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QUESTION: 1

**SECTION I : Single Correct Answer Type**

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONLY ONE is correct.

**Q.** Two identical discs of same radius R are rotating about their axes in opposite directions with the same constant angular speed ω. The discs are in the same horizontal plane. At time t = 0, the points P and Q are facing each other as shown in the figure. The relative speed between the two points P and Q is v_{r}. In one time period (T) of rotation of the discs, vr as a function of time is best represented by

Solution:

In each rotation relative speed becomes zero twice and becomes maximum twice.

QUESTION: 2

A loop carrying current I lies in the x-y plane as shown in the figure. The unit vector is coming

out of the plane of the paper. The magnetic moment of the current loop is

Solution:

QUESTION: 3

An infinitely long hollow conducting cylinder with inner radius R/2 and outer radius R carries a uniform current density along its length. The magnitude of the magnetic field, as a function of the radial distance r from the axis is best represented by

Solution:

QUESTION: 4

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If ρ_{c} is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

Solution:

QUESTION: 5

In the given circuit, a charge of +80 μC is given to the upper plate of the 4 μF capacitor. Then in the steady state, the charge on the upper plate of the 3 μF capacitor is

Solution:

Let ‘q’ be the final charge on 3μF capacitor then

QUESTION: 6

Two moles of ideal helium gas are in a rubber balloon at 30°C. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to 35°C. The amount of heat required in raising the temperature is nearly (take R = 8.31 J/mol.K)

Solution:

QUESTION: 7

Consider a disc rotating in the horizontal plane with a constant angular speed w about its centre O. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an angle towards R. The velocity of projection is in the y-z plane and is same for both pebbles with respect to the disc. Assume that

(i) they land back on the disc before the disc has 1/8 completed rotation,

(ii) their range is less than half the disc radius, and

(iii) ω remains constant throughout. Then

Solution:

QUESTION: 8

A student is performing the experiment of resonance Column. The diameter of the column tube is 4 cm. The frequency of the tuning fork is 512 Hz. The air temperature is 38°C in which the speed of sound is 336 m/s. The zero of the meter scale coincides with the top end of the Resonance Column tube. When the first resonance occurs, the reading of the water level in the column is

Solution:

QUESTION: 9

**SECTION II : Paragraph Type**

This section contains 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 9 and 10**

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 ω, 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 ω in this case

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° 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 ω about the z-axis.

**Q.** Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

Solution:

QUESTION: 10

**Paragraph for Questions 9 and 10**

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 ω, 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 ω in this case

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° 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 ω about the z-axis.

**Q. **Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

Solution:

QUESTION: 11

**Paragraph for Questions 11 and 12**

The β-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 continuous spectrum. Considering a three-body decay process, i.e.

around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino 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 × 10^{6} eV. The kinetic energy carried by the proton is only the recoil energy.

**Q. **If the anti-neutrino had a mass of 3eV/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?

Solution:

QUESTION: 12

**Paragraph for Questions 11 and 12**

The β-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 continuous spectrum. Considering a three-body decay process, i.e.

around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino 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 × 10^{6} eV. The kinetic energy carried by the proton is only the recoil energy.

**Q.** What is the maximum energy of the anti-neutrino?

Solution:

QUESTION: 13

**Paragraph for Questions 13 and 14**

Most materials have the refractive index, n > 1. So, when a light ray from air enters a naturally occurring

material, then by Snell’s law, it is understood that the refracted ray bends towards the normal.

But it never emerges on the same side of the normal as the incident ray. According to electromagnetism,

the refractive index of the medium is given by the relation, where c is the speed of

electromagnetic waves in vacuum, v its speed in the medium, ε_{r} and μ_{r} are the relative permittivity and

permeability of the medium respectively.

In normal materials, both ε_{r} and μ_{r}, are positive, implying positive n for the medium. When both ε_{r} and μ_{r }are negative, one must choose the negative root of n. Such negative refractive index materials can now

be artificially prepared and are called meta-materials. They exhibit significantly different optical behavior,

without violating any physical laws. Since n is negative, it results in a change in the direction of

propagation of the refracted light. However, similar to normal materials, the frequency of light remains

unchanged upon refraction even in meta-materials.

