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*Answer can only contain numeric values

QUESTION: 1

**Section 1**

**Q. No. 1-8 carry 4 marks each.**

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

**Q.**

**Consider a concave mirror and a convex lens (refractive index = 1.5) of focal length 10 cm each, separated by a distance of 50 cm in air (refractive index = 1) as shown in the figure. An object is placed at a distance of 15 cm from the mirror. Its erect image formed by this combination has magnification M1. When the set- up is kept in a medium of refractive index 7/6, the magnification becomes M2. The magnitude ** is

Solution:

Image by mirror is formed at 30 cm from mirror at its right and finally by the combination it is formed at

20 cm on right of the lens. So in air medium, magnification by lens is unity. In second medium μ =7/6

focal length of the lens is given by

So in second medium, final image is formed at 140 cm to the right of the lens. Second medium does not change the magnification by mirror. So

*Answer can only contain numeric values

QUESTION: 2

An infinitely long uniform line charge distribution of charge per unit length λ lies parallel to the y-axis in the

y-z plane at z =a (see figure). If the magnitude of the flux of the electric field through the rectangular

surface ABCD lying in the x-y plane with its center at the origin is = permittivity of free space), then

the value of n is

Solution:

*Answer can only contain numeric values

QUESTION: 3

Consider a hydrogen atom with its electron in the n^{th} orbital. An electromagnetic radiation of wavelength

90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n

is (hc = 1242 eV nm)

Solution:

*Answer can only contain numeric values

QUESTION: 4

A bullet is fired vertically upwards with velocity v from the surface of a spherical planet. When it reaches

its maximum height, its acceleration due to the planet’s gravity is 1/4th of its value at the surface of the

planet. If the escape velocity from the planet is , then the value of N is (ignore energy loss due

to atmosphere)

Solution:

At height R from the surface of planet acceleration due to planet’s gravity is 1/4th in comparison to the value at the surface

*Answer can only contain numeric values

QUESTION: 5

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting

at A and C with linear speeds v_{1} and v_{2}, respectively, and always remain in contact with the surfaces. If

they reach B and D with the same linear speed and v_{1} = 3 m/s, then v_{2} in m/s is (g = 10 m/s^{2})

Solution:

Kinetic energy of a pure rolling disc having velocity of centre of mass

*Answer can only contain numeric values

QUESTION: 6

Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits

10^{4} times the power emitted from B. The ratio (λ_{A}/λ_{B}) of their wavelengths λ_{A} and λ_{B} at which the peaks

occur in their respective radiation curves is

Solution:

*Answer can only contain numeric values

QUESTION: 7

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life T years

as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is

12.5 % of the electrical power available form the plant at that time. If the plant is able to meet the total

power needs of the village for a maximum period of nT years, then the value of n is

Solution:

Where, A_{0} is the initial activity of the radioactive material and A is the activity at t.

*Answer can only contain numeric values

QUESTION: 8

A Young’s double slit interference arrangement with slits S_{1} and S_{2} is immersed in water (refractive index = 4/3) as shown in the figure. The positions of maxima on the surface of water are given by x^{0} = p^{2}m^{2}λ^{2} – d^{2}, where λ is the wavelength of light in air (refractive index = 1), 2d is the separation between the slits and m is an integer. The value of p is

Solution:

*Multiple options can be correct

QUESTION: 9

**Section 2**

**Q. No. 9 - 18 carry 4 marks each and 2 marks is deducted for every wrong answer.**

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

**Q.**

For photo-electric effect with incident photon wavelength λ, the stopping potential is V_{0}. Identify the

correct variation(s) of V_{0} with λ and 1/λ

Solution:

*Multiple options can be correct

QUESTION: 10

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a

screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier

scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the

circular scale moves it by two divisions on the linear scale. Then:

Solution:

For screw gauge,

pitch (p) = 2 main scale division

So least count p/100

So option (B) & (C) are correct.

