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For the benefit of the students, specially the aspiring ones, the question of JEE(advanced), 2013 are also given in this booklet. Keeping the interest of students studying in class XI, the questions based on topics from class XI have been marked with ‘*’, which can be attempted as a test. For this test the time allocated in Physics, Chemistry & Mathematics and Physics are 22 minutes, 21 minutes and 25 minutes respectively.

FIITJEE SOLUTIONS TO JEE(ADVANCED)-2013

CODE

PAPER 2

3

Time: 3 Hours Maximum Marks: 180ase read the instructions carefully. You are allotted 5 minutes specifically for this purpose. INSTRUCTIONS A. General: 1. This booklet is your Question Paper. Do not break the seals of this booklet before being instructed to do so by the invigilators. 2. Blank papers, clipboards, log tables, slide rules, calculators, cameras, cellular phones, pagers and electronic gadgets are NOT allowed inside the examination hall. 3. Write your name and roll number in the space provided on the back cover of this booklet. 4. Answers to the questions and personal details are to be filled on a two-part carbon-less paper, which is provided separately. These parts should only be separated at the end of the examination when instructed by the invigilator. The upper sheet is a machine-gradable Objective Response Sheet (ORS) which will be retained by the invigilator. You will be allowed to take away the bottom sheet at the end of the examination. 5. Using a black ball point pen darken the bubbles on the upper original sheet. Apply sufficient pressure so that the impression is created on the bottom duplicate sheet. B. Question Paper Format 6. The question paper consists of three parts (Physics, Chemistry and Mathematics). Each part consists of three sections. Section 1 contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE OR MORE are correct 7. Section 2 contains 4 paragraphs each describing theory, experiment, data etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of a paragraph has ONLY ONE correct answer among the four choices (A), (B), (C) and (D). 8. Section 3 contains 4 multiple choice questions relate to four paragraphs with two questions on each paragraph. Each question of a paragraph has ONLY ONE CORRECT ANSWER among the four choices (A), (B), (C) and (D). C. Marking Scheme 9. For each question in Section 1, you will be awarded 3 marks if you darken all the bubble(s) corresponding to only the correct answer(s) and zero mark if no bubbles are darkened. In all other cases, minus one (–1) mark will be awarded. 10. For each question Section 2 and 3, you will be awarded 3 marks if you darken the bubble corresponding to only the correct answer and zero mark if no bubbles are darkened. In all other cases, minus one (–1) mark will be awarded.

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JEE(ADVANCED)2013-Paper 2-PCM-2

PART - I: PHYSICS

SECTION – 1 (One or more options correct Type)

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. *1. Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are) (A) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is

GM L (B) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 4 GM . L (C) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 2 2GM L (D) The energy of the mass m remains constant.

Sol.

(B)

2GMm 1 mv 2 0 L 2 GM v2 L Note: The energy of mass ‘m’ means its kinetic energy (KE) only and not the potential energy of interaction between m and the two bodies (of mass M each) – which is the potential energy of the system.

*2.

A particle of mass m is attached to one end of a mass-less spring of force constant k, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time t = 0 with an initial velocity u 0. When the speed of the particle is 0.5 u0. It collides elastically with a rigid wall. After this collision, (A) the speed of the particle when it returns to its equilibrium position is u0. (B) the time at which the particle passes through the equilibrium position for the first time is t (C) the time at which the maximum compression of the spring occurs is t m . k

4 m . 3 k (D) the time at which the particle passes through the equilibrium position for the second time is t 5 m . 3 k

Sol.

(A, D) u0 u 0 cos t1 t1 2 3 2 Now the particle returns to equilibrium position at time t2 = 2t1 i.e. with the same mechanical energy 3 i.e. its speed will u0 . Let t3 is the time at which the particle passes through the equilibrium position for the second time.

v u 0 sin t (suppose t1 is the time of collision)

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JEE(ADVANCED)2013-Paper 2-PCM-3

T 2t1 2 2 5 3 3

t3

5 m 3 k Energy of particle and spring remains conserved.

3.

