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QUESTION: 1

A glass capillary tube is of the shape of truncated cone with an apex angle α so that its two ends have cross

sections of different radii. When dipped in water vertically, water rises in it to a height h, where the radius

of its cross section is b. If the surface tension of water is S, its density is ρ, and its contact angle with glass

is θ, the value of h will be (g is the acceleration due to gravity)

Solution:

If R be the meniscus radius

R cos (θ + α/2) = b

Excess pressure on concave side of meniscus = 2S/R

QUESTION: 2

If λ_{cu} is the wavelength of K_{α} X-ray line of copper (atomic number 29) and λ_{Mo} is the wavelength of the K?

X-ray line of molybdenum (atomic number 42), then the ratio λ_{cu}/λ_{Mo} is close to

Solution:

QUESTION: 3

A planet of radius R = 1/10 x (radius of Earth) has the same mass density as Earth. Scientists dig a well of

depth R/5 on it and lower a wire of the same length and of linear mass density 10^{-3} kgm^{-1} into it. If the wire

is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the

radius of Earth = 6 x 10^{6} m and the acceleration due to gravity of Earth is 10 ms^{-2})

Solution:

QUESTION: 4

A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting

the surface. The force on the ball during the collision is proportional to the length of compression of the

ball. Which one of the following sketches describes the variation of its kinetic energy K with time t most

appropriately? The figures are only illustrative and not to the scale.

Solution:

QUESTION: 5

A metal surface is illuminated by light of two different wavelengths 248 nm and 310 nm. The maximum

speeds of the photoelectrons corresponding to these wavelengths are u_{1} and u_{2}, respectively. If the ratio

u_{1} : u_{2} = 2 : 1 and hc = 1240 eV nm, the work function of the metal is nearly

Solution:

QUESTION: 6

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is

fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it

slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is

Solution:

Initially bead is applying radially inward normal force.

During motion at an instant, N = 0, after that N will act radially outward.

QUESTION: 7

During an experiment with a metre bridge, the galvanometer shows a null point when the jockey is pressed

at 40.0 cm using a standard resistance of 90 , as shown in the figure. The least count of the scale used in

the metre bridge is 1 mm. The unknown resistance is

Solution:

QUESTION: 8

Parallel rays of light of intensity I = 912 Wm^{–2} are incident on a spherical black body kept in surroundings

of temperature 300 K. Take Stefan-Boltzmann constant σ = 5.7×10^{–8} Wm^{–2} K^{–4} and assume that the energy

exchange with the surroundings is only through radiation. The final steady state temperature of the black

body is close to

Solution:

Rate of radiation energy lost by the sphere

= Rate of radiation energy incident on it

QUESTION: 9

A point source S is placed at the bottom of a transparent block of height 10 mm and refractive index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54 mm on the top of the block. The refractive index of the liquid is

Solution:

QUESTION: 10

**Q.No. 11 - 16 carry 3 marks each**

**This section contains 3 paragraphs, each describing theory, experiments, data etc. Six questions relate to the three paragraphs with two questions on each paragraph. Each question has only one correct answer among the four given options (A), (B), (C) and (D).**

**Paragraph For Questions 11 & 12**

**The figure shows a circular loop of radius a with two long parallel wires (numbered 1 and 2) all in the plane of the paper. The distance of each wire from the centre of the loop is d. The loop and the wires are carrying the same current I. The current in the loop is in the counterclockwise direction if seen from above.**

Q.

When d ≈ a but wires are not touching the loop, it is found that the net magnetic field on the axis of the

loop is zero at a height h above the loop. In that case

Solution:

The direction of magnetic field at the given Point due to the loop is normally out of the plane. Therefore,

the net magnetic field due the both wires should be into the plane. For this current in wire I should be along

PQ and that in wire RS should be along SR.

QUESTION: 11

**Paragraph**

**The figure shows a circular loop of radius a with two long parallel wires (numbered 1 and 2) all in the plane of the paper. The distance of each wire from the centre of the loop is d. The loop and the wires are carrying the same current I. The current in the loop is in the counterclockwise direction if seen from above.**

Q.

