speed of sound in air is $320 \mathrm {~m} / \mathrm { s }$. The frequency of the siren heard by the car driver is
(A) 8.50 kHz
(B) 8.25 kHz
(C) 7.75 kHz
(D) 7.50 kHz

ANSWER:A

1. 5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial temperature to be $T _ { 1 }$, the work done in the process is
   (A) $\frac { 9 } { 8 } R T _ { 1 }$
   (B) $\frac { 3 } { 2 } R T _ { 1 }$
   (C) $\frac { 15 } { 8 } R T _ { 1 }$
   (D) $\quad \frac { 9 } { 2 } R T _ { 1 }$

ANSWER:A

1. Consider an electric field $\vec { E } = E _ { 0 } \hat { x }$, where $E _ { 0 }$ is a constant. The flux through the shaded area (as shown in the figure) due to this field is [Figure]
   (A) $2 E _ { 0 } a ^ { 2 }$
   (B) $\sqrt { 2 } E _ { 0 } a ^ { 2 }$
   (C) $E _ { 0 } a ^ { 2 }$
   (D) $\frac { E _ { 0 } a ^ { 2 } } { \sqrt { 2 } }$

ANSWER: C

1. The wavelength of the first spectral line in the Balmer series of hydrogen atom is $6561 \AA$. The wavelength of the second spectral line in the Balmer series of singly-ionized helium atom is
   (A) $1215 \AA$
   (B) $1640 \AA$
   (C) $2430 \AA$
   (D) $4687 \AA$

ANSWER: A

31. A spherical metal shell $A$ of radius $R _ { A }$ and a solid metal sphere $B$ of radius $R _ { B } \left( < R _ { A } \right)$ are kept far apart and each is given charge ' $+ Q$ '. Now they are connected by a thin metal wire. Then
(A) $\quad E _ { A } ^ { \text {mide } } = 0$
(B) $Q _ { A } > Q _ { B }$
(C) $\frac { \sigma _ { A } } { \sigma _ { B } } = \frac { R _ { B } } { R _ { A } }$
(D) $E _ { A } ^ { \text {on surface } } < E _ { B } ^ { \text {on surface } }$

ANSWER: ABCD

1. An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?
   (A) They will never come out of the magnetic field region.
   (B) They will come out travelling along parallel paths.
   (C) They will come out at the same time.
   (D) They will come out at different times.

ANSWER : BC, BD, BCD

33. A composite block is made of slabs $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E of different thermal conductivities (given in terms of a constant K) and sizes (given in terms of length, L) as shown in the figure. All slabs are of same width. Heat 'Q' flows only from left to right through the blocks. Then in steady state [Figure]
(A) heat flow through A and E slabs are same.
(B) heat flow through slab $E$ is maximum.
(C) temperature difference across slab E is smallest.
(D) heat flow through $\mathrm { C } =$ heat flow through $\mathrm { B } +$ heat flow through D .

ANSWER: ACD

1. A metal rod of length ' $L$ ' and mass ' $m$ ' is pivoted at one end. A thin disk of mass ' $M$ ' and radius ' R ' ( $< \mathrm { L }$ ) is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true? [Figure]
   (A) Restoring torque in case $\mathrm { A } =$ Restoring torque in case B
   (B) Restoring torque in case $\mathrm { A } <$ Restoring torque in case B
   (C) Angular frequency for case $A >$ Angular frequency for case $B$.
   (D) Angular frequency for case $A <$ Angular frequency for case $B$.

SECTION - III (Total Marks : 15)

(Paragraph Type)

This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph $\mathbf { 2 }$ multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question Nos. 35 to 37

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is $x ( t )$ vs. $p ( t )$ curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative. [Figure]

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.