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VMC/Paper-2 1 JEE Entrance Exam-2015/Advanced

CODE - 5 PAPER – 2 Time : 3 Hours Maximum Marks : 240

READ THE INSTRUCTIONS CAREFULLY GENERAL: 1. This sealed booklet is your Question Paper. Do not break the seal till you are instructed to do so. 2. The question paper CODE is printed on the left hand top corner of this sheet and the right hand top corner of the

back cover of this booklet. 3. Use the Optical Response Sheet (ORS) provided separately for answering the questions. 4. The ORS CODE is printed on its left part as well as the right part. Ensure that both these codes are identical and

same as that on the question paper booklet. If not, contact the invigilator. 5. Blank spaces are provided within this booklet for rough work. 6. Write your name and roll number in the space provided on the back cover of this booklet. 7. After breaking the seal of the booklet, verify that the booklet contains 32 pages and that all the 60 questions

along with the options are legible. QUESTION PAPER FORMAT AND MARKING SCHEME: 8. The question paper has three parts: Physics, Chemistry and Mathematics. Each part has three sections. 9. Carefully read the instructions given at the beginning of each section. 10. Section 1 contains 8 questions. The answer to each question is a single digit integer ranging from 0 to 9 (both

inclusive). Marking scheme: + 4 for correct answer and 0 in all other cases. 11. Section 2 contains 8 multiple choice questions with one or more than one correct option. Marking scheme: +4 for correct answer, 0 if not attempted and -2 in all other cases. 12. Section 3 contains 2 “paragraph” type questions. Each paragraph describes an experiment, a situation or a

problem. Two multiple choice questions will be asked based on this paragraph. One or more than one option can be correct.

Marking scheme: +4 for correct answer, 0 if not attempted and -2 in all other cases. OPTICAL RESPONSE SHEET: 13. The ORS consists of an original (top sheet) and its carbon-less copy (bottom sheet). 14. Darken the appropriate bubbles on the original by applying sufficient pressure. This will leave an impression at

the corresponding place on the carbon-less copy. 15. The original is machine-gradable and will be collected by the invigilator at the end of the examination. 16. You will be allowed to take away the carbon-less copy at the end of the examination. 17. Do not temper with or mutilate the ORS. 18. Write you name, roll number and the name of the examination center and sign with pen in the space provided for

this purpose on the original. Do not write any of these details anywhere else. Darken the appropriate bubble under each digit of your roll number.

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PART I : PHYSICS

SECTION 1 (Maximum Marks : 32)

This section contains EIGHT questions The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive. For each question, darken the bubble corresponding to the correct integer in the ORS. Marking scheme:

+4 If the bubble corresponding to the answer is darkened 0 In all the other cases

1. The densities of two solid spheres A and B of the same radii R vary with radial distance r as

5

Ar rr k and r kBR R

, respectively, where k is a constant. The moments of inertia of the

individual spheres about axes passing through their centres are IA and IB, respectively. If 10

B

A

I nI

, the value of n

is _________. 2. Four harmonic waves of equal frequencies and equal intensities I0 have phase angle 0, 3/ , 2 3/ and .

When they are superposed, the intensity of the resulting wave in nI0. The value of n is _________.

3. For a radioactive material, its activity A and rate of change of its activity R are defined anddN dAA Rdt dt

,

where N(t) is the number of nuclei at time t. Two radioactive sources P (mean life ) and Q (mean life 2 ) have

the same activity at t = 0. Their rates of change of activities at 2t are RP and RQ, respectively. If P

Q

R nR e

,

then the value of n is _________. 4. A monochromatic beam of light is incident at 60 on one

face of an equilateral prism of refractive index n and emerges from the opposite face making an angle n

with the normal (see the figure). For 3n the value of

is 60 and d mdn . The value of m is _________.

