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QUESTION: 1

**SECTION – I (Single Correct Choice Type)**

This Section contains 6 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.

Q. A block of mass 2 kg is free to move along the x-axis. It is at rest and from t = 0 onwards it is subjected to a time-dependent force F(t) in the x direction. The force F(t) varies with t as shown in the figure. The kinetic

energy of the block after 4.5 seconds is

Solution:

QUESTION: 2

A uniformly charged thin spherical shell of radius R carries uniform surface charge density of σ per unit

area. It is made of two hemispherical shells, held together by pressing them with force F (see figure). F is

proportional to

Solution:

QUESTION: 3

A tiny spherical oil drop carrying a net charge q is balanced in still air with a vertical uniform electric field

of strength When the field is switched off, the drop is observed to fall with terminal

velocity 2 × 10^{−3} ms^{−1}. Given g = 9.8 ms^{−2}, viscosity of the air = 1.8 × 10^{−5} Ns m^{−2} and the density of oil =

900 kg m^{−3}, the magnitude of q is

Solution:

QUESTION: 4

A Vernier calipers has 1 mm marks on the main scale. It has 20 equal divisions on the Vernier scale which

match with 16 main scale divisions. For this Vernier calipers, the least count is

Solution:

QUESTION: 5

A biconvex lens of focal length 15 cm is in front of a plane mirror. The distance between the lens and the mirror is 10 cm. A small object is kept at a distance of 30 cm from the lens. The final image is

Solution:

Object is placed at distance 2 f from the lens. So first image I_{1 }will be formed at distance 2 f on other side. This image I_{1} will behave like a virtual object for mirror. The second image I_{2 }will be formed at distance 20 cm in front of the mirror, or at distance 10 cm to the left hand side of the lens.

Now applying lens formula

Therefore, the final image is at distance 16 cm from the mirror. But, this image will be real.

This is because ray of light is travelling from right to left.

∴ The correct option is (b).

QUESTION: 6

A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform string is vibrating

in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the

wire is 50 N and the speed of sound is 320 ms^{−1}, the mass of the string is

Solution:

QUESTION: 7

**SECTION –II (Integer Type)**

This Section contains 5 questions. The answer to each question is a single-digit integer, ranging from 0 to 9.

Q. A large glass slab (μ = 5/3) of thickness 8 cm is placed over a point source of light on a plane surface. It is

seen that light emerges out of the top surface of the slab from a circular area of radius R cm. What is the

value of R?

Solution:

*Answer can only contain numeric values

QUESTION: 8

Image of an object approaching a convex mirror of radius of curvature 20 m along its optical axis is

observed to move from m in 30 seconds. What is the speed of the object in km per hour ?

Solution:

QUESTION: 9

To determine the half life of a radioactive element, a student plots a graph of is the rate of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 years, the value of p is

Solution:

*Answer can only contain numeric values

QUESTION: 10

A diatomic ideal gas is compressed adiabatically to of its initial volume. If the initial temperature of

the gas is T_{i} (in Kelvin) and the final temperature is aT_{i}, the value of a is

Solution:

*Answer can only contain numeric values

QUESTION: 11

At time t = 0, a battery of 10 V is connected across points A and B in the given circuit. If the capacitors have no charge initially, at what time (in seconds) does the voltage across them becomes 4 volt?

[take ln 5 = 1.6, ln 3 = 1.1]

Solution:

QUESTION: 12

**SECTION –III (Paragraph Type)**

This Section contains 2 paragraphs. Based upon each of the paragraphs 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

**Paragraphs for Question 12 To 14**

When liquid medicine of density ρ is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.

We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension T when the radius of the drop is R. When the force becomes smaller than the weight of the drop, the drop gets detached from the dropper.

**Q.** If the radius of the opening of the dropper is r, the vertical force due to the surface tension on the drop of

radius R (assuming r << R) is

Solution:

QUESTION: 13

**Paragraphs for Question 12 To 14**

When liquid medicine of density ρ is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.

We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension T when the radius of the drop is R. When the force becomes smaller than the weight of the drop, the drop gets detached from the dropper.

**Q. **If r = 5 × 10^{−4} m, ρ = 10^{3} kg m^{−3}, g = 10 m/s^{2}, T = 0.11 Nm^{−1}, the radius of the drop when it detaches from the dropper is approximately

Solution:

QUESTION: 14

**Paragraphs for Question 12 To 14**

When liquid medicine of density ρ is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.

