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QUESTION: 1

**SECTION – I (Total Marks : 24)**

(Single Correct Answer Type)

This Section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONLY ONE is correct.

**Q. **Which of the field patterns given below is valid for electric field as well as for magnetic field?

Solution:

QUESTION: 2

A ball of mass 0.2 kg rests on a vertical post of height 5 m. A bullet of mass 0.01 kg, traveling with a velocity V m/s in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of 20 m and the bullet at a distance of 100 m from the foot of the post. The velocity V of the bullet is

Solution:

v_{ball} = 20 m/s

v_{bullet} = 100 m/s

0.01 V = 0.01 x 100 + 0.2 x 20

v = 100 + 400 = 500 m/s

QUESTION: 3

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with

a screw gauge, whose pitch is 0.5 mm and there are 50 divisions on the circular scale. The reading on the

main scale is 2.5 mm and that on the circular scale is 20 divisions. If the measured mass of the ball has a

relative error of 2 %, the relative percentage error in the density is

Solution:

QUESTION: 4

A wooden block performs SHM on a frictionless surface with frequency, ν_{0}. The block carries a charge +Q on its surface. If now a uniform electric field is switched-on as shown, then the SHM of the block will be

Solution:

QUESTION: 5

A light ray travelling in glass medium is incident on glass-air interface at an angle of incidence θ. The reflected (R) and transmitted (T) intensities, both as function of θ, are plotted. The correct sketch is

Solution:

After total internal reflection, there is no refracted ray.

QUESTION: 6

A satellite is moving with a constant speed ‘V’ in a circular orbit about the earth. An object of mass ‘m’ is

ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its

ejection, the kinetic energy of the object is

Solution:

QUESTION: 7

A long insulated copper wire is closely wound as a spiral of ‘N’ turns. The spiral has inner radius ‘a’ and outer radius ‘b’. The spiral lies in the XY plane and a steady current ‘I’ flows through the wire. The Z-component of the magnetic field at the centre of the spiral is

Solution:

QUESTION: 8

A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, x_{1}(t) = A sinωt and

Adding a third sinusoidal displacement x_{3}(t) = B sin (ωt + φ) brings the mass to a complete rest. The values of B and φ are

Solution:

*Multiple options can be correct

QUESTION: 9

**SECTION – II (Total Marks : 16)**

(Multiple Correct Answer(s) Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONE OR MORE may be correct.

**Q.** Two solid spheres A and B of equal volumes but of different densities d_{A} and d_{B} are connected by a string. They are fully immersed in a fluid of density d_{F}. They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

Solution:

*Multiple options can be correct

QUESTION: 10

A thin ring of mass 2 kg and radius 0.5 m is rolling without on a horizontal plane with velocity 1 m/s. A small ball of mass 0.1 kg, moving with velocity 20 m/s in the opposite direction hits the ring at a height of 0.75 m and goes vertically up with velocity 10 m/s. Immediately after the collision

Solution:

During collision friction is impulsive and immediately after collision the ring will have a clockwise angular

velocity hence friction will be towards left.

*Multiple options can be correct

QUESTION: 11

Which of the following statement(s) is/are correct?

Solution:

(D) is correct if we assume it is work done against electrostatic force

*Multiple options can be correct

QUESTION: 12

A series R-C circuit is connected to AC voltage source. Consider two cases; (A) when C is without a dielectric medium and (B) when C is filled with dielectric of constant 4. The current I_{R} through the resistor and voltage V_{C} across the capacitor are compared in the two cases. Which of the following is/are true?

Solution:

*Answer can only contain numeric values

QUESTION: 13

**SECTION-III (Total Marks : 24)**

(Integer Answer Type)

This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9.

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

**Q.** A series R-C combination is connected to an AC voltage of angular frequency ω = 500 radian/s. If the

impedance of the R-C circuit is the time constant (in millisecond) of the circuit is

Solution:

*Answer can only contain numeric values

QUESTION: 14

A silver sphere of radius 1 cm and work function 4.7 eV is suspended from an insulating thread in freespace.

It is under continuous illumination of 200 nm wavelength light. As photoelectrons are emitted, the

sphere gets charged and acquires a potential. The maximum number of photoelectrons emitted from the

sphere is A x 10^{z }(where 1 < A < 10). The value of ‘Z’ is

Solution:

*Answer can only contain numeric values

QUESTION: 15

A train is moving along a straight line with a constant acceleration ‘a’. A boy standing in the train throws a

ball forward with a speed of 10 m/s, at an angle of 60° to the horizontal. The boy has to move forward by

1.15 m inside the train to catch the ball back at the initial height. The acceleration of the train, in m/s^{2}, is

Solution:

*Answer can only contain numeric values

QUESTION: 16

A block of mass 0.18 kg is attached to a spring of force-constant 2 N/m. The coefficient of friction between the block and the floor is 0.1. Initially the block is at rest and the spring is unstretched. An impulse is given to the block as shown in the figure. The block slides a distance of 0.06 m and comes to rest for the first time. The initial velocity of the block m/s is V = N/10. Then N is:

Solution:

Let ν be the speed of the block just after impulse. At B, the block comes to rest.