5. In the following circuit, the current through the resistor 2ΩR is I Amperes. The value of I is _________.

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6. An electron in an excited state of 2Li ion has angular moment 3 2h / . The de Brogile wavelength of the electron in this state is 0p a (where a0 is the Bohr radius). The value of p is _________.

7. A large spherical mass M is fixed at one position and two

identical point masses m are kept on a line passing through the centre of M (see figure). The point masses are connected by a rigid massless rod of length l and this assembly is free to move along the line connecting them. All three mass interact only through their mutual gravitational interaction. When the point m is nearer to M is at a distance r = 3l from M, the tension in the rod is

zero for 288Mm k

. The value of k is _________.

8. The energy of a system as a function of time t is given as E(t) = A2 exp ( t ), where 10 2. s . The

measurement of A has an error of 1 25. % . If the error in the measurement of time is 1 50. % , the percentage error in the value of E(t) at t = 5s is _________.

SECTION 2 (Maximum Marks : 40) This section contains TEN questions Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are)

correct For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS Marking scheme : +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened 0 If none of the bubbles is darkened –2 In all other cases

9. A parallel plate capacitor having plates of area S and plate separation d, has

capacitance C1 in air. When two dielectriecs of different relative permittivities 1 22 and 4 are introduced between the two plates as

shown in the figure, the capacitance becomes C2. The ratio 2

1

CC

is :

(A) 6/5 (B) 5/3 (C) 7/5 (D) 7/3

10. An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).

Initially the gas is at temperature T1, pressure P1 and volume V1 and the spring is in its relaxed state. The gas is then heated very slowly to temperature T2, pressure P2 and volume V2. During this process the piston moves out by a distance x. Ignoring the friction between the piston and the cylinder, the correct statement(s) is(are) :

(A) If 2 12V V and 2 13T T , then the energy stored in the spring is 1 114

PV

(B) If 2 12V V and 2 13T T , then the change in internal energy is 1 13PV

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(C) If 2 13V V and 2 14T T , then the work done by the gas is 1 173

PV

(D) If 2 13V V and 2 14T T , then the heat supplied to the gas is 1 1176

PV

11. A fission reaction is given by 236 140 9492 54 38U Xe Sr x y , where x and y are two particles. Considering 236

92 U

to be at rest, the kinetic energies of the products are denoted by , , (2MeV)Xe Sr xK K K and (2MeV)yK ,

respectively. Let the binding energies per nucleon of 236 140 9492 54 38U Xe and Sr be 7.5 MeV, 8.5 MeV and 8.5

MeV respectively. Considering different conservation laws, the correct option(s) is(are) (A) Sr Xe, , 129 MeV, 86MeV x n y n K K

(B) Sr Xe, , 129 MeV, 86MeV x p y e K K

(C) Sr Xe, , 129 MeV, 86MeV x p y n K K

(D) Sr Xe, , 86 MeV, 129MeV x n y n K K

12. Two spheres P and Q of equal radii have densities 1ρ and 2ρ ,

respectively. The spheres are connected by a massless string and placed in liquids 1L and 2L of densities 1σ and 2σ and

viscosities 1η and 2η , respectively. They float in equilibrium with

the sphere P in L1 and sphere Q in L2 and the string being taut (see

figure). If sphere P alone in 2L has terminal velocity PV and Q

alone in 1L has terminal velocity QV , then

(A) 1

2

| | ηη| |

PQ

VV

(B) 2

1

| | ηη| |

PQ

VV

(C) . 0 P QV V (D) . 0

P QV V

13. In terms of potential difference V, electric current I, permittivity 0ε , permeability 0μ and speed of light c, the

dimensionally correct equation(s) is(are)

(A) 2 20 oI V (B) 0ε V (C) 0εI cV (D) 0 0μ εcI V

14. Consider a uniform spherical charge distribution of radius R1 centred at the origin O. In this distribution, a

spherical cavity of radius R2 centred at P with distance 1 2OP a R R (see figure) is made. If the electric field

inside the cavity at position r is ( ) E r , then the correct statement(s) is(are)

