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QUESTION: 1

**Q. 1 - 10 carry 3 marks each.**

**Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE THAN ONE are correct.**

**Q.**

**A student is performing an experiment using a resonance column and a tuning fork of frequency**

**244s ^{-1 }. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is (0.350 **

**(Useful information: ** The molar masses M in grams are given in the options. Take the values of for each gas as given there.)

Solution:

Calculations for for gases mentioned in options A, B, C and D, work out to be 0.459 m, 0.363 m 0.340 m & 0.348 m respectively. As . Hence correct option is D.

*Multiple options can be correct

QUESTION: 2

At time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating

current I(t) = I0 cos (ωt), with I0 = 1A and ω = 500 rad/s starts flowing in it with the initial direction shown

in the figure. At the key is switched from B to D. Now onwards only A and D are connected. A

total charge Q flows from the battery to charge the capacitor fully. If C = 20 μF, R = 10 ? and the battery

is ideal with emf of 50 V, identify the correct statement(s).

Solution:

As current leads voltage by π/2 in the given circuit initially, then ac voltage can be represent as

*Multiple options can be correct

QUESTION: 3

A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that

covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the

capacitor is C while that of the portion with dielectric in between is C_{1}. When the capacitor is

charged, the plate area covered by the dielectric gets charge Q_{1} and the rest of the area gets

charge Q_{2}. The electric field in the dielectric is E_{1} and that in the other portion is E_{2}. Choose

the correct option/options, ignoring edge effects.

Solution:

As E = V/d

E_{1}/E_{2} = 1 (both parts have common potential difference)

Assume C_{0} be the capacitance without dielectric for whole capacitor.

*Multiple options can be correct

QUESTION: 4

One end of a taut string of length 3m along the x axis is fixed at x = 0. The speed of the waves in the string

is 100 ms^{-1}. The other end of the string is vibrating in the y direction so that stationary waves are set up in

the string. The possible waveform(s) of these stationary waves is (are)

Solution:

Taking y(t) = A f(x) g(t) & Applying the conditions:

1; here x = 3m is antinode & x = 0 is node

2; possible frequencies are odd multiple of fundamental frequency.

*Multiple options can be correct

QUESTION: 5

A transparent thin film of uniform thickness and refractive index n_{1} = 1.4 is coated on the convex spherical surface of radius R at one end of a long solid glass cylinder of refractive index n_{2} = 1.5, as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance f1 from the film, while rays of light traversing from glass to air get focused at distance f_{2} from the film. Then

Solution:

*Multiple options can be correct

QUESTION: 6

Heater of an electric kettle is made of a wire of length L and diameter d. It takes 4 minutes to raise the

temperature of 0.5 kg water by 40 K. This heater is replaced by a new heater having two wires of the same

material, each of length L and diameter 2d. The way these wires are connected is given in the options.

How much time in minutes will it take to raise the temperature of the same amount of water by 40K?

Solution:

*Multiple options can be correct

QUESTION: 7

Two ideal batteries of emf V_{1} and V_{2} and three resistances R_{1}, R_{2} and R_{3} are connected as shown in the figure. The current in resistance R_{2} would be zero if

Solution:

*Multiple options can be correct

QUESTION: 8

Let E_{1} (r), E_{2} (r) and E_{3} (r) be the respective electric fields at a distance r from a point charge Q, an

infinitely long wire with constant linear charge density λ, and an infinite plane with uniform surface charge

density σ. If E_{1}(r_{0} ) = E_{2} (r_{0} ) = E_{3} (r_{0} ) at a given distance r0, then

Solution:

*Multiple options can be correct

QUESTION: 9

A light source, which emits two wavelengths λ_{1} = 400 nm and λ_{2} = 600 nm, is used in a Young’s double

slit experiment. If recorded fringe widths for λ_{1} and λ_{2} are β_{1} and β_{2} and the number of fringes for them

within a distance y on one side of the central maximum are m_{1} and m_{2}, respectively, then

Solution:

*Multiple options can be correct

QUESTION: 10

In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle

θ with the horizontal floor. The coefficient of friction between the wall and the ladder is μ1 and that

between the floor and the ladder is μ_{2}. The normal reaction of the wall on the ladder is N1 and that of the

floor is N_{2}. If the ladder is about to slip, then

Solution:

Condition of translational equilibrium

Applying torque equation about corner (left) point on the floor

*Answer can only contain numeric values

QUESTION: 11

**Q. No. 11 - 20 carry 3 Marks each**

**Each question, when worked out will result in one integer from 0 to 9 (both inclusive).**

**Q.**

During Searle’s experiment, zero of the Vernier scale lies between 3.20 x 10^{-2} m and 3.25 x 10^{-2} m of the

main scale. The 20th division of the Vernier scale exactly coincides with one of the main scale divisions.