Consider d >> a, and the loop is rotated about its diameter parallel to the wires by 30° from the position

shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its

new position will be (assume that the net field due to the wires is constant over the loop)

Solution:

QUESTION: 12

Paragraph for Questions 13 & 14

In the figure a container is shown to have a movable (without friction) piston on top. The container and

the piston are all made of perfectly insulating material allowing no heat transfer between outside and

inside the container. The container is divided into two compartments by a rigid partition made of a

thermally conducting material that allows slow transfer of heat. The lower compartment of the container

is filled with 2 moles of an ideal monatomic gas at 700 K and the upper compartment is filled with 2

moles of an ideal diatomic gas at 400 K. The heat capacities per mole of an ideal monatomic gas are and those for an ideal diatomic gas are

Q.

Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final

temperature of the gases will be

Solution:

Heat given by lower compartment =

Heat obtained by upper compartment =

equating (i) and (ii)

3 (700 – T) = 7 (T – 400)

2100 – 3T = 7 T – 2800

4900 = 10 T T = 490 K

QUESTION: 13

**In the figure a container is shown to have a movable (without friction) piston on top. The container and
the piston are all made of perfectly insulating material allowing no heat transfer between outside and
inside the container. The container is divided into two compartments by a rigid partition made of a
thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with 2 moles of an ideal monatomic gas at 700 K and the upper compartment is filled with 2 moles of an ideal diatomic gas at 400 K. The heat capacities per mole of an ideal monatomic gas are **

**Q.**

**Now consider the partition to be free to move without friction so that the pressure of gases in both
compartments is the same. Then total work done by the gases till the time they achieve equilibrium will be**

Solution:

Heat given by lower compartment =

Heat obtained by upper compartment =

By equating (i) and (ii)

5(700 -T) = 7(T - 400)

3000 – 5T = 7T– 2800

6300 = 12 T

T = 525K

Work done by lower gas = nRΔT = – 350 R

Work done by upper gas = nRΔT = +250 R

Net work done - 100 R

QUESTION: 14

Paragraph for Questions 15 & 16

A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston

pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are 20 mm and 1 mm respectively. The upper end of the container is open to the atmosphere.

Q.

If the piston is pushed at a speed of 5 mms^{–1}, the air comes out of the nozzle with a speed of

Solution:

QUESTION: 15

If the density of air is ρ_{a} and that of the liquid ρ_{l} , then for a given piston speed the rate (volume per unit

time) at which the liquid is sprayed will be proportional to

Solution:

QUESTION: 16

** Q. no 17-21 cary 3 marks each.**

**Each queation having two matching lists. Choices for the correct combination of elements from List-I and List-II are given as option (A), (B), (C) and (D) out of which one is correct.**

**Q.**

**A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance d of 1.2 m from the person. In the following, state of the lift’s motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the lists.**

Solution:

In P, Q, R no horizontal velocity is imparted to falling water, so d remains same.

In S, since its free fall, a_{eff} = 0

Liquid won’t fall with respect to lift.

QUESTION: 17

Four charges Q1, Q2, Q3 and Q4 of same magnitude are fixed along the x axis at x = -2a, -a, +a and +2a,

respectively. A positive charge q is placed on the positive y axis at a distance b > 0. Four options of the

signs of these charges are given in List I. The direction of the forces on the charge q is given in List II.

Match List I with List II and select the correct answer using the code given below the lists.

Solution:

P: By Q1 and Q4, Q3 and Q2 F is in +y

Q: By Q1 and Q4, Q2 and Q3 F is in +ve x.

R: By Q_{1} and Q_{4}, F is in +ve y

By Q_{2} and Q_{3}, F is in –ve y

But later has more magnitude, since its closer to

(0, b). Therefore net force is in – y

S: By Q1 and Q4, F is in +ve x and by Q_{2} and Q_{3}, F is

in –x, but later is more in magnitude, since its

closer to (0, b). Therefore net force is in –ve x.