Therefore: Loss in K.E. of the block = Gain in P.E. of the spring + Work done against friction

*Answer can only contain numeric values

QUESTION: 17

Two batteries of different emfs and different internal resistances are connected as shown. The voltage across AB in volts is

Solution:

*Answer can only contain numeric values

QUESTION: 18

Water (with refractive index = 4/3 in a tank is 18 cm deep. Oil of refractive index 7/4 lies on water making a convex surface of radius of curvature ‘R = 6 cm’ as shown. Consider oil to act as a thin lens. An object ‘S’ is placed 24 cm above water surface. The location of its image is at ‘x’ cm above the bottom of the tank. Then ‘x’ is

Solution:

V_{2} = 16 cm

x = 18 - 16 = 2 cm

QUESTION: 19

**(Matrix-Match 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.** One mole of a monatomic gas is taken through a cycle ABCDA as shown in the P-V diagram. Column II give the characteristics involved in the cycle. Match them with each of the processes given in Column I.

Solution:

Process A → B → Isobaric compression

Process B → C → Isochoric process

Process C → D → Isobaric expansion

Process D → A → Polytropic with TA = TD

QUESTION: 20

Column I shows four systems, each of the same length L, for producing standing waves. The lowest

possible natural frequency of a system is called its fundamental frequency, whose wavelength is denoted as

λ_{f}. Match each system with statements given in Column II describing the nature and wavelength of the

standing waves.

Solution:

QUESTION: 21

**SECTION – I (Total Marks : 24)**

(Single Correct Answer Type)

This Section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONLY ONE is correct.

**Q.** The freezing point (in ^{o}C) of a solution containing 0.1 g of K_{3}[Fe(CN)_{6} (Mol. Wt. 329) in 100 g of water

(K_{f} = 1.86 K kg mol^{-1}) is

Solution:

QUESTION: 22

Amongst the compounds given, the one that would form a brilliant colored dye on treatment with NaNO_{2} in

dil. HCl followed by addition to an alkaline solution of β-naphthol is

Solution:

QUESTION: 23

The major product of the following reaction is

Solution:

QUESTION: 24

The following carbohydrate is

Solution:

QUESTION: 25

Oxidation states of the metal in the minerals haematite and magnetite, respectively, are

Solution:

Haematite : Fe_{2}O_{3} : 2x + 3 × (-2) = 0

x =3

Magnetite : Fe_{3}O_{4} [an equimolar mixture of FeO and Fe_{2}O_{3}]

FeO : x – 2 = 0 ⇒ x = 2

Fe_{2}O_{3} : x = 3

QUESTION: 26

Among the following complexes (K-P)

K_{3}[Fe(CN)_{6}] (K), [Co(NH_{3})_{6}]Cl_{3} (L), Na_{3}[Co(oxalate)_{3}] (M), [Ni(H_{2}O)_{6}]Cl_{2} (N), K_{2}[Pt(CN)_{4}] (O) and

[Zn(H_{2}O)_{6}](NO_{3})_{2} (P)

Solution:

QUESTION: 27

Passing H_{2}S gas into a mixture of Mn^{2+}, Ni^{2+}, Cu^{2+} and Hg^{2+} ions in an acidified aqueous solution precipitates

Solution:

H_{2}S in presence of aqueous acidified solution precipitates as sulphide of Cu and Hg apart from Pb^{+2}, Bi^{+3}, Cd^{+2}, As^{+3}, Sb^{+3} and Sn^{+2}.

QUESTION: 28

Consider the following cell reaction:

At [Fe^{2+}] = 10^{-3} M, P(O_{2}) = 0.1 atm and pH = 3, the cell potential at 25^{o}C is

Solution:

*Multiple options can be correct

QUESTION: 29

**SECTION – II (Total Marks : 16)**

(Multiple Correct Answer(s) Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONE OR MORE may be correct.

**Q. **Reduction of the metal centre in aqueous permanganate ion involves

Solution:

In acidic medium

In neutral medium

Hence, number of electron loose in acidic and neutral medium 5 and 3 electrons respectively.

*Multiple options can be correct

QUESTION: 30

The correct functional group X and the reagent/reaction conditions Y in the following scheme are

Solution:

Condensation polymers are formed by condensation of a diols or diamine with dicarboxylic acids.