(A) E is uniform, its magnitude is independent of R2 but its direction depends on r

(B) E is uniform, its magnitude depends on R2 and its direction depends on r

(C) E is uniform, its magnitude is independent of a but its direction depends on a

(D) E is uniform and both its magnitude and direction depend on a

15. In plotting stress versus strain curves for two materials

P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is (are)

(A) P has more tensile strength than Q (B) P is more ductile than Q

(C) P is more brittle than Q (D) The Young’s modulus of P is more than that of Q

P Q

Stra

in

Stress

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16. A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If P (r) is the pressure at r (r < R), then the correct option (s) is(are)

(A) P (r = 0) = 0 (B)

3 4 632 3 80

P r R /P r R /

(C)

3 5 162 5 21

P r R /P r R /

(D)

2 203 27

P r R /P r R /

SECTION 3 (Maximum Marks : 16) This section contains TWO questions. Based on each paragraph, there will be TWO questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are)

correct. For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS Marking scheme : +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened 0 If none of the bubbles is darkened –2 In all other cases

Paragraph for Questions 17 - 18

In a thin rectangular metallic strip a constant current I flows along the positive x-direction, as shown in the figure.

The length, width and thickness of the strip are l, w and d, respectively. A uniform magnetic field B

is applied on the strip along the positive y-direction. Due to this the charge carriers experience a net deflection along the z-direction. This results in accumulation of charge carriers on the surface PQRS and appearance of equal and opposite charges on the face opposite to PQRS. A potential difference along the z-direction is thus developed. Charge accumulation continuous until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons. 17. Consider two different metallic strips (1 and 2) of the same material. Their lengths are the same, width are w1

and w2 and thickness are d1 and d2, respectively. Two points K and M are symmetrically located on the opposite faces parallel to the x-y plane (see figure). V1 and V2 are the potential differences between K and M in strips 1 and 2, respectively. Then, for a given current I flowing through them in a given magnetic field strength B, the correct statement (s) is (are)

(A) If w1 = w2 and d1 = 2d2, then V2 = 2V1 (B) If w1 = w2 and d1 = 2d2, then V2 = V1 (C) If w1 = 2w2 and d1 = d2, then V2 = 2V1 (D) If w1 = 2w2 and d1 = d2, then V2 = V1

18. Consider two different metallic strips (1 and 2) of same dimensions (length l, width w and thickness d) with carried densities n1 and n2, respectively. Strip 1 is placed in magnetic field B1 and strip 2 is placed in magnetic field B2, both along positive y-direction. Then V1 and V2 are the potential differences developed between K and M in strips 1 and 2, respectively. Assuming that the current I is the same for both the strips, the correct option(s) is (are)

(A) If B1 = B2 and n1 = 2n2, then V2 = 2V1 (B) If B1 = B2 and n1 = 2n2, then V2 = V1 (C) If B1 = 2B2 and n1 = n2, then V2 = 0.5 V1 (D) If B1 = 2B2 and n1 = n2, then V2 = V1

x

y

z

I

Q P

R K

M d w I

l

S S

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Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index n1 surrounded by a medium of lower refractive index n2. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media n1 and n2 as shown in the figure. All rays with the angle of incidence i less than a particular value im are confined in the medium of refractive index n1. The numerical aperture (NA) of the structure is defined as sin im.

19. For two structures namely S1 with 1 45 4n / and n2 = 3/2, and S2 with n1 = 8/5 and n2 = 7/5 and taking the

refractive index of water to be 4/3 and that of air to be 1, the correct option(s) is (are)

(A) NA of S1 immersed in water is the same as that of S2 immersed in a liquid of refractive index 163 15

(B) NA of S1 immersed in liquid of refractive index 615

is the same as that of S2 immersed in water

(C) NA of S1 placed in air is the same as that of S2 immersed in liquid of refractive index 415

(D) NA of S1 placed in air is the same as that of S2 placed in water 20. If two structures of same cross-sectional area, but different numerical apertures NA1 and NA2 (NA2 < NA1) are

joined longitudinally, the numerical aperture of the combined structure is.