QUESTION: 18

Four combinations of two thin lenses are given in List I. The radius of curvature of all curved surfaces is r

and the refractive index of all the lenses is 1.5. Match lens combinations in List I with their focal length in

List II and select the correct answer using the code given below the lists.

Solution:

QUESTION: 19

A block of mass m_{1} = 1 kg another mass m_{2} = 2kg, are placed together (see figure) on an inclined plane

with angle of inclination θ. Various values of θ are given in List I. The coefficient of friction between the

block m1 and the plane is always zero. The coefficient of static and dynamic friction between the block m_{2}

and the plane are equal to μ = 0.3. In List II expressions for the friction on the block m_{2} are given. Match

the correct expression of the friction in List II with the angles given in List I, and choose the correct option.

The acceleration due to gravity is denoted by g.

[Useful information: tan (5.5^{o}) ≈ 0.1; tan (11.5^{o}) ≈ 0.2; tan (16.5^{o}) ≈ 0.3]

Solution:

QUESTION: 20

**Inst.**

**Q. No 1- 10 carry 3 marks each and 1 as negative marking for incorrect answer**

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

**Q.**

Charges Q, 2Q and 4Q are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of radii R/2, R

and 2R respectively, as shown in figure. If magnitudes of the electric fields at point P at a distance R from

the centre of spheres 1, 2 and 3 are E_{1}, E_{2} and E_{3} respectively, then

Solution:

QUESTION: 21

**Q. No 21- 40 carry 3 marks each and 1 as negative marking for incorrect answer**

Q.

Assuming 2s – 2p mixing is NOT operative, the paramagnetic species among the following is

Solution:

QUESTION: 22

For the process

at T = 100^{o}C and 1 atmosphere pressure, the correct choice is

Solution:

At 100^{0}C and 1 atmosphere pressure is at equilibrium. For equilibrium ΔS_{total} = 0

QUESTION: 23

For the elementary reaction M → N, the rate of disappearance of M increases by a factor of 8 upon

doubling the concentration of M. The order of the reaction with respect to M is

Solution:

QUESTION: 24

For the identification of β-naphthol using dye test, it is necessary to use

Solution:

QUESTION: 25

Isomers of hexane, based on their branching, can be divided into three distinct classes as shown in the

figure.

The correct order of their boiling point is

Solution:

III > II > I

More the branching in an alkane, lesser will be the surface area, lesser will be the boiling point

QUESTION: 26

The major product in the following reaction is

[Figure

Solution:

QUESTION: 27

Under ambient conditions, the total number of gases released as products in the final step of the reaction

scheme shown below is

Solution:

QUESTION: 28

The product formed in the reaction of SOCl2 with white phosphorous is

Solution:

QUESTION: 29

Hydrogen peroxide in its reaction with KIO_{4} and NH_{2}OH respectively, is acting as a

Solution:

QUESTION: 30

The acidic hydrolysis of ether (X) shown below is fastest when

Figure

Solution:

When two phenyl groups are replaced by two para methoxy group, carbocation formed will be more stable

QUESTION: 31

**Q. No 31- 36 carry 3 marks each and 1 as negative marking for incorrect answer**

**Paragraph For Questions 31 and 32**

**X and Y are two volatile liquids with molar weights of 10 g mol ^{-1} and 40 g mol^{-1} respectively. Two cotton plugs, one soaked in X and the other soaked in Y, are simultaneously placed at the ends of a tube of length L = 24 cm, as shown in the figure. The tube is filled with an inert gas at 1 atmosphere pressure and a temperature of 300 K. Vapours of X and Y react to form a product which is first observed at a distance d cm from the plug soaked in X.Take X and Y to have equal molecular diameters and assume ideal behaviour for the inert gas and the two vapours.**

Q.

The value of d in cm (shown in the figure), as estimated from Graham’s law, is

Solution:

QUESTION: 32

**X and Y are two volatile liquids with molar weights of 10 g mol ^{-1} and 40 g mol^{-1} respectively. Two cotton plugs, one soaked in X and the other soaked in Y, are simultaneously placed at the ends of a tube of length L = 24 cm, as shown in the figure. The tube is filled with an inert gas at 1 atmosphere pressure and a temperature of 300 K. Vapours of X and Y react to form a product which is first observed at a distance d cm from the plug soaked in X.Take X and Y to have equal molecular diameters and assume ideal behaviour for the inert gas and the two vapours.**

Q.

