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QUESTION: 1

**Inst.**

**Q. 1 to 5 carry 3 marks each**

**Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.**

**Q.**

In a historical experiment to determine Planck’s constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectron energies were measured by applying a stopping potential. The relevant data for the wavelength (λ) of incident light and the corresponding stopping potential (V^{0}) are given below:

Given that c = 3 x10^{8} ms^{-1} and e = 1.6 x 10^{-19}C, Planck’s constant (in units of J s) found from such an

experiment is

Solution:

**Alternate Solution
From first two date points.
⇒ 6.4 * 10**

We get same value from second and third data points.

QUESTION: 2

A uniform wooden stick of mass 1.6 kg and length rests in an inclined manner on a smooth, vertical wall of height h (<) such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of 30^{0} with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio h/ and the frictional force* f *at the bottom of the stick are (g = 10 ms^{-2})

Solution:

On the basis of given data, it is not possible for the rod to stay at rest.

Note: If we take the normal reaction at the wall equal to the normal reaction at the floor, the answer

will be D.

QUESTION: 3

A water cooler of storage capacity 120 litres can cool water at a constant rate of P watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load). The temperature of water fed into the device cannot exceed 30^{0}C and the entire stored 120 litres of water is initially cooled to 10^{0}C. The entire system is thermally insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is

(Specific heat of water is 4.2 kJ kg^{-1 }K^{-1} and the density of water is 1000 kg m^{-3})

Solution:

QUESTION: 4

A parallel beam of light is incident from air at an angle α on the side PQ of a right angled triangular prism of refractive index n = √2. Light undergoes total internal reflection in the prism at the face PR when α has a minimum value of 450. The angle θ of the prism is

Solution:

For TIR on face PR,

QUESTION: 5

An infinite line charge of uniform electric charge density λ lies along the axis of an electrically conducting infinite cylindrical shell of radius R. At time t = 0, the space inside the cylinder is filled with a material of permittivity ε and electrical conductivity σ. The electrical conduction in the material follows Ohm’s law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j(t) at any point in the material?

Solution:

For infinite line,

Current through an elemental shell;

This current is radially outwards so;

*Multiple options can be correct

QUESTION: 6

**Q. 6 to 13 carry 4 marks each**

**Each questions has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.**

**Q. Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the following statement(s) is (are) true?**

Solution:

*Multiple options can be correct

QUESTION: 7

Two loudspeakers M and N are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at 60 km/hr along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let v(t) represent the beat frequency measured by a person sitting in the car at time t. Let v_{P}, v_{Q} and v_{R} be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is 330 m s^{-1}. Which of the following statement(s) is(are) true regarding the sound

heard by the person?

Solution:

Now, v (between P and Q)

Now, sin θ = 1 at Q (maximum)

∴ rate of change in beat frequency is maximum at Q

*Multiple options can be correct

QUESTION: 8

An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true?

Solution:

When filament breaks up, the temperature of filament will be higher so according to wein’s law

, the filament emits more light at higher band of frequencies. As voltage is constant, so consumed electrical power is P=V^{2}/R

As R increases with increase in temperature so the filament consumes less electrical power towards the end of the life of the bulb.

*Multiple options can be correct

QUESTION: 9

A plano-convex lens is made of a material of refractive index n. When a small object is placed 30 cm away

in front of the curved surface of the lens, an image of double the size of the object is produced. Due to

reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away

from the lens. Which of the following statement(s) is(are) true?

Solution:

For refraction through lens,

The faint image is erect and virtual.

*Multiple options can be correct

QUESTION: 10

A length-scale () depends on the permittivity (ε) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression(s) for is(are) dimensionally correct?

Solution:

k_{B}T ≈ FL,ε≈Q^{2}/FL^{2} and n ≈L^{-3}; where F is force, Q is charge and L is length So (B) and (D) are correct

*Multiple options can be correct

QUESTION: 11

A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the 90^{0 }vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at a constant rate of 10 A s^{-1}. Which of the following statement(s) is(are) true?