We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension T when the radius of the drop is R. When the force becomes smaller than the weight of the drop, the drop gets detached from the dropper.

**Q. **After the drop detaches, its surface energy is

Solution:

Surface energy = T(4πR^{2}) = 2.7 × 10^{−6} J

QUESTION: 15

**Paragraph for questions 15 to 17.**

The key feature of Bohr’s theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr’s quantization condition.

**Q. **A diatomic molecule has moment of inertia I. By Bohr’s quantization condition its rotational energy in the

n^{th} level (n = 0 is not allowed) is

Solution:

QUESTION: 16

**Paragraph for questions 15 to 17.**

The key feature of Bohr’s theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr’s quantization condition.

Q. It is found that the excitation frequency from ground to the first excited state of rotation for the CO

molecule is close to . Then the moment of inertia of CO molecule about its centre of mass is

close to

Solution:

QUESTION: 17

**Paragraph for questions 15 to 17.**

The key feature of Bohr’s theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr’s quantization condition.

**Q. **In a CO molecule, the distance between C (mass = 12 a.m.u) and O (mass = 16 a.m.u.),

where 1 a.m.u. is close to

Solution:

QUESTION: 18

**SECTION – IV (Matrix Type)**

This Section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I and five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching with one or more statement(s) given in Column II.

**Q.** Two transparent media of refractive indices μ_{1} and μ_{3} have a solid lens shaped transparent material of

refractive index μ2 between them as shown in figures in Column II. A ray traversing these media is also

shown in the figures. In Column I different relationships between μ_{1}, μ_{2} and μ_{3} are given. Match them to

the ray diagram shown in Column II.

Solution:

QUESTION: 19

You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage

source (the first two circuits) or an AC voltage source of 50 Hz frequency (the next three circuits) in

different ways as shown in Column II. When a current I (steady state for DC or rms for AC) flows through

the circuit, the corresponding voltage V_{1} and V_{2}. (indicated in circuits) are related as shown in Column I.

Match the two

Solution:

QUESTION: 20

**SECTION – I (Single Correct Choice Type)**

This Section contains 6 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.

Q. The compounds P, Q and S

were separately subjected to nitration using HNO_{3}/H_{2}SO_{4} mixture. The major product formed in each case

respectively, is

Solution:

QUESTION: 21

Assuming that Hund’s rule is violated, the bond order and magnetic nature of the diatomic molecule B_{2} is

Solution:

QUESTION: 22

The packing efficiency of the two-dimensional square unit cell shown below is

Solution:

QUESTION: 23

The complex showing a spin-only magnetic moment of 2.82 B.M. is

Solution:

QUESTION: 24

In the reaction

Product T is

Solution:

QUESTION: 25

The species having pyramidal shape is

Solution:

*Answer can only contain numeric values

QUESTION: 26

**SECTION-II (Integer Type)**

This Section contains 5 questions. The answer to each question is a single-digit integer, ranging from 0 to 9.

**Enter only the numerical value in the space provided below.**

**Q.** Silver (atomic weight = 108 g mol^{−1}) has a density of 10.5 g cm^{−3}. The number of silver atoms on a surface of area 10^{−12} m^{2} can be expressed in scientific notation as y × 10^{x}. The value of x is

Solution:

*Answer can only contain numeric values

QUESTION: 27

Among the following , the number of elements showing only one non-zero oxidation state is

O, Cl, F, N, P, Sn, Tl, Na, Ti

Solution:

Na, F show only one non-zero oxidation state.