A steady current I flows along an infinitely long hollow cylindrical conductor of radius R. This cylinder is placed coaxially inside an infinite solenoid of radius 2R. The solenoid has n turns per unit length and carries a steady current I. Consider a point P at a distance r from the common axis. The correct statement(s) is (are) (A) In the region 0 < r < R, the magnetic field is non-zero (B) In the region R < r < 2R, the magnetic field is along the common axis. (C) In the region R < r < 2R, the magnetic field is tangential to the circle of radius r, centered on the axis. (D) In the region r > 2R, the magnetic field is non-zero. (A, D) Due to field of solenoid is non zero in region 0 < r < R and non zero in region r>2R due to conductor. Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f1 . An observer in the other vehicle hears the frequency of the whistle to be f 2 . The speed of sound in still air is V. The correct statement(s) is (are) (A) If the wind blows from the observer to the source, f 2 f1 . (B) If the wind blows from the source to the observer, f 2 f1 . (C) If the wind blows from observer to the source, f 2 f1 . (D) If the wind blows from the source to the observer f 2 f1 .

Sol.

*4.

Sol.

(A, B) If wind blows from source to observer Vwu f 2 f1 Vwu When wind blows from observer towards source Vwu f 2 f1 Vwu In both cases, f 2 f1 . Using the expression 2d sin = , one calculates the values of d by measuring the corresponding angles in the range to 90°. The wavelength is exactly known and the error in is constant for all values of . As increases from 0°, (A) the absolute error in d remains constant. (B) the absolute error in d increases (C) the fractional error in d remains constant. (D) the fractional error in d decreases. (D)

2sin ln d ln ln sin 2 d

*5.

Sol.

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JEE(ADVANCED)2013-Paper 2-PCM-4

d cos d 0 d sin d = cot d max Also (d)max d cot cot 2sin cos 2 sin 2

As increases cot decreases and 6.

cos sin 2

also decreases.

Two non-conducting spheres of radii R1 and R2 and carrying uniform volume charge densities + and –, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region, (A) the electrostatic field is zero (B) the electrostatic potential is constant (C) the electrostatic field is constant in magnitude (D) the electrostatic field has same direction (C, D) In triangle PC1C2 r2 d r1 The electrostatic field at point P is 4 3 4 K R1 r2 K R 3 r1 2 3 3 E 3 3 R1 R2 4 E K (r2 r1 ) 3 E d 30

R1

– R2

Sol.

r1 – r2 P C2 C1 R2 R1 d *7.

The figure shows the variation of specific heat capacity (C) of a solid as a function of temperature (T). The temperature is increased continuously from 0 to 500 K at a constant rate. Ignoring any volume change, the following statement(s) is (are) correct to a reasonable approximation. (A) the rate at which heat is absorbed in the range 0-100 K varies linearly with temperature T. (B) heat absorbed in increasing the temperature from 0-100 K is less than the heat required for increasing the temperature from 400 – 500 K. (C) there is no change in the rate of heat absorption in range 400 – 500 K. (D) the rate of heat absorption increases in the range 200 – 300 K.

C

100

200

300 400 T (K)

500

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JEE(ADVANCED)2013-Paper 2-PCM-5

Sol. (A, B, C, D) Option (A) is correct because the graph between (0 – 100 K) appears to be a straight line upto a reasonable approximation. Option (B) is correct because area under the curve in the temperature range (0 100 K) is less than in range (400 500 K.) Option (C) is correct because the graph of C versus T is constant in the temperature range (400 500 K) Option (D) is correct because in the temperature range (200 – 300 K) specific heat capacity increases with temperature. The radius of the orbit of an electron in a Hydrogen-like atom is 4.5 a0 where a0 is the Bohr radius. Its 3h orbital angular momentum is . It is given that h is Planck’s constant and R is Rydberg constant. The 2 possible wavelength(s), when the atom de-excites, is (are) 9 9 9 4 (A) (B) (C) (D) 32R 16R 5R 3R (A, C) Given data

4.5a 0 a 0 n2 Z

8.

Sol.

…(i)

nh 3h …(ii) 2 2 So n = 3 and z = 2 So possible wavelength are 1 9 1 1 RZ2 2 2 1 1 32R 1 3

1 1 1 1 RZ 2 2 2 2 2 3R 1 2 1 9 1 1 RZ2 2 2 3 3 5R 2 3

SECTION – 2 : (Paragraph Type)

This section contains 4 paragraphs each describing theory, experiment, date etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of paragraph has only one correct answer along the four choice (A), (B), (C) and (D). Paragraph for Questions 9 to 10 A small block of mass 1 kg is released from rest at the top of a rough track. The track is circular arc of radius 40 m. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point Q, as shown in the figure, below, is 150 J. (Take the acceleration due to gravity, g = 10 m/s2).

R 30 P

Q

R

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JEE(ADVANCED)2013-Paper 2-PCM-6

*9.

