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QUESTION: 1

**SECTION I**

**Q. No. 1 - 10 carry 3 marks each and 1 mark is deducted for every wrong answer.**

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

**Q.**

In the determination of Young’s modulus by using Searle’s method, a wire of length L = 2m

and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension = 0.25 mm in the length of the wire

is observed. Quantities d and are measured using a screw gauge and a micrometer, respectively. They

have the same pitch of 0.5 mm. The number of divisions on their circular scale is 100. The contributions to

the maximum probable error of the Y measurement

Solution:

QUESTION: 2

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The

mass is undergoing circular motion in the x-y plane with centre at O and constant angular speed ω. If the

angular momentum of the system, calculated about O and P are denoted by respectively, then

Solution:

QUESTION: 3

A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index n of

the first lens is 1.5 and that of the second lens is 1.2. Both the curved surface are of the same radius of

curvature R = 14 cm. For this bi-convex lens, for an object distance of 40 cm, the image distance will be

Solution:

QUESTION: 4

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed ω, as shown in the figure. At time t = 0, a small insect starts from O and moves with constant speed v, with respect to the

rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains ω throughout. The magnitude of the torque about O, as a function of time is best represented by which plot?

Solution:

QUESTION: 5

A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40

amu) is kept at 300 K in a container. The ratio of the rms speeds is

Solution:

QUESTION: 6

Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage

source of potential difference X. A proton is released at rest midway between the two plates. It is found to

move at 450 to the vertical JUST after release. Then X is nearly

Solution:

QUESTION: 7

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal

black surfaces and have very high thermal conductivity. The first and third plates are maintained at

temperatures 2T and 3T respectively. The temperature of the middle (i.e. second) plate under steady state

condition is

Solution:

QUESTION: 8

A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of

the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is

stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular

frequency ω = π/3 rad/s. Simultaneously at t = 0, a small pebble is projected with speed v form point P at

an angle of 450 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits

the block at t = 1 s, the value of v is (take g = 10 m/s^{2})

Solution:

QUESTION: 9

Young’s double slit experiment is carried out by using green, red and blue light, one color at a time. The

fringe widths recorded are respectively. Then

Solution:

QUESTION: 10

Consider a thin spherical shell of radius R with centre at the origin, carrying uniform positive surface

charge density. The variation of the magnitude of the electric field and the electric potential V(r)

with the distance r from the centre, is best represented by which graph?

Solution:

*Multiple options can be correct

QUESTION: 11

**SECTION II**

**Q. No. 11 -15 carry 4 mark each**

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

**Q.**

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric

and magnetic fields At time t = 0, this charge has velocity in the x-y plane, making an angle q with the x-axis. Which of the following option(s) is (are) correct for time t > 0?

Solution:

If θ = 90°,

If θ = 0° 10° ;the charge particle moves in helix with increasing pitch due to

*Multiple options can be correct

QUESTION: 12

A cubical region of side a has its centre at the origin. It encloses three fixed point charges, -q at

(0, -a/4, 0), +3q at (0, 0, 0) and -q at (0, +a/4, 0). Choose the correct options(s)

Solution:

Net flux through the cubical region

The flux passing through the faces are same due to symmetry

*Multiple options can be correct

QUESTION: 13

A person blows into open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe.

When this pulse reaches the other end of the pipe,

Solution:

At the open end, the phase of a pressure wave changes by π radian due to reflection. At the closed end, there is no change in the phase of a pressure wave due to reflection.

*Multiple options can be correct

QUESTION: 14

A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle θ with the

horizontal. A horizontal force of 1 N acts on the block through its centre of mass as shown in the figure.

The block remains stationary if (take g = 10 m/s^{2})

Solution:

At θ = 45° , mg sin q = 1xcosθ

At θ > 45° , mg sin q > 1xcosθ (friction acts upward)

At θ < 45° , mg sin q < 1xcosθ (friction acts downward)

*Multiple options can be correct

QUESTION: 15

For the resistance network shown in the figure, choose the correct option(s)

Solution:

Nodes P and Q are equipotential and nodes S

and T are equipotential from wheatstone bridge, no current passes through PQ and ST.