*Multiple options can be correct

QUESTION: 31

For the first order reaction

2N_{2}O_{5}(g) → 4NO_{2}(g) + O_{2}(g)

Solution:

For first order reaction

[A] = [A]_{0}e^{–kt}

Hence concentration of [NO_{2}] decreases exponentially.

Also, t_{1/2} =. Which is independent of concentration and t_{1/2} decreases with the increase of

temperature.

*Multiple options can be correct

QUESTION: 32

The equilibrium

in aqueous medium at 25^{o}C shifts towards the left in the presence of

Solution:

Cu^{2+} ions will react with CN^{-} and SCN^{-} forming [Cu(CN)^{4}]^{3-} and [Cu(SCN)^{4}]^{3-} leading the reaction in the backward direction.

Cu^{2+} also combines with CuCl_{2} which reacts with Cu to produce CuCl pushing the reaction in the backward direction.

*Answer can only contain numeric values

QUESTION: 33

**SECTION-III (Total Marks : 24)**

**(Integer Answer Type)**

This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9.

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

Q. The maximum number of isomers (including stereoisomers) that are possible on monochlorination of the

following compound is

Solution:

Total = 2 + 4 + 1 + 1 = 8

*Answer can only contain numeric values

QUESTION: 34

The total number of contributing structure showing hyperconjugation (involving C-H bonds) for the

following carbocation is

Solution:

6 x H-atoms are there

*Answer can only contain numeric values

QUESTION: 35

Among the following, the number of compounds than can react with PCl_{5} to give POCl_{3} is O_{2}, CO_{2}, SO_{2}, H_{2}O, H_{2}SO_{4}, P_{4}O_{10}

Solution:

*Answer can only contain numeric values

QUESTION: 36

The volume (in mL) of 0.1 M AgNO_{3} required for complete precipitation of chloride ions present in 30 mL

of 0.01 M solution of [Cr(H_{2}O)_{5}Cl]Cl_{2}, as silver chloride is close to

Solution:

Number of ionisable Cl^{-} in [Cr(H_{2}O)_{5}Cl]Cl_{2} is 2

Millimoles of Cl^{-} = 30 x 0.01 ´ 2 = 0.6

Millimoles of Ag^{+} required = 0.6

0.6 = 0.1 V

V = 6 ml

*Answer can only contain numeric values

QUESTION: 37

In 1 L saturated solution of AgCl [K_{sp}(AgCl) = 1.6 x 10^{-10}], 0.1 mol of CuCl [K_{sp}(CuCl) = 1.0 x 10^{-6}] is

added. The resultant concentration of Ag+ in the solution is 1.6 x 10^{-x}. The value of “x” is

Solution:

*Answer can only contain numeric values

QUESTION: 38

The number of hexagonal faces that are present in a truncated octahedron is

Solution:

QUESTION: 39

**SECTION-IV (Total Marks : 16)**

**(Matrix-Match 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 transformations in column I with appropriate options in column II

Solution:

QUESTION: 40

Match the reactions in column I with appropriate types of steps/reactive intermediate involved in these

reactions as given in column II

Solution:

QUESTION: 41

**SECTION – I (Total Marks : 24)**

(Single Correct Answer Type)

This Section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONLY ONE is correct.

**Q.** If

Solution:

QUESTION: 42

Let f : [-1, 2] → [0, ∞) be a continuous function such that f(x) = f(1 - x) for all x ε[-1, 2]. Let and R_{2} be the area of the region bounded by y = f(x), x = -1, x = 2, and the x-axis. Then

Solution:

QUESTION: 43

Let f(x) = x^{2} and g(x) = sinx for all x ε R. Then the set of all x satisfying (f o g o g o f) (x) = (g o g o f) (x),

where (f o g) (x) = f(g(x)), is

Solution:

(fogogof) (x) = sin^{2} (sin x^{2})

(gogof) (x) = sin (sin x^{2})

sin^{2} (sin x^{2}) = sin (sin x^{2})

⇒ sin (sin x^{2}) [sin (sin x^{2}) - 1] = 0

⇒ sin (sin x^{2}) = 0 or 1

⇒ sin x^{2} = nπ or 2mπ + π/2, where m, n ε I

⇒ sin x^{2} = 0

⇒ x^{2} = np x = ± np , n Î {0, 1, 2, …}.