The speed of the block when it reaches the point Q is (A) 5 ms1 (B) 10 ms1 (C) 10 3 ms1 (B) Using work energy theorem 1 mg R sin 30 + Wf = mv 2 2 2 v 200 150 = 2 v = 10 m/s The magnitude of the normal reaction that acts on the block at the point Q is (A) 7.5 N (B) 8.6 N (C) 11.5 N (A) N mg cos 60 =

N 5 mv 2 R

(D) 20 ms1

Sol.

*10.

(D) 22.5 N

Sol.

5 7.5 Newton. 2

Paragraph for Questions 11 to 12 A thermal power plant produces electric power of 600 kW at 4000 V, which is to be transported to a place 20 km away from the power plant for consumers’ usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers’ end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with the power factor unity. All the currents and voltage mentioned are rms values. 11. If the direct transmission method with a cable of resistance 0.4 km1 is used, the power dissipation (in %) during transmission is (A) 20 (B) 30 (C) 40 (D) 50 (B) For direct transmission P i 2 R (150) 2 (0.4 20) = 1.8 105 w fraction(in %) 1.8 105 100 30% 6 105

Sol.

12.

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is 1 : 10. If the power to the consumers has to be supplied at 200 V, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is (A) 200 : 1 (B) 150 : 1 (C) 100 : 1 (D) 50 : 1 (A) 40000 200 200

Sol.

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JEE(ADVANCED)2013-Paper 2-PCM-7

Paragraph for Questions 13 to 14 A point Q is moving in a circular orbit of radius R in the x-y plane with an angular velocity . This can be Q considered as equivalent to a loop carrying a steady current . A uniform magnetic field along the positive z-axis 2 is now switched on, which increases at a constant rate from 0 to B in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant . 13. The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change, is BR BR (A) (B) (C) BR (D) 2BR 4 2 (B) E(2R) = R 2

E RB 2 dB dt

Sol.

14.

The change in the magnetic dipole moment associated with the orbit, at the end of time interval of the magnetic field change, is BQR 2 BQR 2 (A) BQR 2 (B) (C) (D) BQR 2 2 2 (B) L dt

R Q B R(1) 2 QR 2 B , in magnitude 2 = L BQR 2 = – (taking into account the direction) 2 v Sol.

Paragraph for Questions 15 to 16 The mass of nucleus A X is less than the sum of the masses of (A-Z) number of neutrons and Z number of protons in Z the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass M can break into two light nuclei of mass m 1 and m2 only if (m1 + m2) < M. Also two light nuclei of masses m3 and m4 can undergo complete fusion and form a heavy nucleus of mass M only if (m3 + m4) > M. The masses of some neutral atoms are given in the table below:

1 1 6 3

H Li Gd

1.007825 u 6.015123 u 151.919803 u

2 1 7 3

H Li Pb

2.014102 u 7.016004 u 205.974455 u

3 1

H Zn Bi

3.016050 u 69.925325 u 208.980388 u

4 2

He Se Po

4.002603 u 81.916709 u 209.982876 u

70 30

82 34

152 64

206 82

209 83

210 84

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JEE(ADVANCED)2013-Paper 2-PCM-8

15.

The correct statement is 6 (A) The nucleus 3 Li can emit an alpha particle

210 (B) The nucleus 84 Po can emit a proton. (C) Deuteron and alpha particle can undergo complete fusion. 70 (D) The nuclei 30 Zn and 82 Se can undergo complete fusion. 34

Sol.

(C) 6 4 2 3 Li 2 He 1 H

Q 6.015123 4.002603 2.014102 C2 0 0.001582 0 So no -decay is possible 210 1 209 84 P0 1 H 83 Bi Q 209.9828766 1.007825 208.980388 0.005337 0 C2 So, this reaction is not possible 2 4 6 1 H 2 He 3 Li Q 2.014102 4.002603 6.015123 0.001582 0 C2 So, this reaction is possible 70 82 152 30 Zn 34 Se 64 Gd Q 69.925325 81.916709 151.919803 0.077769 0 C2 So this reaction is not possible

16.

The kinetic energy (in keV) of the alpha particle, when the nucleus (A) 5319 (B) 5422 (C) 5707 (A) 210 4 206 84 Po 2 He 82 Pb Q = (209.982876 4.002603 205.97455)C2 = 5.422 MeV from conservation of momentum 2K1 (4) 2K 2 (206) 4K1 = 206K2 103 K1 K2 2 K1 + K2 = 5.422 2 K1 K1 5.422 103 105 K1 5.422 103 K1 = 5.319 MeV = 5319 KeV

210 84

Po at rest undergoes alpha decay, is (D) 5818

Sol.