**Q.** For light incident from air on a meta-material, the appropriate ray diagram is

Solution:

QUESTION: 14

**Paragraph for Questions 13 and 14**

Most materials have the refractive index, n > 1. So, when a light ray from air enters a naturally occurring

material, then by Snell’s law, it is understood that the refracted ray bends towards the normal.

But it never emerges on the same side of the normal as the incident ray. According to electromagnetism,

the refractive index of the medium is given by the relation, where c is the speed of

electromagnetic waves in vacuum, v its speed in the medium, ε_{r} and μ_{r} are the relative permittivity and

permeability of the medium respectively.

In normal materials, both ε_{r} and μ_{r}, are positive, implying positive n for the medium. When both ε_{r} and μ_{r }are negative, one must choose the negative root of n. Such negative refractive index materials can now

be artificially prepared and are called meta-materials. They exhibit significantly different optical behavior,

without violating any physical laws. Since n is negative, it results in a change in the direction of

propagation of the refracted light. However, similar to normal materials, the frequency of light remains

unchanged upon refraction even in meta-materials.

**Q. **Choose the correct statement.

Solution:

*Multiple options can be correct

QUESTION: 15

**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.

15. In the given circuit, the AC source has ω = 100 rad/s. Considering the inductor and capacitor to be ideal, the correct choice(s) is (are)

Solution:

*Multiple options can be correct

QUESTION: 16

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 , which of the following statement(s) is (are) correct?

Solution:

Line PR is perpendicular bisector of all

the dipoles.

*Multiple options can be correct

QUESTION: 17

Two spherical planets P and Q have the same uniform density r, masses M_{p} and M_{Q} and surface

areas A and 4A respectively. A spherical planet R also has uniform density ρ and its mass is (M_{P}

+ M_{Q}). The escape velocities from the planets P, Q and R are V_{P}, V_{Q} and V_{R}, respectively. Then

Solution:

By calculation, if Mass of P = m

and Radius of P = R

Then Mass of Q = 8M

and radius of Q = 2R

and Mass of R = 9M

and radius of R = 9^{1/3}R

*Multiple options can be correct

QUESTION: 18

The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed ω and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed ω/2. The ring and disc are separated by frictionless ball bearings. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of 30^{0} with the horizontal. Then with respect to the horizontal surface,

Solution:

*Multiple options can be correct

QUESTION: 19

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?

Solution:

*Multiple options can be correct

QUESTION: 20

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)

Solution:

QUESTION: 21

**SECTION I : Single Correct Answer Type**

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONLY ONE is correct.

**Q. **The major product H of the given reaction sequence is

Solution:

QUESTION: 22

NiCl_{2}{P(C_{2}H_{5})_{2}(C_{6}H_{5})}_{2} exhibits temperature dependent magnetic behaviour (paramagnetic/diamagnetic).

The coordination geometries of Ni^{2+} in the paramagnetic and diamagnetic states are respectively

Solution:

In both complexes Ni exists as Ni^{2+}.

In sp^{3} (tetrahedral)

In dsp^{2 }(square planar geometry)

QUESTION: 23

In the cyanide extraction process of silver from argentite ore, the oxidising and reducing agents used are

Solution:

The reactions involved in cyanide extraction process are:

QUESTION: 24

The reaction of white phosphorous with aqueous NaOH gives phosphine along with another phosphorous

containing compound. The reaction type; the oxidation states of phosphorus in phosphine and the other

product are respectively

Solution:

The balanced disproportionation reaction involving white phosphorus with aq. NaOH is

QUESTION: 25

The shape of XeO_{2}F_{2} molecule is

Solution:

Hybridization = sp^{3}d

Shape = see – saw

QUESTION: 26

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°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_{b} = 0.76 K kg mol^{-1})

Solution:

B → Solute; A → Solvent

W_{B} = 2.5 g, W_{A} = 100 g

QUESTION: 27

The compound that undergoes decarboxylation most readily under mild condition is

Solution:

QUESTION: 28

Using the data provided, calculate the multiple bond energy (kJ mol^{-1}) of a CºC bond in C_{2}H_{2}. That energy

is (take the bond energy of a C–H bond as 350 kJ mol^{-1})

Solution:

QUESTION: 29

**SECTION II : Paragraph Type**

This section contains 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**

In the following reaction sequence, the compound J is an intermediate.