When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between 3.20 x

10^{-2} m and 3.25 x10^{-2} m of the main scale but now the 45th division of Vernier scale coincides with one of

the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is 8 x 10^{-7}

m^{2}. The least count of the Vernier scale is 1.0 x 10^{-5} m. The maximum percentage error in the Young’s

modulus of the wire is

Solution:

change in lengths = 25(LC)

and the maximum permissible error in elongation is one LC

*Answer can only contain numeric values

QUESTION: 12

Airplanes A and B are flying with constant velocity in the same vertical plane at angles 30^{0} and 60^{0} with

respect to the horizontal respectively as shown in the figure. The speed of A is 100√3 ms^{-1}. At time t = 0

s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity

perpendicular to the line of motion of A. If at t = t_{0}, A just escapes being hit by B, t_{0} in seconds is

Solution:

The relative velocity of B with respect to A is perpendicular to line of motion of A.

*Answer can only contain numeric values

QUESTION: 13

A thermodynamic system is taken from an initial state i with internal energy Ui = 100 J to the final state f

along two different paths *iaf and ibf*, as schematically shown in the figure. The work done by the system

along the paths *af, ib* and *bf* are W_{af} = 200 J, W_{ib} = 50 J and W_{bf} = 100 J respectively. The heat supplied to

the system along the path iaf, ib and bf are iaf ib Q , Q and Q_{bf} respectively. If the internal energy of the

system in the state b is U_{b} = 200 J and Q_{iaf} = 500 J, the ratio Q_{bf} / Q_{ib} is

Solution:

U_{b} = 200 J, U_{i} =100 J

Process iaf

*Answer can only contain numeric values

QUESTION: 14

Two parallel wires in the plane of the paper are distance X_{0} apart. A point charge is moving with speed u

between the wires in the same plane at a distance X_{1} from one of the wires. When the wires carry current of

magnitude I in the same direction, the radius of curvature of the path of the point charge is R_{1}. In contrast,

if the currents I in the two wires have directions opposite to each other, the radius of curvature of the path is R_{2}. id X_{0}/X_{1 } =3 the value of R_{1}/ R_{2 }is

Solution:

*Answer can only contain numeric values

QUESTION: 15

To find the distance d over which a signal can be seen clearly in foggy conditions, a railways engineer uses

dimensional analysis and assumes that the distance depends on the mass density ρ of the fog, intensity

(power/area) S of the light from the signal and its frequency f. The engineer finds that d is proportional to

S^{1/n}. The value of n is

Solution:

*Answer can only contain numeric values

QUESTION: 16

A rocket is moving in a gravity free space with a constant acceleration of 2 ms^{–2} along + x direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in + x direction with a speed of 0.3 ms^{–1} relative to the rocket. At the same time, another ball is thrown in +x direction with a speed of 0.2 ms^{–1} from its right end relative to the rocket. The time in seconds when the two balls hit each other is

Solution:

Maximum displacement of the left ball from the left wall of the chamber is 2.25 cm, so the right ball has to travel almost the whole length of the chamber (4m) to hit the left ball. So the time taken by the right ball is

1.9 sec (approximately 2 sec)

*Answer can only contain numeric values

QUESTION: 17

A galvanometer gives full scale deflection with 0.006 A current. By connecting it to a 4990 resistance, it can be converted into a voltmeter of range 0 – 30 V. If connected to a 2n/249 resistance, it becomes an ammeter of range 0 – 1.5 A. The value of n is

Solution:

*Answer can only contain numeric values

QUESTION: 18

A uniform circular disc of mass 1.5 kg and radius 0.5 m isinitially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along the three sides of an equilateral

triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad s^{–1} is

Solution:

*Answer can only contain numeric values

QUESTION: 19

A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance 0.25 m from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have horizontal speed of 9 ms^{–1} with respect to the ground. The rotational speed of the platform in rad s^{–1} after the balls leave the platform is

Solution:

Since net torque about centre of rotation is zero, so we can apply conservation of angular momentum of the

system about center of disc

L_{i} = L_{f}

0 = Iω + 2mv (r/2); comparing magnitude

*Answer can only contain numeric values

QUESTION: 20

Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is (n×10) Joules. The

value of n is (take acceleration due to gravity = 10 ms^{–2})

Solution:

*Multiple options can be correct

QUESTION: 21

**Q. No. 21- 30 carry 3 mark each.**

**Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE THAN ONE are correct.**

**Q. **The correct combination of names for isomeric alcohols with molecular formula C_{4}H_{10}O is/are

Solution:

*Multiple options can be correct

QUESTION: 22

An ideal gas in a thermally insulated vessel at internal pressure = P_{1}, volume = V_{1} and absolute

temperature = T_{1} expands irreversibly against zero external pressure, as shown in the diagram. The final

internal pressure, volume and absolute temperature of the gas are P_{2}, V_{2} and T_{2}, respectively. For this

expansion,

Solution:

Since container is thermally insulated. So, q = 0, and it is a case of free expansion therefore W = 0 and

ΔE = 0

So, T_{1} = T_{2}

Also, P_{1}V_{1} = P_{2}V_{2}

*Multiple options can be correct

QUESTION: 23

Hydrogen bonding plays a central role in the following phenomena:

Solution:

(A) Ice has cage-like structure in which each water molecule is surrounded by four other water molecules tetrahedrally through hydrogen boding, due to this density of ice is less than water and it floats in

water.