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JEE(ADVANCED)2013-Paper 2-PCM-9 SECTION – 3 (Matching List Type)

This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 17. A right angled prism of refractive index 1 is placed in a rectangular block of refractive index 2, which is surrounded by a medium of refractive index 3, as shown in the figure. A ray of light ‘e’ enters the rectangular block at normal incidence. Depending upon the relationships between 1, 2 and 3, it takes one of the four possible paths ‘ef’, ‘eg’, ‘eh’, or ‘ei’. f e i 2 3 450 1 h g

Match the paths in List I with conditions of refractive indices in List II and select the correct answer using the codes given below the lists: P. Q. R. S. List I ef eg eh ei 1. 2. 3. 4. List II 1 > 2 2 2 > 1 and 2 > 3 1 = 2 2 < 1 < 2 2 and 2 > 3

Codes: (A) (B) (C) (D) Sol. P 2 1 4 2 Q 3 2 1 3 R 1 4 2 4 S 4 3 3 1

*18.

(D) P. (2) ; Q. (3); R. (4); S. (1) P. 2 > 1… (towards normal) 2 > 3 … (away from normal) Q. 1 = 2… (No change in path) i = 0 r = 0 on the block. R. 1 > 2 … (Away from the normal) 2 > 3 … (Away from the normal) 1 1 1 2 sin r sin r = . Since sin r < 1 1 < 2 2 2 2 2 1 S. For TIR : 450 > C sin 450 > sin C 2 1 2 2 2 1 Match List I with List II and select the correct answer using the codes given below the lists: List I P. Boltzmann Constant Q. Coefficient of viscosity R. Plank Constant S. Thermal conductivity Codes: P Q R (A) 3 1 2 (B) 3 2 1 (C) 4 2 1 (D) 4 1 2 List II [ML2T-1] [ML-1T-1] [MLT-3K-1] [ML2T-2K-1]

1. 2. 3. 4. S 4 4 3 3

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JEE(ADVANCED)2013-Paper 2-PCM-10

Sol.

(C) P. (4) ; Q. (2); R. (1); S. (3) 3 P. KE = K T [ML2T-2] = K[K] K = [ML2T-2K-1] 2 Q. F = 6rv [MLT-2] = [L][LT-1] = [ML-1T-1] h R. E = hf [ML2T-2] = h = [ML2T-1] [T] S. dQ KA( [ML2T 2 ] k[L2 ][K ] dt x [T] [L] K = [MLT-3K-1]

*19.

One mole of mono-atomic ideal gas is taken along two cyclic processes EFGE and EFHE as shown in the PV diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic.

P 32P0 F

P0

E V0

H

G

V

Match the paths in List I with the magnitudes of the work done in List II and select the correct answer using the codes given below the lists. P. Q. R. S. List I GE GH FH FG 1. 2. 3. 4. List II 160 P0V0 ln2 36 P0V0 24 P0V0 31 P0 V0

Codes: (A) (B) (C) (D) Sol. P 4 4 3 1 Q 3 3 1 3 R 2 1 2 2 S 1 2 4 4

(A) P. (4) ; Q. (3); R. (2); S. (1) Apply PV1+2/3 = constant for F to H. 5/3 5/3 (32P0) V0 = P0 VH VH = 8V0 For path FG PV = constant (32P0)V0 = P0VG VG = 32V0 Work done in GE = 31 P0V0 Work done in GH = 24 P0V0 P V PF VF Work done in FH = H H = 36P0V0 ( 2 / f )

V Work done in FG = RT ln G VF = 160P0V0ln2.

P 32P0 F n =1 f =3

Isothermal

Adiabatic P0 E V0 H 8V0 G 32V0 V

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JEE(ADVANCED)2013-Paper 2-PCM-11

20.

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists: List I Alpha decay + decay Fission Proton emission List II 1. 2. 3. 4.

15 15 8 O 7 N ..... 238 234 92 U 90 Th ..... 185 184 83 Bi 82 Pb ..... 239 140 94 Pu 57 La .....

P. Q. R. S.

Codes: (A) (B) (C) (D) P 4 1 2 4 Q 2 3 1 3 R 1 2 4 2 S 3 4 3 1

Sol.