(A) 1 2

1 2

NA NANA NA

(B) 1 2NA NA

(C) NA1 (D) NA2

i

Core

Cladding

n1

n2 Air

n1 > n2

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PART I : CHEMISTRY


This section contains EIGHT questions The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive For each question, darken the bubble corresponding to the correct integer in the ORS Marking scheme:


21. In the complex acetylbromidodicarbonylbis (triethylphosphine) iron(II), the number of Fe C bond(s) is ______.

22. Among the complex ions, 2 2 2 2 22Co NH CH CH NH Cl

, 3

2 2 4 2CrCl C O

,

2 4 2Fe H O OH

, 3 2 4Fe NH CN

,

22 2 2 2 32Co NH CH CH NH NH Cl

and

2

3 24Co NH H O Cl

, the number of complex ions(s) that show(s) cis-trans isomerism is ______.

23. Three moles of 2 6B H are completely reacted with methanol. The number of moles of boron containing products

formed is ______. 24. The molar conductivity of a solution of a weak acid HX 0.01M is 10 times smaller than the molar conductivity

of a solution of a week acid HY 0.10 M . If y

0 0x , the difference in their apK values,

a apK HX pK HY , is ______. (Consider degree of ionization of both acids to be << 1)

25. A closed vessel with rigid walls contains 1 mol of 23892 U and 1 mol of air at 298 K. Considering complete decay

of 238 20692 82U to Pb , the ratio of the final pressure to the initial pressure of the system at 298 K is ______.

26. In dilute aqueous 2 4H SO , the complex diaquodixalatoferrate(II) is oxidized by 4MnO . For this reaction, the

ratio of the rate of change of H

to the rate of change of 4MnO

is ______.

27. The number of hydroxyl group(s) in Q is :

28. Among the following, the number of reaction(s) that produce(s) benzaldehyde is ______.

I II

III IV d

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29. Under hydrolytic conditions, the compounds used for preparation of linear polymer and for chain termination,

respectively, are : (A) 3 3 3 4CH SiCl and Si CH (B) 3 2 32 3CH SiCl and CH SiCl

(C) 3 2 3 32CH SiCl and CH SiCl (D) 4 3 3SiCl and CH SiCl

30. When 2O is adsorbed on a metallic surface, electron transfer occurs from the metal to 2O . The TRUE

statement(s) regarding this adsorption is (are) : (A) 2O is physisorbed (B) Heat is released

(C) Occupancy of *2p of 2O is increased (D) Bond length of 2O is increased

31. One mole of a monoatomic real gas satisfies the equation p V b RT where b is a constant. The relationship

of interatomic potential V r and interatomic distance r for the gas is given by :

(A) (B) (C) (D) 32. In the following reactions, the products S is :

(A) (B) (C) (D)

33. The major product U in the following reactions is :

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(A) (B) (C) (D)

34. In the following reactions, the major product W is

35. The correct statement(s) regarding, (i) HClO , (ii) 2HClO , (iii) 3HClO and (iv) 4HClO , is (are)

(A) The number of Cl O bonds in (ii) and (iii) together is two (B) The number of lone pairs of electrons on Cl in (ii) and (iii) together is three

(C) The hybridization of Cl in (iv) is 3sp

(D) Amongst (i) to (iv), the strongest acid is (i) 36. The pair(s) of ions where BOTH the ions are precipitated upon passing 2H S gas in presence of dilute

HCl, is(are)

(A) 2 2Ba , Zn (B) 3 3Bi , Fe (C) 2 2Cu , Pb (D) 2 3Hg , Bi

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In the following reactions

2 642 2 2 2

2 24 2 4

i. B HPd-BaSO8 6 8 8H ii. H O , NaOH, H O

H O i. EtMgBr, H O8 8HgSO , H SO ii. H , heat

C H C H X

C H O Y

37. Compound X is :

38. The major compound Y is :


When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of 5.7 C was measured for the beaker and its contents (Expt.1). Because the enthalpy of

neutralization of a strong acid with a strong base is a constant 157.0kJ mol , this experiment could be used to

measure the calorimeter constant.