*Multiple options can be correct

QUESTION: 11

Planck’s constant h, speed of light c and gravitational constant G are used to form a unit of length L and a

unit of mass M. Then the correct option(s) is(are)

Solution:

*Multiple options can be correct

QUESTION: 12

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular

frequencies ω_{1} and ω_{2} and have total energies E_{1} and E_{2}, respectively. The variations of their momenta p

with positions x are shown in the figures. If a/b =n^{2 } and a/R =n, then the correct equation(s) is(are)

Solution:

*Multiple options can be correct

QUESTION: 13

A ring of mass M and radius R is rotating with angular speed ω about a fixed vertical axis passing through its centre O with two point masses each of mass M/8 at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed

of the system is 8/9 ω and one of the masses is at a distance of 3/5 R from O. At this instant the distance of the other mass from O is

Solution:

*Multiple options can be correct

QUESTION: 14

The figures below depict two situations in which two infinitely long static line charges of constant positive

line charge density λ are kept parallel to each other. In their resulting electric field, point charges q and -q

are kept in equilibrium between them. The point charges are confined to move in the x direction only. If

they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are)

Solution:

Hence +q, charge will performs SHM with time period

*Multiple options can be correct

QUESTION: 15

Two identical glass rods S1 and S2 (refractive index = 1.5) have one convex end of radius of

curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light P is placed inside rod S1 on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside S2. The distance d is

Solution:

*Multiple options can be correct

QUESTION: 16

A conductor (shown in the figure) carrying constant current I is kept in the x-y plane in a uniform magnetic

field If F is the magnitude of the total magnetic force acting on the conductor, then the correct

statement(s) is(are)

Solution:

*Multiple options can be correct

QUESTION: 17

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium

at temperature T. Assuming the gases are ideal, the correct statement(s) is(are)

Solution:

*Multiple options can be correct

QUESTION: 18

In an aluminium (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as

shown in the figure. The electrical resistivities of Al and Fe are 2.7 × 10^{–8} m and 1.0 × 10^{–7} m,

respectively. The electrical resistance between the two faces P and Q of the composite bar is

Solution:

QUESTION: 19

**SECTION 3**

**Q. No. 19 -20 carry 2 marks each and 1 mark is deducted for every wrong answer.**

**Each question contains two columns, Column I and Column II
Column I has four entries (A), (B), (C) and (D)
Column II has five entries (P), (Q), (R), (S) and (T)
Match the entries in Column I with the entries in Column II**

**Q.**

**Match the nuclear processes given in column I with the appropriate option(s) in column II**

Solution:

QUESTION: 20

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is

conserved. Four possible forms of the potential energy of the particle are given in column I (a and U0 are

constants). Match the potential energies in column I to the corresponding statement(s) in column II.

Solution:

*Answer can only contain numeric values

QUESTION: 21

**SECTION – 1**

**Q. No. - 21 - 28 carry 4 marks each.**

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

**Q.**

If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride-ammonia complex(which

behaves as a strong electrolyte) is – 0.0558^{0}C, the number of chloride(s) in the coordination sphere of the

complex is

[Kf of water = 1.86 K kg mol^{–1}]

Solution:

*Answer can only contain numeric values

QUESTION: 22

The total number of stereoisomers that can exist for M is

Solution:

*Answer can only contain numeric values

QUESTION: 23

The number of resonance structures for N is

Solution:

*Answer can only contain numeric values

QUESTION: 24

The total number of lone pairs of electrons in N_{2}O_{3} is

Solution:

*Answer can only contain numeric values

QUESTION: 25

For the octahedral complexes of Fe^{3+} in SCN^{–} (thiocyanato-S) and in CN^{–} ligand environments, the

difference between the spin-only magnetic moments in Bohr magnetons (When approximated to the nearest

integer) is

[Atomic number of Fe = 26]

Solution:

*Answer can only contain numeric values

QUESTION: 26

Among the triatomic molecules/ions the total

number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution

from the d-orbital(s) is

[Atomic number: S = 16, Cl = 17, I = 53 and Xe = 54]

Solution:

So among the following only four (4) has linear shape and no d-orbital is involved in hybridization

*Answer can only contain numeric values

QUESTION: 27

Not considering the electronic spin, the degeneracy of the second excited state( n = 3) of H atom is 9, while

the degeneracy of the second excited state of H^{–} is

Solution:

Single electron species don’t follow the rule but multi electron species do

*Answer can only contain numeric values

QUESTION: 28

All the energy released from the reaction

is used for oxidizing

Under standard conditions, the number of moles of M^{+} oxidized when one mole of X is converted to Y is

[F = 96500 C mol^{–1}]

Solution:

So the number of moles of M^{+} oxidized using X→Y will be =193/48.25 = 4 mole

QUESTION: 29

**SECTION 2**

**Q. No 29 - 38 carry 4 marks each and 2 marks are deducted for every wrong answer.**

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

**Q.**

If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of

octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium

ions, m and n, respectively, are

Solution:

*Multiple options can be correct

QUESTION: 30

Compound(s) that on hydrogenation produce(s) optically inactive compound(s) is (are)

Solution:

*Multiple options can be correct

QUESTION: 31

The major product of the following reaction is

Solution:

*Multiple options can be correct

QUESTION: 32

In the following reaction, the major product is

Solution:

*Multiple options can be correct

QUESTION: 33

The structure of D-(+)-glucose is

The structure of L-(–)-glucose is

Solution:

*Multiple options can be correct

QUESTION: 34

The major product of the reaction is

Solution:

*Multiple options can be correct

QUESTION: 35

The correct statement(s) about Cr^{2+} and Mn^{3+} is(are)

[Atomic numbers of Cr = 24 and Mn = 25]

Solution:

(1) Cr^{2+} is a reducing agent because Cr^{3+} is more stable.

(2) Mn^{3+} is an oxidizing agent because Mn^{2+} is more stable.

(3) Cr^{2+} and Mn^{3+} exhibit d^{4} electronic configuration.

*Multiple options can be correct

QUESTION: 36

Copper is purified by electrolytic refining of blister copper. The correct statement(s) about this process

is(are)

Solution:

(1) Impure Cu strip is used as anode and impurities settle as anode mud.

(2) Pure Cu deposits at cathode.

(3) Acidified aqueous CuSO_{4} is used as electrolyte

*Multiple options can be correct

QUESTION: 37

Fe^{3+} is reduced to Fe^{2+} by using

Solution:

*Multiple options can be correct

QUESTION: 38

The % yield of ammonia as a function of time in the reaction

If this reaction is conducted at (P, T_{2}), with T_{2} > T_{1}, the % yield of ammonia as a function of time is

represented by

Solution:

Increasing the temperature lowers equilibrium yield of ammonia.

However, at higher temperature the initial rate of forward reaction would be greater than at lower

temperature that is why the percentage yield of NH_{3} too would be more initially.

QUESTION: 39

**SECTION 3**

**Q. No. 39 - 40 carry 4 marks each and 2 marks deducted for every wrong answer.**

**Each question contains two columns, Column I and Column II
Column I has four entries (A), (B), (C) and (D)
Column II has five entries (P), (Q), (R), (S) and (T)
Match the entries in Column I with the entries in Column II
One or more entries in Column I may match with one or more entries in Column II**

**Q.**

Match the anionic species given in Column I that are present in the ore(s) given in Column II

Solution:

QUESTION: 40

Match the thermodynamic processes given under Column I with the expression given under Column II:

Solution:

*Answer can only contain numeric values

QUESTION: 41

**Section 1**

**Q. No. 41- 48 carry 4 marks each.**

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

Q.

Let be a continuous function. For if F'(a) + 2 is the area of the region bounded by x = 0, y = 0, y = f (x) and x = a, then f (0) is

Solution:

*Answer can only contain numeric values

QUESTION: 42

The number of distinct solutions of the equation

in the interval [0, 2π] is

Solution:

*Answer can only contain numeric values

QUESTION: 43

Let the curve C be the mirror image of the parabola y^{2} = 4x with respect to the line x + y + 4 = 0. If A and B

are the points of intersection of C with the line y = -5, then the distance between A and B is

Solution:

Image of y = –5 about the line x + y + 4 = 0 is x = 1

Distance AB = 4

*Answer can only contain numeric values

QUESTION: 44

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two

heads is at least 0.96 is

Solution:

*Answer can only contain numeric values

QUESTION: 45

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls

stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a

queue in such a way that exactly four girls stand consecutively in the queue. Then the value of m/n is

Solution:

(4 out of 5 girls together arranged with others – number of cases all 5 girls are together)

*Answer can only contain numeric values

QUESTION: 46

If the normals of the parabola y^{2} = 4x drawn at the end points of its latus rectum are tangents to the circle

(x - 3)^{2} + (y + 2)^{2} = r^{2}, then the value of r^{2} is

Solution:

*Answer can only contain numeric values

QUESTION: 47

Let f : R → R be a function defined by where [x] is the greatest integer less than or

equal to x. then the value of (4I - 1) is

Solution:

*Answer can only contain numeric values

QUESTION: 48

A cylindrical container is to be made from certain solid material with the following constraints: It has a

fixed inner volume of V mm3, has a 2 mm thick solid wall and is open at the top. The bottom of the

container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the

container.

If the volume of the material used to make the container is minimum when the inner radius of the container

is 10 mm, then the value of V/250π is

Solution:

*Multiple options can be correct

QUESTION: 49

**Section 2**

**Q. No. 49 - 58 carry 4 marks each and 2 marks are deducted for every wrong answer**

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

**Q.**

**Let ΔPQR be a triangle. then which of the following is (are) true ? **

Solution:

*Multiple options can be correct

QUESTION: 50

Let X and Y be two arbitrary, 3 x 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 x 3, nonzero,

symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

Solution:

*Multiple options can be correct

QUESTION: 51

Which of the following values of α satisfy the equation

Solution:

*Multiple options can be correct

QUESTION: 52

In , consider the planes P_{1} : y = 0 and P_{2} : x + z = 1. Let P_{3} be a plane, different from P_{1} and P_{2}, which

passes through the intersection of P_{1} and P_{2}. If the distance of the point (0, 1, 0) from P_{3} is 1 and the

distance of a point (α, β, γ) from P_{3} is 2, then which of the following relations is (are) true ?

Solution:

*Multiple options can be correct

QUESTION: 53

In , let L be a straight line passing through the origin. Suppose that all the points on L are at a constant

distance from the two planes P1 : x + 2y - z + 1 = 0 and P2 : 2x - y + z - 1 = 0. Let M be the locus of the

feet of the perpendiculars drawn from the points on L to the plane P_{1}. Which of the following points lie(s)

on M ?

Solution:

Equation of projection of line L on plane P1 is

Clearly points and satisfy the line of projection i.e. M

*Multiple options can be correct

QUESTION: 54

Let P and Q be distinct points on the parabola y^{2} = 2x such that a circle with PQ as diameter passes through

the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle ΔOPQ is (3sqrt2) , then which of the following is (are) the coordinates of P ?

Solution:

*Multiple options can be correct

QUESTION: 55

Let y(x) be a solution of the differential equation (1 + e^{x} )y' + ye^{x} = 1. If y(0) = 2, then which of the

following statements is (are) true ?

Solution:

*Multiple options can be correct

QUESTION: 56

Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is

represented by the differential equation where P, Q are functions of x, y and then which of the following statements is (are) true ?

Solution:

*Multiple options can be correct

QUESTION: 57

Let be a differential function with g(0) = 0, g'(0) = 0 and g'(1) 0.

and for all x R. Let (f o h)(x) denote f(h(x)) and (h o f)(x) denote h(f(x)). Then which of the following is (are) true?

Solution:

*Multiple options can be correct

QUESTION: 58

and (g o f)(x) denote g(f(x)). Then which of the following is (are) true ?

Solution:

QUESTION: 59

**SECTION 3**

**Q. No. 59 - 60 carry 2 marks each and 1 mark is deducted for every wrong answer.**

**Each question contains two columns, Column I and Column II
Column I has four entries (A), (B), (C) and (D)
Column II has five entries (P), (Q), (R), (S) and (T)
Match the entries in Column I with the entries in Column II
One or more entries in Column I may match with one or more entries in Column II**

**Q.**

Solution:

QUESTION: 60

Solution:

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