(C) P. (2) ; Q. (1); R. (4); S. (3) 15 15 0 8 O 7 N 1 (Beta decay)

238 234 4 92 U 90 Th 2 He 185 184 1 83 Bi 82 Pb 1 H 239 140 99 94 Ph 57 La 37 Rb

(Alpha decay) (Proton emission) (fission)

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JEE(ADVANCED)2013-Paper 2-PCM-12

PART - II: CHEMISTRY

SECTION –1 (One or more options correct Type)

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. *21. The Ksp of Ag2CrO4 is 1.1 10–12 at 298K. The solubility (in mol/L) of Ag2CrO4 in a 0.1M AgNO3 solution is (A) 1.1 10–11 (B) 1.1 10–10 –12 (C) 1.1 10 (D) 1.1 10–9 (B)

K sp 1.11012 Ag CrO 2 4

1.1 10 12 0.1 s

2

Sol.

2

s 1.1 1010

22. In the following reaction, the product(s) formed is(are) OH

CHCl3 ? OH

CH3 OH OHC CHO O OH OH CHO

CH3 P (A) P(major) (C) R(minor) (B, D) OH

H3C Q

CHCl 2

H3C

CHCl 2

CH3 S

R (B) Q(minor) (D) S(major)

Sol.

O

OH CHO

OH

CHCl3

+

CH3

H3C CHCl 2 (Minor)

CH3 (major)

CHCl3 OH : CCl 2 H 2O Cl

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JEE(ADVANCED)2013-Paper 2-PCM-13

OH O

+

OH

H 2O

CH3

CH3

O

O CCl 2

: CCl 2

O CHCl 2

OH

OH CHO

H

CH3

O

CH3

O

CH3

O

CH3 (major)

H 2O

CH3

:CCl 2 H3C

CCl 2

H3C

CHCl 2 (minor)

23.

The major product(s) of the following reaction is (are) OH

2 ? aqueous Br 3.0 equivalents

SO3H OH Br Br Br

OH Br

OH

OH Br

Br SO3H P (A) P (C) R Sol. (B) Br Q

Br Br R (B) Q (D) S

Br

Br SO3H S

Br

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JEE(ADVANCED)2013-Paper 2-PCM-14

OH Br

2

Br 3equivalents

OH Br

SO3H 24.

Br (Q)

After completion of the reactions (I and II), the organic compound(s) in the reaction mixtures is(are) O

Br2 1.0 mol Reaction I : H C CH3 aqueous/ NaOH 3 1.0 mol

O Reaction II :

Br2 1.0 mol H3C CH3 CH3COOH 1.0 mol

O H3C P CH2Br H3C

O CBr3 Br 3C Q

O CBr3 BrH2C R

O CH2Br S H3C

O ONa T CHBr3 U

(A) (B) (C) (D) Sol.

Reaction I: P and Reaction II : P Reaction I: U, acetone and Reaction II:Q, acetone Reaction I : T, U, acetone and Reaction II : P Reaction I : R, acetone and Reaction II : S, acetone

(C) Solve as per law of limiting reagent. OH O C H3C CH3

H H3C

OH CH3

H

C

C H2C CH3

O

Br BrCH2

C

Br2 CH3

25.

The correct statement(s) about O3 is(are) (A) O–O bond lengths are equal. (C) O3 is diamagnetic in nature. (A, C, D)

1.21Å 1170

(B) Thermal decomposition of O3 is endothermic. (D) O3 has a bent structure.

Sol.

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JEE(ADVANCED)2013-Paper 2-PCM-15

*26. In the nuclear transmutation

Be X 4 Be Y (X, Y) is (are) (A) (, n) (C) (n, D)

9 4 8

(B) (p, D) (D) (, p)

Sol.

(A, B) 8 9 1 4 Be 4 Be 0 n

9 4 2 Be 1 P 4 Be 1 H 1 8

Hence (A) and (B) are correct 27. The carbon–based reduction method is NOT used for the extraction of (A) tin from SnO2 (B) iron from Fe2O3 (C) aluminium from Al2O3 (D) magnesium from MgCO3.CaCO3 (C, D) Fe2O3 and SnO2 undergoes C reduction. Hence (C) and (D) are correct. The thermal dissociation equilibrium of CaCO3(s) is studied under different conditions. CaCO3 s CaO s CO 2 g For this equilibrium, the correct statement(s) is(are) (A) H is dependent on T (B) K is independent of the initial amount of CaCO3 (C) K is dependent on the pressure of CO2 at a given T (D) H is independent of the catalyst, if any Sol. (A, B, D) For the equilibrium CaCO3 s CaO S CO 2 g . The equilibrium constant (K) is independent of initial amount of CaCO3 where as at a given temperature is independent of pressure of CO2. ∆H is independent of catalyst and it depends on temperature. Hence (A), (B) and (D) are correct.