*Answer can only contain numeric values

QUESTION: 16

**SECTION III**

**Q. No. 16 - 20 carry 4 marks each**

**The answer to each question is single digit integer, ranging from 0 to 9 (both inclusive).**

**A circular wire loop of radius R is placed in the x-y plane centered at the origin O. A square loop of side a(a<<R) having two turns is placed with its centre at z = along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of 45° with respect to the z-axis. If the mutual inductance between the loops is given by then the value of p is**

Solution:

QUESTION: 17

An infinitely long solid cylinder of radius R has a uniform volume charge density r. It has a spherical

cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the

electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the

expression The value of k is

Solution:

*Answer can only contain numeric values

QUESTION: 18

A proton is fired from very far away towards a nucleus with charge Q = 120 e, where e is the electronic

charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm) of

the proton at its start is: (take the proton mass, m_{p} = (5/3) x 10^{-27} kg; h/e = 4.2 x 10^{-15} J.s / C;

Solution:

*Answer can only contain numeric values

QUESTION: 19

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and

radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing though O and P

is I_{O} and I_{P} respectively. Both these axes are perpendicular to the plane of the lamina. The ratio I_{P} / I_{O} to

the nearest integer is

Solution:

*Answer can only contain numeric values

QUESTION: 20

A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the

cylinder and the cavity are infinity long. A uniform current density J flows along the length. If the

magnitude of the magnetic field at the point P is given by then the value of N is

Solution:

QUESTION: 21

**SECTION – I**

**Q. No. 21- 30 caeey 3 marks each and 1 mark is deducted for every wrong answer**

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

**Q.**

A compound M_{p}X_{q} has cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below.

The empirical formula of the compound is

Solution:

So, unit cell formula of the compound is M_{2}X_{4} and the empirical formula of the compound is MX_{2}.

QUESTION: 22

The carboxyl functional group (–COOH) is present in

Solution:

QUESTION: 23

As per IUPAC nomenclature, the name of the complex [Co(H_{2}O)_{4}(NH_{3})_{2}]Cl_{3} is

Solution:

[Co(H_{2}O)_{4}(NH_{3})_{2}]Cl_{3}

Diamminetetraaquacobalt (III) chloride

QUESTION: 24

In allene (C_{3}H_{4}), the type(s) of hybridization of the carbon atoms is (are)

Solution:

QUESTION: 25

The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a_{0} is Bohr radius]

Solution:

QUESTION: 26

Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?

Solution:

QUESTION: 27

For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. 1/V plot is shown below. The

value of the van der Waals constant a (atm.litre^{2}mol^{–2}) is

Solution:

QUESTION: 28

The number of aldol reaction(s) that occurs in the given transformation is

Solution:

QUESTION: 29

The colour of light absorbed by an aqueous solution of CuSO_{4} is

Solution:

Aqueous solution of copper sulphate absorbs orange red light and appears blue (complementary colour).

QUESTION: 30

The number of optically active products obtained from the complete ozonolysis of the given compound is

Solution:

*Multiple options can be correct

QUESTION: 31

**SECTION II**

**Q. No. 31 -35 carry 4 marks each**

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

**Q.**

**Which of the following hydrogen halides react(s) with AgNO _{3}(aq) to give a precipitate that dissolves in Na- 2S_{2}O_{3}(aq)?**

Solution:

*Multiple options can be correct

QUESTION: 32

Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction, as

shown in the given scheme

Solution:

(A) Both are soluble in NaOH, hence inseparable.

(B) Only benzoic acid (C_{6}H_{5}COOH) is soluble in NaOH and NaHCO3, while benzyl alcohol (C_{6}H_{5}CH_{2}OH)

is not. Hence, separable.

(C) Although NaOH can enable separation between benzyl alcohol (C_{6}H_{5}CH_{2}OH) and phenol (C6H5OH) as

only the later is soluble in NaOH. However, in NaHCO_{3}, both are insoluble. Hence, inseparable.

(D) a-phenyl acetic acid (C_{6}H_{5}CH_{2}OH) is soluble in NaOH and NaHCO_{3}. While benzyl alcohol

(C_{6}H_{5}CH_{2}OH) is not. Hence, separable.

*Multiple options can be correct

QUESTION: 33

For an ideal gas, consider only P-V work in going from an initial state X to the final state Z. The final state

Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is(are)

correct? [Take DS as change in entropy and w as work done]

Solution:

*Multiple options can be correct

QUESTION: 34

Which of the following molecules, in pure form, is (are) unstable at room temperature?

Solution:

Compound being antiaromatic are unstable at room temperature

*Multiple options can be correct

QUESTION: 35

Choose the correct reason(s) for the stability of the lyophobic colloidal particles.

Solution:

Lyophobic colloids are stable due to preferential adsorption of ions on their surface from solution and potential difference between the fixed layer and the diffused layer of opposite charges around the colloidal

particles that makes lyophobic sol stable

*Answer can only contain numeric values

QUESTION: 36

**SECTION III**

**Q. No. 36 - 40 carry 4 marks each**

**The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).**

**Q.**

29.2 % (w/w) HCl stock solution has density of 1.25 g mL^{-1}. The molecular weight of HCl is 36.5 g mol^{-1}.

The volume (mL) of stock solution required to prepare a 200 mL solution of 0.4 M HCl is

Solution:

*Answer can only contain numeric values

QUESTION: 37

The substituents R_{1} and R_{2} for nine peptides are listed in the table given below. How many of these

peptides are positively charged at pH = 7.0?