In a second experiment (Expt. 2) 100mL of 2.0 M ACETIC ACID 5aK 2.0 10 was mixed with 100mL of 1.0 M

NaOH (under identical conditions to Expt.1) where a temperature rise of 5.6 C was measured. (Consider heat capacity of

all solutions as 1 14.2Jg K and density of all solutions as 11.0g mL )

39. Enthalpy of dissociation 1in kJ mol of acetic acid obtained from the Expt. 2 is

(A) 1.0 (B) 10.0 (C) 24.5 (D) 51.4 40. The pH of the solution after Expt.2 is (A) 2.8 (B) 4.7 (C) 5.0 (D) 7.0

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PART I : MATHEMATICS


This section contains EIGHT questions The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive For each question, darken the bubble corresponding to the correct integer in the ORS Marking scheme:


41. The coefficient of 9x in the expansion of 2 3 1001 1 1 ...... 1x x x x is _________.

42. Suppose that the foci of the ellipse 2 2

19 5x y

are (f1, 0) and (f2, 0) where f1 > 0 and f2 < 0.

Let P1 and P2 be two parabolas with a common vertex at (0, 0) and with foci at (f1, 0) and (2f2, 0), respectively. Let T1 be a tangent to P1 which passes through (2f2, 0) and T2 be a tangent to P2 which passes through (f1, 0). If

m1 is the slope of T1 and m2 is the slope of T2, then the value of 222

1

1 mm

is _________.

43. Let m and n be two positive integers greater than 1. If cos( )

lim2

n

me e e

then the value of

mn

is

44. If 1

1 29 3tan

20

12 91

x x xe dxx

where 1tan x takes only principal values, then the value of 3log 14e

is _________.

45. Let :f be a continuous odd function, which vanishes exactly at one point and 112

f . Suppose that

1

( ) ( )x

F x f t dt

for all 1

1, 2 ( ) ( ( ))x

x and G x t f f t dt

for all 1

( ) 11,2 . lim ,( ) 14x

F xx IfG x

then

the value of 12

f

is _________.

46. Suppose that p, q and r and three non-coplanar vectors in 3 . Let the components of a vector s along

,p q and r be 4, 3 and 5, respectively. If the components of this vector s along

p q r , p q r and p q r are x, y and z, respectively, then the value of 2x y z is

_________.

47. For any integer k, let cos sin , 1.7 7k

k ki where i

The value of the expression

12

11

3

4 1 4 21

k kk

k kk

is _________.

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48. Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140 , then the common difference of this A.P is _________.



49. Let , : 1, 2f g be continuous functions which are twice differentiable on the interval 1, 2 . Let the

values of f and g at the points 1, 0 and 2 be as given in the following table:

1x 0x 2x

f x 3 6 0

g x 0 1 1

In each of the intervals 1,0 and 0,2 the function 3 "f g never vanishes. Then the correct statements(s) is(are) : (A) ' 3 ' 0f x g x has exactly three solution in 1,0 0,2

(B) ' 3 ' 0f x g x has exactly one solution in 1,0

(C) ' 3 ' 0f x g x has exactly one solution in 0,2

(D) ' 3 ' 0f x g x has exactly two solutions in 1,0 and exactly two solutions in 0,2

50. Let 8 6 4 27 tan 7 tan 3 tan 3 tanf x x x x x for all , .2 2

x Then the correct expression (s) is (are)

(A) /4

0

1( )12

xf x dx

(B) /4

00

f x dx

(C) /4

0

16

xf x dx

(D) 0

1f x dx

51. Let 3

4192'( )

2 sinxf x

x

for all x with 1 0.