Solution:

*Multiple options can be correct

QUESTION: 12

The position vector of a particle of mass m is given by the following equation , where α = 10/3 m s^{-3}, β= 5 m s^{-2} and m = 0.1 kg. At t = 1 s, which of the following statement(s) is(are) true about the particle?

Solution:

*Multiple options can be correct

QUESTION: 13

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices n1 and n2 (> n1), as shown in the figure. A ray of light is incident with angle θ_{i} from medium 1 and emerges in medium 2 with refraction angle θ_{f} with a lateral displacement *l*.

Which of the following statement (s) is (are) true?

Solution:

From Snell's Law

The deviation of ray in the slab will depend on n (z)

Hence, l will depend on n (z) but not on n_{2}

*Multiple options can be correct

QUESTION: 14

**Q. 14 to 18 carry 3 marks each**

**The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.**

**Q.**

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P) by the metal. The sensor has a scale that displays log2 (P/P_{0}), where P_{0} is a constant. When the metal surface is at a temperature of 487°C, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to 2767 °C?

Solution:

*Answer can only contain numeric values

QUESTION: 15

The isotope having a mass 12.014 u undergoes β-decay to . has an excited state of the nucleus at 4.041 MeV above its ground state. If decays to , the maximum kinetic energy of the β- particle in units of MeV is (1 u = 931.5 MeV/c^{2}, where c is the speed of light in vacuum).

Solution:

Hence, β particle will have a maximum KE of 9 MeV

*Answer can only contain numeric values

QUESTION: 16

Consider two solid spheres P and Q each of density 8 gm cm^{–3} and diameters 1cm and 0.5cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm^{–3} and viscosity *η *= 3 poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm^{–3} and viscosity η = 2 poiseulles. The ratio of the terminal velocities of P and Q is

Solution:

Terminal velocity where ρ is the density of the solid sphere and σ is the density of the liquid

*Answer can only contain numeric values

QUESTION: 17

A hydrogen atom in its ground state is irradiated by light of wavelength 970 Å. Taking hc/e = 1.237×10^{–6 }eV m and the ground state energy of hydrogen atom as –13.6 eV, the number of lines present in the emission spectrum is

Solution:

Note : Unit of hc/e in original paper is incorrect Energy available (4^{th} energy level of hydrogen atom). Hence, the number of lines present in the emission spectrum = ^{4}C_{2} = 6

*Answer can only contain numeric values

QUESTION: 18

Two inductors L_{1} (inductance 1 mH, internal resistance 3) and L_{2} (inductance 2 mH, internal resistance 4), and a resistor R (resistance 12) are all connected in parallel across a 5V battery. The circuit is switched on at time t = 0. The ratio of the maximum to the minimum current (I_{max} / I_{min}) drawn from the battery is

Solution:

At t = 0, current will flow only in 12Ω resistance

At t → ∝ both L_{1 }and L_{2} behave as conducting wires

QUESTION: 19

**Q. 19 to 23 carry 3 marks each**

**Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.**

**Q.**

P is the probability of finding the 1s electron of hydrogen atom in a spherical shell of infinitesimal thickness, dr, at a distance r from the nucleus. The volume of this shell is 4πr^{2}dr. The qualitative sketch of the dependence of P on r is

Solution:

The probability distribution curve for 1s electron of hydrogen atom.

QUESTION: 20

One mole of an ideal gas at 300 K in thermal contact with surroundings expands isothermally from 1.0 L to 2.0 L against a constant pressure of 3.0 atm. In this process, the change in entropy of surrounding (ΔS_{surr})in JK^{–1} is (1L atm = 101.3 J)

Solution:

QUESTION: 21

The increasing order of atomic radii of the following Group 13 elements is

Solution:

Increasing order of atomic radius of group 13 elements Ga < Al < In < Tl.

Due to poor shielding of d-electrons in Ga, its radius decreases below Al.