The cation (I) more stabilized through hydrogen boding than cation (II) . So, R–NH_{2} is better base than (R)_{3}N in aqueous solution.

(C) HCOOH is stronger acid than CH_{3}COOH due to inductive effect and not due to hydrogen bonding.

(D) Acetic acid dimerizes in benzene through intermolecular hydrogen bonding.

*Multiple options can be correct

QUESTION: 24

In a galvanic cell, the salt bridge

Solution:

*Multiple options can be correct

QUESTION: 25

For the reaction:

The correct statement(s) in the balanced equation is/are:

Solution:

The balanced equation is,

*Multiple options can be correct

QUESTION: 26

The reactivity of compound Z with different halogens under appropriate conditions is given below:

The observed pattern of electrophilic substitution can be explained by

Solution:

*Multiple options can be correct

QUESTION: 27

The correct statement(s) for orthoboric acid is/are

Solution:

(A) H_{3}BO_{3} is a weak monobasic Lewis acid.

(B) Equilibrium (i) is shifted in forward direction by the addition of syn-diols like ethylene glycol which forms a stable complex with

(C) It has a planar sheet like structure due to hydrogen bonding.

(D) H_{3}BO_{3} is a weak electrolyte in water.

*Multiple options can be correct

QUESTION: 28

Upon heating with Cu_{2}S, the reagent(s) that give copper metal is/are

Solution:

*Multiple options can be correct

QUESTION: 29

The pair(s) of reagents that yield paramagnetic species is/are

Solution:

(A) sodium (Na) when dissolved in excess liquid ammonia, forms a blue coloured paramagnetic solution.

(B)

(c)

(D)

*Multiple options can be correct

QUESTION: 30

In the reaction shown below, the major product(s) formed is/are

Solution:

Only amines undergo acetylation and not acid amides

*Answer can only contain numeric values

QUESTION: 31

Q. No. 31 - 40 Carry 3 marks each

Each question, when worked out will result in one integer from 0 to 9 (both inclusive).

Q.

Among PbS, CuS, HgS, MnS, Ag_{2}S, NiS, CoS, Bi_{2}S_{3} and SnS_{2}, the total number of BLACK coloured

sulphides is

Solution:

Black coloured sulphides (PbS,CuS,HgS,Ag_{2}S,NiS,CoS)

* Bi_{2}S_{3} in its crystalline form is dark brown but Bi_{2}S_{3} precipitate obtained is black in colour.

*Answer can only contain numeric values

QUESTION: 32

The total number(s) of stable conformers with non-zero dipole moment for the following compound is(are)

Solution:

*Answer can only contain numeric values

QUESTION: 33

Consider the following list of reagents:

Acidified K_{2}Cr_{2}O_{7} , alkaline KMnO_{4} ,CuSO_{4},H_{2}O_{2} ,Cl_{2} ,O_{3},FeCl_{3},HNO_{3} and Na_{2}S_{2}O_{3} .

The total number of reagents that can oxidise aqueous iodide to iodine is

Solution:

*Answer can only contain numeric values

QUESTION: 34

A list of species having the formula XZ4 is given below.

Defining shape on the basis of the location of X and Z atoms, the total number of species having a square

planar shape is

Solution:

*Answer can only contain numeric values

QUESTION: 35

Consider all possible isomeric ketones, including stereoisomers of MW = 100. All these isomers are

independently reacted with NaBH_{4} (NOTE: stereoisomers are also reacted separately). The total number of

ketones that give a racemic product(s) is/are

Solution:

*Answer can only contain numeric values

QUESTION: 36

In an atom, the total number of electrons having quantum numbers n = 4, |m_{l}| = 1 and m_{s} = –1/2 is

Solution:

*Answer can only contain numeric values

QUESTION: 37

If the value of Avogadro number is 6.023 x 1023 mol^{–1} and the value of Boltzmann constant is

1.380x10^{-23} JK^{-1} , then the number of significant digits in the calculated value of the universal gas constant

is

Solution:

*Answer can only contain numeric values

QUESTION: 38

MX_{2} dissociates in M^{2+} and X^{-} ions in an aqueous solution, with a degree of dissociation (α) of 0.5. The

ratio of the observed depression of freezing point of the aqueous solution to the value of the depression of

freezing point in the absence of ionic dissociation is

Solution:

*Answer can only contain numeric values

QUESTION: 39

The total number of distinct naturally occurring amino acids obtained by complete acidic hydrolysis of the

peptide shown below is

Solution:

This peptide on complete hydrolysis produced 4 distinct amino acids which are given below:

*Answer can only contain numeric values

QUESTION: 40

A compound H_{2}X with molar weight of 80g is dissolved in a solvent having density of 0.4 gml^{–1}. Assuming

no change in volume upon dissolution, the molality of a 3.2 molar solution is

Solution:

*Multiple options can be correct

QUESTION: 41

**Q. No. 41 to 50 carry 3 marks each.**

**Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE THAN ONE are correct.**

**Q.**

Let be given by

Solution:

Hence, an odd function

*Multiple options can be correct

QUESTION: 42

Let a R and let f : R → R be given by f (x) = x^{5} - 5x + a, then

Solution:

*Multiple options can be correct

QUESTION: 43

For every pair of continuous functions , g : [0, 1] → R such that max{ f (x) : x [0, 1]} = max{g(x) : x

[0, 1]}, the correct statement(s) is(are)

Solution:

Let f (x) and g (x) achieve their maximum value at x_{1} and x_{2} respectively

h (x) = f (x) - g (x)

h (x1) = f (x_{1}) - g (x_{1}) 0

h (x2) = f (x_{2}) - g (x_{2}) 0

h (c) = 0 where c [0, 1] f (c) = g (c).

*Multiple options can be correct

QUESTION: 44

A circle S passes through the point (0, 1) and is orthogonal to the circles (x - 1)^{2} + y^{2} = 16 and x^{2} + y^{2} = 1.

Then

Solution:

*Multiple options can be correct

QUESTION: 45

Let be three vectors each of magnitude √2 and the angle between each pair of them is π/3. If is a non-zero vector perpendicular to is a non-zero vector perpendicular to and

Solution:

*Multiple options can be correct

QUESTION: 46

From a point P(λ, λ, λ), perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x, z = -1. If P is such that QPR is a right angle, then the possible value(s) of λ is(are)

Solution:

*Multiple options can be correct

QUESTION: 47

Let M be a 2 x 2 symmetric matrix with integer entries. Then M is invertible if

Solution:

*Multiple options can be correct

QUESTION: 48

Let M and N be two 3 x 3 matrices such that MN = NM. Further, if M N^{2} and M^{2} = N^{4}, then

Solution:

*Multiple options can be correct

QUESTION: 49

Let be a continuous function and let g : R → R be defined as

Then

Solution:

Hence g(x) is not differentiable at x = a

Similarly LHD at x = b is greater than 1

g(x) is not differentiable at x = b

*Multiple options can be correct

QUESTION: 50

Let be given by f(x) = (log(sec x + tan x))^{3}. Then

Solution:

*Answer can only contain numeric values

QUESTION: 51

**Each question, when worked out will result in one integer from 0 to 9 (both inclusive).**

**Q.**

Let n_{1} < n_{2} < n_{3} < n_{4} < n_{5} be positive integers such that n_{1} + n_{2} + n_{3} + n_{4} + n_{5} = 20. Then the number of such distinct arrangements (n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) is _________

Solution:

When n_{5} takes value from 10 to 6 the carry forward moves from 0 to 4 which can be arranged in

*Answer can only contain numeric values

QUESTION: 52

Let n 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment.

Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of

red and blue line segments are equal, then the value of n is _________

Solution:

*Answer can only contain numeric values

QUESTION: 53

Let f : R → R and g : R → R be respectively given by by

Then number of points at which h(x) is not differentiable is __________

Solution:

*Answer can only contain numeric values

QUESTION: 54

Let a, b, c be positive integers such that b/a is an integer. If a, b, c are in geometric progression and the

arithmetic mean of a, b, c is b + 2, then the value of

Solution:

*Answer can only contain numeric values

QUESTION: 55

Let be three non-coplanar unit vectors such that the angle between every pair of them is π/3 . if where p, q and r are scalars, then the value of is ___________________

Solution:

*Answer can only contain numeric values

QUESTION: 56

The slope of the tangent to the curve at the point (1, 3) is ______

Solution:

*Answer can only contain numeric values

QUESTION: 57

The value of

Solution:

*Answer can only contain numeric values

QUESTION: 58

The largest value of the non-negative integer a for which

Solution:

*Answer can only contain numeric values

QUESTION: 59

Let be defined by f (x) = cos^{-1}(cos x). The number of points x [0, 4π] satisfying the

equation

Solution:

*Answer can only contain numeric values

QUESTION: 60

For a point P in the plane, let d_{1}(P) and d_{2}(P) be the distances of the point P from the lines x - y = 0 and x +

y = 0 respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane

and satisfying is ___________________

Solution:

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