The experimental value of d is found to be smaller than the estimate obtained using Graham’s law. This is

due to

Solution:

As the collision frequency increases then molecular speed decreases than the expected.

QUESTION: 33

**Paragraph For Questions 33 and 34**

Schemes 1 and 2 describe sequential transformation of alkynes M and N. Consider only the major products formed in each step for both schemes

**Q.**

**The product X is**

Solution:

QUESTION: 34

Schemes 1 and 2 describe sequential transformation of alkynes M and N. Consider only the major products formed in each step for both schemes

Q.

The correct statement with respect to product Y is

Solution:

QUESTION: 35

Paragraph For Questions 35 and 36

An aqueous solution of metal ion M1 reacts separately with reagents Q and R in excess to give tetrahedral and square planar complexes, respectively. An aqueous solution of another metal ion M2 always forms tetrahedral complexes with these reagents. Aqueous solution of M2 on reaction with reagent S gives white precipitate which dissolves in excess of S. The reactions are summarized in the scheme given below.

Q.

M1, Q and R, respectively are

Solution:

QUESTION: 36

An aqueous solution of metal ion M1 reacts separately with reagents Q and R in excess to give tetrahedral and square planar complexes, respectively. An aqueous solution of another metal ion M2 always forms tetrahedral complexes with these reagents. Aqueous solution of M2 on reaction with reagent S gives white precipitate which dissolves in excess of S. The reactions are summarized in the scheme given below.

Q.

Reagent S is

Solution:

QUESTION: 37

**Q. No 37- 40 carry 3 marks each and 1 as negative marking for incorrect answer**

**This section contains four questions, each having two matching lists. Choices for the correct combination of elements from List-I and List-II are given as options (A), (B), (C) and (D), out of which one is correct.**

Q.

Match each coordination compound in List-I with an appropriate pair of characteristics from List-II and

select the correct answer using the code given below the lists

{en = H_{2}NCH_{2}CH_{2}NH_{2}; atomic numbers: Ti = 22, Cr = 24; Co= 27; Pt = 78)

Solution:

(P) [Cr(NH_{3})_{4}Cl_{2}]Cl → Paramagnetic and exhibits cis-trans isomerism

(Q) [Ti(H_{2}O)_{5}Cl](NO_{3})_{2} → Paramagnetic and exhibits ionization isomerism

(R) [Pt(en)(NH_{3})Cl]NO_{3} → Diamagnetic and exhibits ionization isomerism

(S) [Co(NH_{3})_{4}(NO_{3})_{2}]NO_{3} → Diamagnetic and exhibits cis-trans isomerism

QUESTION: 38

Match the orbital overlap figures shown in List-I with the description given in List-II and select the correct

answer using the code given below the lists.

Solution:

QUESTION: 39

Different possible thermal decomposition pathways for peroxyesters are shown below. Match each

pathway from List I with an appropriate structure from List II and select the correct answer using the code

given below the lists.

Solution:

QUESTION: 40

Match the four starting materials (P, Q, R, S) given in List I with the corresponding reaction schemes

(I, II, III, IV) provided in List II and select the correct answer using the code given below the lists.

Solution:

QUESTION: 41

Q. No 41- 50 carry 3 marks each and 1 as negative marking for incorrect answer

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

Q.

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at

least one more than the number of girls ahead of her, is

Solution:

Either a girl will start the sequence or will be at second position and will not acquire the last position as

well.