QUESTION: 44

Let (x, y) be any point on the parabola y^{2} = 4x. Let P be the point that divides the line segment from (0, 0)

to (x, y) in the ratio 1 : 3. Then the locus of P is

Solution:

y^{2} = 4x and Q will lie on it

(4k)^{2} = 4 ´ 4h

k^{2} = h

y^{2} = x (replacing h by x and k by y)

QUESTION: 45

Let P(6, 3) be a point on the hyperbola If the normal at the point P intersects the x-axis at (9,

0), then the eccentricity of the hyperbola is

Solution:

QUESTION: 46

A value of b for which the equations

x^{2} + bx - 1 = 0

x^{2} + x + b = 0,

have one root in common is

Solution:

x^{2} + bx - 1 = 0

x^{2} + x + b = 0 … (1)

Common root is

(b - 1) x - 1 - b = 0

QUESTION: 47

Let ω =1 be a cube root of unity and S be the set of all non-singular matrices of the form where each of a, b, and c is either ω or ω^{2}. Then the number of distinct matrices in the set S is

Solution:

QUESTION: 48

The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point

Solution:

Circle touching y-axis at (0, 2) is (x - 0)^{2} + (y - 2)^{2} + λx = 0 passes through (- 1, 0)

1 + 4 - λ = 0 ⇒ λ = 5

x^{2} + y^{2} + 5x - 4y + 4 = 0

Put y = 0 ⇒ x = - 1, - 4

Circle passes through (- 4, 0)

*Multiple options can be correct

QUESTION: 49

**SECTION – II (Total Marks : 16)**

**(Multiple Correct Answer(s) Type)**

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONE OR MORE may be correct.

**Q. **

Solution:

Clearly, f (x) is not differentiable at x = 0 as f´(0^{-}) = 0 and f´(0^{+}) = 1.

f (x) is differentiable at x = 1 as f' (1^{–}) = f'(1^{+}) = 1.

*Multiple options can be correct

QUESTION: 50

where be is a constant such that 0 < b < 1. Then

Solution:

*Multiple options can be correct

QUESTION: 51

Let L be a normal to the parabola y^{2} = 4x. If L passes through the point (9, 6), then L is given by

Solution:

y^{2} = 4x

Equation of normal is y = mx – 2m – m^{3}.

It passes through (9, 6)

⇒ m^{3} – 7m + 6 = 0

⇒ m = 1, 2, – 3

⇒ y – x + 3 = 0, y + 3x – 33 = 0, y – 2x + 12 = 0.

*Multiple options can be correct

QUESTION: 52

Let E and F be two independent events. The probability that exactly one of them occurs is and the

probability of none of them occurring is If P(T) denotes the probability of occurrence of the event T,

then

Solution:

*Answer can only contain numeric values

QUESTION: 53

**SECTION-III (Total Marks : 24)**

(Integer Answer Type)

This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9.

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

**Q. **Let be three given vectors. If and then the value of is

Solution:

*Answer can only contain numeric values

QUESTION: 54

The straight line 2x - 3y = 1 divides the circular region x2 + y2 __<__ 6 into two parts. If

then the number of point(s) in S lying inside the smaller part is

Solution:

*Answer can only contain numeric values

QUESTION: 55

Let ω = e^{iπ/3}, and a, b, c, x, y, z be non-zero complex numbers such that

a + b + c = x

a + bω + cω^{2} = y

a + bω^{2} + cω = z.

Solution:

The expression may not attain integral value for all a, b, c

If we consider a = b = c, then

x = 3a

*Answer can only contain numeric values

QUESTION: 56

The number of distinct real roots of x^{4} - 4x^{3} + 12x^{2} + x - 1 = 0 is

Solution:

Let f (x) = x^{4} - 4x^{3} + 12x^{2} + x - 1 = 0

f'(x) = 4x^{3} - 12x^{2} + 24x + 1= 4 (x^{3} - 3x^{2} + 6x) + 1

f''(x) = 12x^{2} - 24x + 24 = 12 (x^{2} - 2x + 2)

f''(x) has 0 real roots

f (x) has maximum 2 distinct real roots as f (0) = –1.

*Answer can only contain numeric values

QUESTION: 57

Let y'(x) + y(x)g'(x) = g(x)g'(x), y(0) = 0, x ε R, where f'(x) denotes and g(x) is a given nonconstant

differentiable function on R with g(0) = g(2) = 0. Then the value of y(2) is

Solution:

y'(x) + y(x) g'(x) = g(x) g'(x)

⇒ e^{g(x)} y'(x) + e^{g(x)} g'(x) y(x) = e^{g(x)} g(x) g'(x)

*Answer can only contain numeric values

QUESTION: 58

Let M be a 3 x 3 matrix satisfying

Then the sum of the diagonal entries of M is

Solution:

QUESTION: 59

**SECTION-IV (Total Marks : 16)**

**(Matrix-Match 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 given in Column I with the values given in Column II

Solution:

QUESTION: 60

Match the statements given in Column I with the intervals/union of intervals given in Column II

Solution:

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