*Answer can only contain numeric values

QUESTION: 28

One mole of an ideal gas is taken from a to b along two paths denoted by the solid and the dashed lines as

shown in the graph below. If the work done along the solid line path is ws and that along the dotted line

path is w_{d}, then the integer closest to the ratio w_{d}/w_{s} is

Solution:

*Answer can only contain numeric values

QUESTION: 29

The total number of dirpotic acids among the following is

Solution:

H_{2}SO_{4}, H_{2}CO_{3}, H_{2}S_{2}O_{7}, H_{2}CrO_{4}, H_{3}PO_{3}, H_{2}SO_{3}

*Answer can only contain numeric values

QUESTION: 30

Total number of geometrical isomers for the complex [RhCl(CO)(PPh_{3})(NH_{3})] is

Solution:

QUESTION: 31

**SECTION-III (Paragraph Type)**

This Section contains 2 paragraphs. Based upon each of the paragraphs 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

**Paragraph for questions 31 to 33**

Two aliphatic aldehydes P and Q react in the presence of aqueous K_{2}CO_{3} to give compound R, which upon

treatment with HCN provides compound S. On acidification and heating, S gives the product shown below:

**Q. **The compounds P and Q respectively are

Solution:

QUESTION: 32

**Paragraph for questions 31 to 33**

Two aliphatic aldehydes P and Q react in the presence of aqueous K_{2}CO_{3} to give compound R, which upon

treatment with HCN provides compound S. On acidification and heating, S gives the product shown below:

The compound R is

Solution:

QUESTION: 33

**Paragraph for questions 31 to 33**

Two aliphatic aldehydes P and Q react in the presence of aqueous K_{2}CO_{3} to give compound R, which upon

treatment with HCN provides compound S. On acidification and heating, S gives the product shown below:

The compound S is

Solution:

QUESTION: 34

**Paragraph for questions 34 to 36**

The hydrogen-like species Li^{2+} is in a spherically symmetric state S_{1} with one radial node. Upon absorbing light the ion undergoes transition to a state S_{2}. The state S_{2} has one radial node and its energy is equal to the ground state energy of the hydrogen atom.

**Q. **The state S_{1} is

Solution:

QUESTION: 35

**Paragraph for questions 34 to 36**

The hydrogen-like species Li^{2+} is in a spherically symmetric state S_{1} with one radial node. Upon absorbing light the ion undergoes transition to a state S_{2}. The state S_{2} has one radial node and its energy is equal to the ground state energy of the hydrogen atom.

**Q. **Energy of the state S_{1} in units of the hydrogen atom ground state energy is

Solution:

QUESTION: 36

**Paragraph for questions 34 to 36**

The hydrogen-like species Li^{2+} is in a spherically symmetric state S_{1} with one radial node. Upon absorbing light the ion undergoes transition to a state S_{2}. The state S_{2} has one radial node and its energy is equal to the ground state energy of the hydrogen atom.

**Q. **The orbital angular momentum quantum number of the state S_{2} is

Solution:

Azimuthal quantum number for S_{2} = l =1

QUESTION: 37

**SECTION-IV (Matrix Type)**

This Section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I and five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching with one or more statement(s) given in Column II.

**Q. **Match the reactions in Column I with appropriate options in Column II.

Solution:

(A) It is an example of electrophilic substitution reaction which results in coupled product hence it is coupling reaction also.

(B) Pinacole-pinacolone rearrangement. In this reaction intermediate is carbocation.

(C) It is an example of addition reaction by carbonyl compounds and both enantiomers will be formed.

Hence, racemic mixture will be obtained.

(D) It is an example of nucleophilic substitution.

QUESTION: 38

All the compounds listed in Column I react with water. Match the result of the respective reactions with the

appropriate options listed in Column II.

Solution:

QUESTION: 39

**SECTION – I (Single Correct Choice Type)**

This Section contains 6 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.

**Q. **If the distance of the point P (1, − 2, 1) from the plane x + 2y − 2z = α, where α > 0, is 5, then the foot of

the perpendicular from P to the plane is

Solution:

QUESTION: 40

A signal which can be green or red with probability 4/5 and 1/5 respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is 3/4. If the signal received at station B is green, then the probability that the original signal was green is

Solution:

Event G = original signal is green

E_{1} = A receives the signal correct

E_{2} = B receives the signal correct

E = signal received by B is green

QUESTION: 41

Two adjacent sides of a parallelogram ABCD are given by The side AD is rotated by an acute angle α in the plane of the parallelogram so that AD becomes AD′. If AD′ makes a right angle with the side AB, then the cosine of the angle α is given by

Solution:

QUESTION: 42

For r = 0, 1, …, 10, let A_{r}, B_{r} and C_{r} denote, respectively, the coefficient of x_{r} in the expansions of (1 + x)^{10},

(1 + x)^{20} and (1 + x)^{30}. is equal to

Solution:

QUESTION: 43

Let f be a real-valued function defined on the interval (− 1, 1) such that for all x ∈ (−1 , 1) and let f^{−1 }be the inverse function of f. Then (f^{-1})′ (2) is equal to

Solution:

QUESTION: 44

Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to

Solution:

Total number of unordered pairs of disjoint subsets

*Answer can only contain numeric values

QUESTION: 45

**SECTION – II (Integer Type)**

This Section contains 5 questions. The answer to each question is a single-digit integer, ranging from 0 to 9.