**Q. **The compound I is

Solution:

QUESTION: 30

**SECTION II : Paragraph Type**

This section contains 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**

In the following reaction sequence, the compound J is an intermediate.

**Q.** The compound K is

Solution:

29- 30

QUESTION: 31

**Paragraph for Questions 31 and 32**

The electrochemical cell shown below is a concentration cell.

M|M^{2+} (saturated solution of a sparingly soluble salt, MX_{2}) || M^{2+} (0.001 mol dm^{–3}) | M

The emf of the cell depends on the difference in concentrations of M^{2+} ions at the two electrodes. The emf of the cell at 298 K is 0.059 V.

**Q. **The value of ΔG (kJ mol^{–1}) for the given cell is (take 1F = 96500 C mol^{–1})

Solution:

QUESTION: 32

**Paragraph for Questions 31 and 32**

The electrochemical cell shown below is a concentration cell.

M|M^{2+} (saturated solution of a sparingly soluble salt, MX_{2}) || M^{2+} (0.001 mol dm^{–3}) | M

The emf of the cell depends on the difference in concentrations of M^{2+} ions at the two electrodes. The emf of the cell at 298 K is 0.059 V.

**Q.** The solubility product (Ksp; mol^{3} dm^{–9}) of MX_{2} at 298 K based on the information available for the given

concentration cell is (take 2.303 x R x 298/F = 0.059 V)

Solution:

QUESTION: 33

**Paragraph for Questions 33 and 34**

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.

**Q.** Bleaching powder contains a salt of an oxoacid as one of its components. The anhydride of that oxoacid is

Solution:

QUESTION: 34

**Paragraph for Questions 33 and 34**

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.

**Q. **25 mL of household 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 N Na_{2}S_{2}O_{3} was used to reach the end point. The molarity of

the household bleach solution is

Solution:

*Multiple options can be correct

QUESTION: 35

**SECTION III : Multiple Correct Answer(s) Type**

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

**Q. **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?

Solution:

T_{1} = T_{2} because process is isothermal.

Work done in adiabatic process is less than in isothermal process because area covered by isothermal curve is more than the area co vered by the adiabatic curve.

In adiabatic process expansion occurs by using internal energy hence it decreases while in isothermal process temperature remains constant that’s why no change in internal energy.

*Multiple options can be correct

QUESTION: 36

For the given aqueous reactions, which of the statement(s) is (are) true?

Solution:

*Multiple options can be correct

QUESTION: 37

With reference to the scheme given, which of the given statement(s) about T, U, V and W is (are) correct?

Solution:

*Multiple options can be correct

QUESTION: 38

Which of the given statement(s) about N, O, P and Q with respect to M is (are) correct?

Solution:

*Multiple options can be correct

QUESTION: 39

With respect to graphite and diamond, which of the statement(s) given below is (are) correct?

Solution:

⇒ Diamond is harder than graphite.

⇒ Graphite is good conductor of electricity as each carbon is attached to three C-atoms leaving one valency free, which is responsible for electrical conduction, while in diamond, all the four valencies of

carbon are satisfied, hence insulator.

⇒ Diamond is better thermal conductor than graphite. Whereas electrical conduction is due to availability of free electrons; thermal conduction is due to transfer of thermal vibrations from atom to atom. A compact and precisely aligned crystal like diamond thus facilitates fast movement of heat.

⇒ In graphite, C – C bond acquires double bond character, hence higher bond order than in diamond.

*Multiple options can be correct

QUESTION: 40

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?