Sol.

*28.

SECTION-2 (Paragraph Type)

This section contains 4 paragraphs each describing theory, experiment, data etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of a paragraph has only one correct answer among the four choices (A), (B), (C) and (D).

Paragraph for Question Nos. 29 and 30

An aqueous solution of a mixture of two inorganic salts, when treated with dilute HCl, gave a precipitate (P) and a filtrate (Q). The precipitate P was found to dissolve in hot water. The filtrate (Q) remained unchanged, when treated with H2S in a dilute mineral acid medium. However, it gave a precipitate (R) with H2S in an ammoniacal medium. The precipitate R gave a coloured solution (S), when treated with H2O2 in an aqueous NaOH medium. 29. The precipitate P contains (A) Pb2+ (C) Ag+ (A) (B) Hg 2 2 (D) Hg2+

Sol.

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JEE(ADVANCED)2013-Paper 2-PCM-16

30. The coloured solution S contains (A) Fe2(SO4)3 (C) ZnSO4

(B) CuSO4 (D) Na2CrO4

Sol. (D) Solution for the Q. No. 29 to 30. Hot Pb 2 2HCl PbCl 2 soluble. Water

H2 S Cr 3 Cr OH 3 ammoniacal

NaOH Cr OH 3 Na 2 CrO 4 H 2 O2

Yellow solution

Paragraph for Question Nos. 31 to 32

P and Q are isomers of dicarboxylic acid C4H4O4. Both decolorize Br2/H2O. On heating, P forms the cyclic anhydride. Upon treatment with dilute alkaline KMnO4, P as well as Q could produce one or more than one from S, T and U. COOH H H OH OH COOH S *31. H HO COOH OH H COOH T HO H COOH H OH COOH U

Compounds formed from P and Q are, respectively (A) Optically active S and optically active pair (T, U) (B) Optically inactive S and optically inactive pair (T, U) (C) Optically active pair (T, U) and optically active S (D) Optically inactive pair (T, U) and optically inactive S (B)

H C C H P COOH COOH

4 OH

Sol.

COOH

KMnO

H H

OH OH COOH

H C C HOOC Q

COOH

KMnO4 OH

S optically inactive COOH

COOH OH H H OH COOH U

H HO

OH H COOH T

H

optically inactive pair

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JEE(ADVANCED)2013-Paper 2-PCM-17

*32. In the following reaction sequences V and W are, respectively H2 / Ni Q V

3 V W 1. Zn HHg / HCl 2. 3 PO 4

AlCl anhydrous

(A)

O O and

(B)

CH2OH and CH2OH V W

V (C)

O

O

W

O (D)

HOH2C and CH2 OH V W CH2OH

O and

V

O

W

Sol.

(A)

O

H 2 / Ni Q

O

O O

AlCl3

O C

+

O

Zn Hg, HCl

C O HO O OH

C O H3PO4

O

Paragraph for Question Nos. 33 to 34

A fixed mass ‘m’ of a gas is subjected to transformation of states from K to L to M to N and back to K as shown in the figure

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JEE(ADVANCED)2013-Paper 2-PCM-18

K

L

Pressure

N Volume

M

*33.

The succeeding operations that enable this transformation of states are (A) Heating, cooling, heating, cooling (B) Cooling, heating, cooling, heating (C) Heating, cooling, cooling, heating (D) Cooling, heating, heating, cooling (C) The pair of isochoric processes among the transformation of states is (A) K to L and L to M (B) L to M and N to K (C) L to M and M to N (D) M to N and N to K

Sol. *34.

Sol. (B) Solution for the Q. No. 33 to 34.

K P N L

M

V K – L heating, isobaric L – M cooling, isochoric M – N cooling, isobaric N – K heating, isochoric

Paragraph for Question Nos. 35 to 36

The reactions of Cl2 gas with cold-dilute and hot-concentrated NaOH in water give sodium salts of two (different) oxoacids of chlorine, P and Q, respectively. The Cl2 gas reacts with SO2 gas, in presence of charcoal, to give a product R. R reacts with white phosphorus to give a compound S. On hydrolysis, S gives an oxoacid of phosphorus, T. 35. P and Q, respectively, are the sodium salts of (A) hypochlorus and chloric acids (C) chloric and perchloric acids (A) R, S and T, respectively, are (A) SO2Cl2, PCl5 and H3PO4 (C) SOCl2, PCl3 and H3PO2 (A)

(B) hypochlorus and chlorus acids (D) chloric and hypochlorus acids

Sol. 36.