2f

If 1

1/2,m f x dx M then the possible values of m

and M are :

(A) 13, 24m M (B) 1 1,4 2

m M

(C) 11, 0m M (D) 1, 12m M 52. Let S be the set of all non-zero real numbers such that the quadratic equation 2 0ax x has two

distinct real roots 1x and 2x satisfying the inequality 1 2| | 1.x x Which of the following intervals is(are) a

subset(s) of ?S

(A) 1 1,2 5

(B) 1 , 05

(C) 10,5

(D) 1 1,25

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53. If 1 16 4sin and cos ,11 9

where the inverse trigonometric functions take only the principal

values, then the correct option(s) is(are) : (A) cos (B) sin (C) cos 0 (D) cos

54. Let E1 and E2 be two ellipses whose centers are at the origin. The major axes of E1 and E2 lie along the x-axis

and the y-axis, respectively, Let S be the circle 22 1 2.x y The straight line 3x y touches the curves

S, E1 and E2 at P, Q and R, respectively. Suppose that 2 23

PQ PR If 1 2ande e are the eccentricities of E1

and E2, respectively, then the correct expression(s) is(are) :

(A) 2 21 2

4340

e e (B) 1 27

2 10e e (C) 2 2

1 258

e e (D) 1 23

4e e

55. Consider the hyperbola 2 2: 1H x y and a circle S with center 2( , 0)N x . Suppose that H and S touch each

other at a point 1 1( , )P x y with 1 1x and 1 0y . The common tangent to H and S at P intersects the x-axis at point M. If ( , )l m is the centroid of the triangle PMN , then the correct expression(s) is(are) :

(A) 121 1

11 13

dl for xdx x

(B) 11

21 1

13 1

xdm for xdx x

(C) 121 1

11 13

dl for xdx x

(D) 11

1 03

dm for ydy

56. The option(s) with the values of a and L that satisfy the following equation is(are) :

4 6 40

6 40

(sin cos )?

(sin cos )

t

t

e at at dtL

e at at dt

(A) 4 12,

1ea Le

(B) 4 12,

1ea Le

(C) 4 14,

1ea Le

(D) 4 14,

1ea Le

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Let n1 and n2 be the number of red and black balls, respectively, in box I. Let n3 and n4 be the number of red and black balls, respectively, in box II.

57. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box.

The ball was found to be red. If the probability that this red ball was drawn from box II is 13

, then the correct

option(s) with the possible values of n1, n2, n3 and n4 is(are) : (A) 1 2 3 43 3 5 15n , n , n , n (B) 1 2 3 43 6 10 50n , n , n , n

(C) 1 2 3 48 6 5 20n , n , n , n (D) 1 2 3 46 12 5 20n , n , n , n

58. A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box

I, after this transfer, is 13

, then the correct option(s) with the possible values of n1 and n2 is(are)

(A) 1 24 6n and n (B) 1 22 3n and n

(C) 1 210 20n and n (D) 1 23 6n and n


Let :F be a thrice differentiable function. Suppose that F(1) = 0, F(3) = -4 and F’(x) < 0 for all (1 2 3)x / , .

Let ( ) ( )f x xF x for all x .

59. The correct statement(s) is (are) (A) ' (1) 0f (B) (2) 0f

(C) '( ) 0 (1, 3)f x for any x (D) '( ) 0 (1, 3)f x for some x

60. If 3 32 3

1 1'( ) 12 ''( ) 40x F x dx and x F x dx , then the correct expression(s) is(are) :

(A) 9 '(3) '(1) 32 0f f (B) 3

1( ) 12f x dx

(C) 9 '(3) '(1) 32 0f f (D) 3

1( ) 12f x dx

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