**Enter only the numerical value in the space provided below.**

**Q. **Let a_{1}, a_{2}, a_{3}, …, a_{11} be real numbers satisfying a_{1} = 15, 27 − 2a_{2} > 0 and a_{k} = 2a_{k−1} − a_{k−2} for k = 3, 4, …,

11.

Solution:

*Answer can only contain numeric values

QUESTION: 46

Let f be a function defined on R (the set of all real numbers) such that f′(x) = 2010 (x − 2009) (x − 2010)^{2}

(x − 2011)^{3} (x − 2012)^{4}, for all x ∈ R. If g is a function defined on R with values in the interval (0, ∞) such

that f (x) = ln (g (x)), for all x ∈ R, then the number of points in R at which g has a local maximum is

Solution:

*Answer can only contain numeric values

QUESTION: 47

Let k be a positive real number and let . If det (adj A) + det (adj B) = 10^{6}, then [k] is equal to

[Note : adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].

Solution:

*Answer can only contain numeric values

QUESTION: 48

Two parallel chords of a circle of radius 2 are at a distance apart. If the chords subtend at the centre,

angles of , where k > 0, then the value of [k] is

[Note : [k] denotes the largest integer less than or equal to k].

Solution:

*Answer can only contain numeric values

QUESTION: 49

Consider a triangle ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C

respectively. Suppose a = 6, b = 10 and the area of the triangle is . If ∠ACB is obtuse and if r denotes

the radius of the incircle of the triangle, then r^{2} is equal to

Solution:

QUESTION: 50

**SECTION – III (Paragraph Type)**

This Section contains 2 paragraphs. Based upon each of the paragraphs 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

**Paragraph for questions 50 to 52.**

Consider the polynomial f(x) = 1 + 2x + 3x^{2} + 4x^{3}. Let s be the sum of all distinct real roots of f(x) and let t = |s|.

**Q. **The real number s lies in the interval

Solution:

QUESTION: 51

**Paragraph for questions 50 to 52.**

Consider the polynomial f(x) = 1 + 2x + 3x^{2} + 4x^{3}. Let s be the sum of all distinct real roots of f(x) and let t = |s|.

**Q. **The area bounded by the curve y = f(x) and the lines x = 0, y = 0 and x = t, lies in the interval

Solution:

QUESTION: 52

**Paragraph for questions 50 to 52.**

Consider the polynomial f(x) = 1 + 2x + 3x^{2} + 4x^{3}. Let s be the sum of all distinct real roots of f(x) and let t = |s|.

**Q. **The function f′(x) is

Solution:

f″(x) = 2[12x + 3] = 0 ⇒ x = − 1/4.

QUESTION: 53

**Paragraph for questions 53 to 55.**

Tangents are drawn from the point P(3, 4) to the ellipse touching the ellipse at points A and B.

**Q.** The coordinates of A and B are

Solution:

QUESTION: 54

**Paragraph for questions 53 to 55.**

Tangents are drawn from the point P(3, 4) to the ellipse touching the ellipse at points A and B.

**Q.** The orthocentre of the triangle PAB is

Solution:

QUESTION: 55

**Paragraph for questions 53 to 55.**

Tangents are drawn from the point P(3, 4) to the ellipse touching the ellipse at points A and B.

**Q. **The equation of the locus of the point whose distances from the point P and the line AB are equal, is

Solution:

QUESTION: 56

**SECTION – IV (Matrix Type)**

This Section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I and five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching with one or more statement(s) given in Column II.

**Q.** Match the statements in column-I with those in column-II.

[Note: Here z takes the values in the complex plane and Im z and Re z denote, respectively, the imaginary

part and the real part of z]

Solution:

QUESTION: 57

**Match the statements in column-I with those in column-II.**

Solution:

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