Solution:

Graph (I) and (III) represent physiosorption because, in physiosorption, the amount of adsorption decreases with the increase of temperature and increases with the increase of pressure.

Graph (II) represent chemisorption, because in chemisorption amount of adsorption increase with the increase of temperature. Graph (IV) is showing the formation of a chemical bond, hence chemisorption.

QUESTION: 41

**SECTION I : Single Correct Answer Type**

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONLY ONE is correct.

**Q.** Let a_{1}, a_{2}, a_{3}, ….. be in harmonic progression with a_{1} = 5 and a_{20} = 25. The least positive integer n for

which a_{n} < 0

Solution:

QUESTION: 42

The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x – y + z

= 3 and at a distance from the point (3, 1, –1) is

Solution:

-5x + 11y - z + 17 = 0.

QUESTION: 43

Let PQR be a triangle of area where a, b, and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively. Then equals

Solution:

QUESTION: 44

If then a possible

value of

Solution:

QUESTION: 45

If P is a 3 × 3 matrix such that P^{T} = 2P + I, where P^{T} is the transpose of P and I is the 3 × 3 identity matrix,

then there exists a column matrix such that

Solution:

QUESTION: 46

Let a (a) and b (a) be the roots of the equation where

a > – 1. Then are

Solution:

QUESTION: 47

Four fair dice D_{1}, D_{2}, D_{3} and D_{4}, each having six faces numbered 1, 2, 3, 4, 5, and 6, are rolled simultaneously. The probability that D_{4 }shows a number appearing on one of D_{1}, D_{2} and D_{3} is

Solution:

QUESTION: 48

The value of the integral

Solution:

QUESTION: 49

**SECTION II : Paragraph Type**

paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

**Paragraph for Questions 49 and 50**

A tangent PT is drawn to the circle x^{2} + y^{2} = 4 at the point . A straight line L, perpendicular to PT is a tangent to the circle (x -3)^{2} + y^{2} =1.

**Q.** A possible equation of L is

Solution:

QUESTION: 50

**Paragraph for Questions 49 and 50**

A tangent PT is drawn to the circle x^{2} + y^{2} = 4 at the point . A straight line L, perpendicular to PT is a tangent to the circle (x -3)^{2} + y^{2} =1.

**Q.** A common tangent of the two circles is

Solution:

QUESTION: 51

**Paragraph for Questions 51 and 52**

**Q. **Consider the statements:

P : There exists some x ε such that f(x) + 2x = 2(1 + x^{2})

Q : There exists some x ε such that 2f(x) + 1 = 2x(1 + x)

Then

Solution:

QUESTION: 52

**Paragraph for Questions 51 and 52**

**Q. **Which of the following is true?

Solution:

QUESTION: 53

**Paragraph for Questions 53 and 54**

Let an denote the number of all n–digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_{n} = the number of such n–digit integers ending with digit 1 and c_{n} = the number of such n–digit integers ending with digit 0.

**Q. **The value of b_{6} is

Solution:

QUESTION: 54

**Paragraph for Questions 53 and 54**

Let an denote the number of all n–digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_{n} = the number of such n–digit integers ending with digit 1 and c_{n} = the number of such n–digit integers ending with digit 0.

**Q. **Which of the following is correct?

Solution:

*Multiple options can be correct

QUESTION: 55

**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.

55. For every integer n, let a_{n} and b_{n} be real numbers. Let function f : IR → IR be given by

for all integers n. If f is continuous, then which of the following hold(s) for all n?

Solution:

*Multiple options can be correct

QUESTION: 56

If the straight lines are coplanar, then the plane(s) containing these two lines is(are)

Solution:

For given lines to be coplanar, we get

*Multiple options can be correct

QUESTION: 57

If the adjoint of a 3 × 3 matrix P is , then the possible value(s) of the determinant of P is (are)

Solution:

*Multiple options can be correct

QUESTION: 58

Solution:

*Multiple options can be correct

QUESTION: 59

Let X and Y be two events such that

Which of the following is (are) correct?

Solution:

*Multiple options can be correct

QUESTION: 60

Solution:

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