QUESTION: 22

Among [Ni(CO)_{4}], [NiCl_{4}]^{2–}, [Co(NH_{3})_{4}Cl_{2}]Cl, Na_{3}[CoF_{6}], Na_{2}O_{2} and CsO_{2}, the total number of paramagnetic compounds is

Solution:

Number of paramagnetic compounds are 3.

Following compounds are paramagnetic.

QUESTION: 23

On complete hydrogenation, natural rubber produces

Solution:

*Multiple options can be correct

QUESTION: 24

**Q. 24 to 31 carry 4 marks each**

**Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.**

**Q.**

**According to the Arrhenius equation**

Solution:

*Multiple options can be correct

QUESTION: 25

A plot of the number of neutrons (N) against the number of protons (P) of stable nuclei exhibits upward deviation from linearity for atomic number, Z > 20. For an unstable nucleus having N/P ratio less than 1, the possible mode(s) of decay is(are)

Solution:

Positron, emission

Both process cause an increase in n/p ratio towards 1 thus stabilising the nucleus.

*Multiple options can be correct

QUESTION: 26

The crystalline form of borax has

Solution:

The structure of anion of borax is

*Multiple options can be correct

QUESTION: 27

The compound(s) with TWO lone pairs of electrons on the central atom is(are)

Solution:

CIF_{3} abd XeF_{4} contain two lone pair of electrons on the central atom.

*Multiple options can be correct

QUESTION: 28

The reagent(s) that can selectively precipitate S^{2–} from a mixture of S^{2–} and in aqueous solution is(are)

Solution:

The reagent that can selectively precipitate S^{2–} and in aqueous solution is CuCl_{2.}

*Multiple options can be correct

QUESTION: 29

Positive Tollen’s test is observed for

Solution:

Beside aldehyde α-hydroxy ketones can also show Tollen’s test due to rearrangement in aldehyde via ene

diol intermediate. (however this needs a terminal α-carbon)

*Multiple options can be correct

QUESTION: 30

The product(s) of the following reaction sequence is (are)

Solution:

*Multiple options can be correct

QUESTION: 31

The correct statement(s) about the following reaction sequence is(are)

Solution:

Q (not R) is steam volatile due to intramolecular hydrogen bonding.

Q gives violet colouration due to phenolic functional group.

S gives yellow precipitate due to aldehydic group with 2, 4 –DNP.

S does not have free phenolic group to respond to FeCl_{3} test.

*Answer can only contain numeric values

QUESTION: 32

**Inst.**

**Q, No- 32 to 36 carry 3 marks each**

**The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.**

**Q.**

The mole fraction of a solute in a solution is 0.1. At 298 K, molarity of this solution is the same as its molality. Density of this solution at 298 K is 2.0 g cm^{–3}. The ratio of the molecular weights of the solute

and solvent,

Solution:

*Answer can only contain numeric values

QUESTION: 33

The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute temperature of an ideal gas is increased 4 times and its pressure is increased 2 times. As a result, the diffusion coefficient of this gas increases x times. The value of x is

Solution:

Diffusion coefficient (D) is proportional to both mean free path and mean speed

If T_{1} is increased 4 times and P_{1} is increased 2 times.

*Answer can only contain numeric values

QUESTION: 34

In neutral or faintly alkaline solution, 8 moles of permanganate anion quantitatively oxidize thiosulphate anions to produce X moles of a sulphur containing product. The magnitude of X is

Solution:

*Multiple options can be correct

QUESTION: 35

The number of geometric isomers possible for the complex

Solution:

Number of geometrical isomers = 5.