Solution:

Peptides with isoelectric point (pI) > 7, would exist as cation in neutral solution (pH = 7).

IV, VI, VIII and IX

*Answer can only contain numeric values

QUESTION: 38

An organic compound undergoes first-order decomposition. The time taken for its decomposition to 1/8 and

1/10 of its initial concentration are t_{1/8} and t_{1/10} respectively. What is the value of

0.3

Solution:

*Answer can only contain numeric values

QUESTION: 39

When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its

pyranose form is

Solution:

Hence total number of stereoisomers in pyranose form of D-configuration = 2^{3} = 8

*Answer can only contain numeric values

QUESTION: 40

The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes a

nuclear reaction yielding element X as shown below. To which group, element X belongs in the periodic

table?

Solution:

QUESTION: 41

**SECTION I**

**Q. No 42- 50 carry 3 marks each and 1 mark is deducted for every wrong answer.**

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

**Q.**

if , then

Solution:

QUESTION: 42

Let P = [a_{ij}] be a 3 × 3 matrix and let If the determinant of P is 2, then the determinant of the matrix Q is

Solution:

QUESTION: 43

The locus of the mid–point of the chord of contact of tangents drawn from points lying on the straight line

4x – 5y = 20 to the circle x^{2} + y^{2} = 9 is

Solution:

QUESTION: 44

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that

each person gets at least one ball is

Solution:

QUESTION: 45

The integral equals (for some arbitrary constant K)

Solution:

QUESTION: 46

The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, –1, 4) with the plane

5x – 4y – z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length

of the line segment PS is

Solution:

QUESTION: 47

Let

Solution:

QUESTION: 48

Let z be a complex number such that the imaginary part of z is nonzero and a = z^{2} + z + 1 is real. Then a

cannot take the value

Solution:

Given equation is z^{2} + z + 1 - a = 0

Clearly this equation do not have real roots if

D < 0

1 - 4(1 - a) < 0

4a < 3

a<3/4

QUESTION: 49

The ellipse is inscribed in a rectangle R whose sides are parallel to the coordinate axes.

Another ellipse E_{2} passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the

ellipse E_{2} is

Solution:

**Alternate**

Let the ellipse be as it is passing through (0, 4) and (3, 2).

QUESTION: 50

The function f : [0, 3] → [1, 29], defined by f(x) = 2x^{3} – 15x^{2} + 36x + 1, is

Solution:

*Multiple options can be correct

QUESTION: 51

**SECTION II**

**Q. No. 51 -55 carry 4 marks each.**

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

**Q.**

**Tangents are drawn to the hyperbola ** parallel to the straight line 2x - y = 1. The points of contact of the tangents on the hyperbola are

Solution:

Slope of tangent = 2

*Multiple options can be correct

QUESTION: 52

Let be such that and Then φ cannot satisfy

Solution:

*Multiple options can be correct

QUESTION: 53

If y(x) satisfies the differential equation y' - ytanx = 2x secx and y(0) = 0, then

Solution:

*Multiple options can be correct

QUESTION: 54

A ship is fitted with three engines E_{1}, E_{2} and E_{3}. The engines function independently of each other with

respective probabilities 1/2, 1/4, and 1/4 For the ship to be operational at least two of its engines must

function. Let X denote the event that the ship is operational and let X_{1}, X_{2} and X_{3} denote respectively the

events that the engines E_{1}, E_{2} and E_{3} are functioning. Which of the following is(are) true ?

Solution:

(B) P [exactly two engines of the ship are functioning

*Multiple options can be correct

QUESTION: 55

Let S be the area of the region enclosed by

Then

Solution:

*Answer can only contain numeric values

QUESTION: 56

**SECTION III**

**Q. No. 56 - 60 carry 4 marks each**

**The answer to each question is single digit integer, ranging from 0 to 9 (both inclusive).**

**Q.**

if are unit vectors satisfying is

Solution:

*Answer can only contain numeric values

QUESTION: 57

Let f : IR → IR be defined as The total number of points at which f attains either a local

maximum or a local minimum is

Solution:

so, total number of points of local maximum or minimum is 5.

*Answer can only contain numeric values

QUESTION: 58

Let S be the focus of the parabola y^{2} = 8x and let PQ be the common chord of the circle x^{2} + y^{2} - 2x - 4y =

0 and the given parabola. The area of the triangle PQS is

Solution:

The parabola is x = 2t^{2}, y = 4t

Solving it with the circle we get :

so, the points P and Q are (0, 0) and (2, 4) which are also diametrically opposite points on the circle. The

focus is S º (2, 0).

*Answer can only contain numeric values

QUESTION: 59

Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x

= 3. If p(1) = 6 and p(3) = 2, then p'(0) is

Solution:

*Answer can only contain numeric values

QUESTION: 60

The value of

Solution:

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