QUESTION: 42

In a triangle the sum of two sides is x and the product of the same two sides is y. If x^{2} - c^{2} = y, where c is

the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

Solution:

QUESTION: 43

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that

each envelope contains exactly one card and no card is placed in the envelope bearing the same number and

moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can

be done is

Solution:

QUESTION: 44

The common tangents to the circle x^{2} + y^{2} = 2 and the parabola y^{2} = 8x touch the circle at the points P, Q

and the parabola at the points R, S. Then the area of the quadrilateral PQRS is

Solution:

QUESTION: 45

The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation

p(p(x)) = 0 has

Solution:

Hence real or purely imaginary number can not satisfy P(P(x)) = 0.

QUESTION: 46

The following integral is equal to

Solution:

QUESTION: 47

The function y = f (x) is the solution of the differential equation satisfying

Solution:

QUESTION: 48

Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1.

Solution:

QUESTION: 49

Coefficient of x^{11} in the expansion of (1 + x^{2})^{4} (1 + x^{3})^{7} (1 + x^{4})^{12} is

Solution:

2x_{1} + 3x_{2} + 4x_{3} = 11

Possibilities are (0, 1, 2); (1, 3, 0); (2, 1, 1); (4, 1, 0).

Required coefficients

QUESTION: 50

For the equation sinx + 2sin2x - sin3x = 3 has

Solution:

which is not possible at same time

Hence, no solution

QUESTION: 51

**SECTION – 2 : Comprehension Type (Only One Option Correct)**

**Q. No 51- 56 carry 3 marks each and 1 as negative marking for incorrect answer**

**This section contains 3 paragraph, each describing theory, experiments, data etc. Six questions relate to the three paragraphs with two questions on each paragraph. Each question has only one correct answer among the four given options (A), (B), (C) and (D).**

**Paragraph For Questions 51 and 52**

Box 1 contains three cards bearing numbers 1, 2, 3 ; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5 ; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let x_{i} be the number on the card drawn from the i^{th} box, i = 1, 2, 3.

Q.

The probability that x_{1} + x_{2} + x_{3} is odd, is

Solution:

Case I : One odd, 2 even

Total number of ways = 2 x 2 x 3 + 1 x 3 x 3 + 1 x 2 x 4 = 29.

Case II: All 3 odd

Number of ways = 2 x 3 x 4 = 24

Favourable ways = 53

QUESTION: 52

**Box 1 contains three cards bearing numbers 1, 2, 3 ; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5 ; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let x _{i} be the number on the card drawn from the i^{th} box, i = 1, 2, 3.**

**Q.**

The probability that x_{1} , x_{2} , x_{3} are in an arithmetic progression, is

Solution:

Here 2x_{2} = x_{1} + x_{3}

x_{1} + x_{3} = even

Hence number of favourable ways =

QUESTION: 53

**Paragraph For Questions 53 and 54**

**Let a, r, s, t be non-zero real numbers. Let P(at ^{2}, 2at), Q, R(ar^{2}, 2ar) and S(as^{2}, 2as) be distinct points on the parabola y^{2} = 4ax. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a, 0).**

**Q.**

**The value of r is**

Solution:

QUESTION: 54

**Let a, r, s, t be non-zero real numbers. Let P(at ^{2}, 2at), Q, R(ar^{2}, 2ar) and S(as^{2}, 2as) be distinct points on the parabola y^{2} = 4ax. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a, 0).**

Q.

If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

Solution:

QUESTION: 55

**Paragraph For Questions 55 and 56**

**Given that for each exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1)**

**Q.**

The value of g(1/2) is

Solution:

QUESTION: 56

**Given that for each exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1)**

**Q.**

**The value of g'(1/2) is **

Solution:

We have g (a) = g (1 - a) and g is differentiable

Hence g'(1/2) =0

QUESTION: 57

**SECTION – 3 : Matching List Type (Only One Option Correct)**

**Q. No 57- 60 carry 3 marks each and 1 as negative marking for incorrect answer**

**This section contains four questions, each having two matching list. Choices for the correct combination of elements from List-I and List-II are given as options (A), (B), (C) and (D), out of which ONE is correct.**

**Q.**

**Match the following**

A

Solution:

QUESTION: 58

Match the following:

Solution:

QUESTION: 59

Let be defined by

Solution:

QUESTION: 60

Let

Solution:

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