(B) SO2Cl2, PCl3 and H3PO3 (D) SOCl2, PCl5 and H3PO4

Sol.

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JEE(ADVANCED)2013-Paper 2-PCM-19

Solution for the Q. No. 35 to 36 2NaOH Cl 2 NaCl NaClO H 2O

Cold dil

P

6NaOH 3Cl2 5NaCl NaClO3 3H 2 O

Hot conc.

Q

Charcoal

SO 2 Cl2 SO 2Cl 2

R

SO 2Cl 2 P4 PCl5 SO 2

S

PCl5 H 2O H 3 PO 4 HCl

T

SECTION – 3: (Matching List Type) This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 37. The unbalanced chemical reactions given in List – I show missing reagent or condition (?) which are provided in List – II. Match List – I with List – II and select the correct answer using the code given below the lists: List – I List - II ? (P) (1) NO PbO H SO PbSO O other product

2 2 4 4 2

(Q) (R) (S) Codes: (A) (B) (C) (D) Sol. (D)

Na 2S2 O3 H 2 O NaHSO 4 other product N 2 H 4 N 2 other product XeF2 Xe other product

? ?

?

(2) (3) (4)

I2 Warm Cl2

P 4 3 1 3

Q 2 2 4 4

R 3 1 2 2

S 1 4 3 1

1 (P) PbO 2 H 2SO4 PbSO 4 H 2 O O 2 2 (Q) 2Na 2S2 O3 Cl 2 2H 2 O 2NaCl 2NaHSO4 2S

(R) N 2 H 4 2I 2 N 2 4HI (S) XeF2 2NO Xe 2NOF *38. Match the chemical conversions in List – I with appropriate reagents in List – II and select the correct answer using the code given below the lists: List – I List - II (P) (1) (i) Hg(OAc)2; (ii) NaBH4 Cl

(Q) ONa OEt

(2)

NaOEt

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JEE(ADVANCED)2013-Paper 2-PCM-20

(R)

OH

(3)

Et-Br

(S)

(4)

(i) BH3; (ii) H2O2/NaOH

OH Codes: (A) (B) (C) (D) Sol. (A) (P)

NaOEt Cl

P 2 3 2 3

Q 3 2 3 2

R 1 1 4 4

S 4 4 1 1

(Q)

ONa

EtBr

OEt

(R)

2 ii NaBH 4

(i) HgOAc

OH

(S)

3 2 2

i BH (ii) H O / NaOH

OH

39.

An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List – I. The variation in conductivity of these reactions in List – II. Match List – I with List – II and select the correct answer using the code given below the lists: List – I List - II (P) (1) Conductivity decreases and then increases C2 H5 N CH3COOH

3 X Y

(Q) (R) (S) Codes: (A) (B) (C) (D) (A)

KI 0.1M AgNO3 0.01M

X Y

(2) (3) (4)

CH3 COOH KOH

X Y

NaOH HI

X Y

Conductivity decreases and then does not change much Conductivity increases and then does not change much Conductivity does not change much and then increases

P 3 4 2 1

Q 4 3 3 4

R 2 2 4 3

S 1 1 1 2

Sol.

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JEE(ADVANCED)2013-Paper 2-PCM-21

C2 H 5 3 N CH3COOH C2 H 5 3 NH CH 3COO

X Y

(P)

Initially conductivity increases due to ion formation after that it becomes practically constant because X alone can not form ions. Hence (3) is the correct match.