*Answer can only contain numeric values

QUESTION: 36

In the following monobromination reaction, the number of possible chiral products is

(enantiomerically pure)

Solution:

QUESTION: 37

**Inst. **

**Q No. 37 - 41 carry 3 marks each**

**Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.**

**Q.**

Let Suppose α_{1} and β_{1} are the roots of the equation x^{2} – 2x secθ + 1 = 0 and α_{2} and β_{2} are the roots of the equation x^{2} + 2x tanθ – 1 = 0. If α_{1} > β_{1} and α_{2} > β_{2}, then α_{1} + β_{2} equals

Solution:

QUESTION: 38

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Solution:

If a boy is selected then number of ways = ^{4}C_{1}.^{6}C_{3}

If a boy is not selcected then number of ways = ^{6}C_{4}

Captain can be selected in ^{4}C_{1} ways

Required number of ways =

QUESTION: 39

Let The sum of all distinct solutions of the equation in the set S is equal to

Solution:

QUESTION: 40

A computer producing factory has only two plants T_{1} and T_{2}. Plant T_{1} produces 20% and plant T_{2} produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P(computer turns out to be defective given that it is produced in plant T_{1}) = 10 P(computer turns out to be defective given that it is produced in plant T_{2}), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T_{2} is

Solution:

E_{1} : Computer is produced by plant T_{1}

E_{2} : Computer is produced by plant T_{2}

A : Computer is defective

QUESTION: 41

The least value of for which for all x > 0, is

Solution:

*Multiple options can be correct

QUESTION: 42

**Q. No. 42 - 49 carry 4 marks each**

**Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.**

**Q.**

**Consider a pyramid OPQRS located in the first octant (x **** 0, y **** 0, z **** 0) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The bases OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3. Then**

Solution:

Points O, P, Q, R, S are respectively.

Angle between OQ and OS is

Equation of plane containing the points O, Q and S is x ? y = 0

Perpendicular distance from P(3, 0, 0) to the plane

perpenicular distance from O (0,0,0) to the line RS:

*Multiple options can be correct

QUESTION: 43

Let be a differentiable function such that Then

Solution:

(D) for f(x) is unbounded function for

*Multiple options can be correct

QUESTION: 44

Let where Suppose Q = [q_{ij}] is a matrix such that PQ = kI, where and I is the identity matrix of order 3. If

Solution:

*Multiple options can be correct

QUESTION: 45

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and and area of incircle of the triangle XYZ is 8π/3 , then

Solution:

*Multiple options can be correct

QUESTION: 46

A solution curve of the differential equation passes through the point (1, 3). Then the solution curve

Solution:

Hence no solution

*Multiple options can be correct

QUESTION: 47

Let be differentiable functions such that f(x) = x^{3} + 3x + 2, , then

Solution:

*Multiple options can be correct

QUESTION: 48

The circle C_{1} : x^{2} + y^{2} = 3, with centre at O, intersects the parabola x^{2} = 2y at the point P in the first quadrant. Let the tangent to the circle C_{1} at P touches other two circles C_{2} and C_{3} at R_{2} and R_{3}, respectively. Suppose C_{2} and C_{3} have equal radii 2√3 and centres Q_{2} and Q_{3}, respectively. If Q_{2} and Q_{3} lie on the y-axis, then

Solution:

Equation of tangent at P(√2,1) is √2 x+y-3=0

If centre of C_{2} at (0, α) and radius equal to 2√3

*Multiple options can be correct

QUESTION: 49

Let RS be the diameter of the circle x^{2} + y^{2} = 1, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point(s)

Solution:

*Answer can only contain numeric values

QUESTION: 50

**Q. No. 50 - 55 carry 3 marks each.**

**The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.**

**Q.**

The total number of distinct x R for which is

Solution:

*Answer can only contain numeric values

QUESTION: 51

Let m be the smallest positive integer such that the coefficient of x_{2} in the expansion of (1 + x)^{2} + (1 + x)^{3} + for some positive integer n. Then the value of n is

Solution:

Least positive integer m for which n is an integer is m = 16 for which n = 5

*Answer can only contain numeric values

QUESTION: 52

The total number of distinct x [0, 1] for which

Solution:

*Answer can only contain numeric values

QUESTION: 53

Let α,β be such that Then 6(α + β) equals

Solution:

*Answer can only contain numeric values

QUESTION: 54

Let and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P^{2} = – I is

Solution:

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