(Q) KI 0.1 M AgNO3 0.01M AgI KNO3

X Y

Number of ions in the solution remains constant until all the AgNO3 precipitated as AgI. Thereafter conductance increases due to increases in number of ions. Hence (4) is the correct match. (R) Initially conductance decreases due to the decrease in the number of OH ions thereafter it slowly increases due to the increases in number of H+ ions. Hence (2) is the correct match. (S) Initially it decreases due to decrease in H+ ions and then increases due to the increases in OH ions. Hence (1) is the correct match. 40. The standard reduction potential data at 25C is given below:

E Fe 2 , Fe 0.44V E Cu 2 , Cu 0.34V; E Cu , Cu 0.52V

E Fe3 , Fe 2 0.77V;

E O2 g 4H 4e 2H 2 O 1.23V; E O2 g 2H 2 O 4e 4OH 0.40V

E Cr 2 , Cr 0.91V

E Cr3 , Cr 0.74V;

Match E0 of the redox pair in List – I with the values given in List – II and select the correct answer using the code given below the lists: (P) (1) 0.18 V E Fe3 , Fe (Q) (R) (S) Codes: (A) (B) (C) (D) Sol. P 4 2 1 3 Q 1 3 2 4 R 2 4 3 1 S 3 1 4 2

E 4H 2 O 4H 4OH

(2) (3) (4)

0.4 V 0.04 V 0.83 V

E Cr3 , Cr 2

E Cu 2 Cu 2Cu

(D) o (P) G o 3 / Fe G o 3 /Fe2 G Fe2 / Fe Fe Fe

3 FE oFe 3 /Fe 1 FE oFe3 / Fe2 2 FE o 2 / Fe Fe

E o 3 / Fe 0.04 V Fe

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JEE(ADVANCED)2013-Paper 2-PCM-22

(Q) O 2 g 2H 2 O 4e 4O H

2H 2 O O 2 g 4H 4e

So 4H 2 O 4H 4 O H

E o 0.40 V o ... (i) .. (ii) ... (iii)

E 1.23 V

E for III reduction = 0.40 – 1.23 = - 0.83 V. (R) G oCu 2 /Cu G oCu2 /Cu G oCu /Cu

o

rd

2 FE

o Cu 2 /Cu

1 FE

o Cu 2 /Cu

o 1 F E Cu /Cu

E

o Cu 2 /Cu

0.18 V . o o G Cr 3 /Cr G Cr /Cr2

(S) G

o Cr 3 /Cr 2

o o 1 F E o 3 /Cr 2 3 F E Cr3 / Cr 2 F E Cr /Cr 2 Cr

E

o Cr 3 /Cr 2

0.4 V .

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JEE(ADVANCED)2013-Paper 2-PCM-23

PART - III: MATHEMATICS

SECTION 1 : (One or more option correct Type)

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

41.

For a R (the set of all real numbers), a 1, lim Then a = (A) 5 15 (C) 2

1

n 1 a 1

a

2a ... n a

n

na 1 na 2 ... na n

1 . 60

(B) 7 17 (D) 2

Sol.

(B, D)

1

x dx

Required limit =

0 1

a

a x dx

0

2 2 2a 1 (a 1) 120

17 a = 7 or . 2

*42.

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 2 7 on y-axis is (are) (A) x2 + y2 6x + 8y + 9 = 0 (B) x2 + y2 6x + 7y + 9 = 0 2 2 (C) x + y 6x 8y + 9 = 0 (D) x2 + y2 6x 7y + 9 = 0 (A), (C) Equation of circle can be written as (x 3)2 + y2 + (y) = 0 x2 + y2 6x + y + 9 = 0. Now, (radius)2 = 7 + 9 = 16

Sol.

2 9 16 4 2 = 64 = 8. Equation is x2 + y2 6x 8y + 9 = 0.

9 43. Two lines L1 : x = 5, (A) 1 (C) 3 Sol. (A, D) x 5 y 0 z 0 0 3 2 y z y z and L2 : x = , are coplanar. Then can take value(s) 3 2 1 2 (B) 2 (D) 4

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JEE(ADVANCED)2013-Paper 2-PCM-24 x y z 0 1 2 will be coplanar if shortest distance is zero 5 0 0

0 0

2

3 2 1 2

0

(5 – )( – 5 + 4) = 0, = 1, 4, 5 so = 1, 4 Alternate Solution: As x = 5 and x = are parallel planes so the remaining two planes must be coplanar. 3 2 So, 2 – 5 + 4 = 0 = 1, 4. 1 2 *44.

1 . Further the incircle of the triangle touches the sides 3 PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are) (A) 16 (B) 18 (C) 24 (D) 22

In a triangle PQR, P is the largest angle and cos P =

Sol.

(B), (D) Let s a = 2k 2, s b = 2k, s c = 2k + 2, k I, k > 1 Adding we get, s = 6k So, a = 4k + 2, b = 4k, c = 4k 2 1 Now, cos P = 3

P sa N sc M

Q

sb

L

R

b2 c2 a 2 1 3 [(4k)2 + (4k 2)2 (4k + 2)2] = 2 4k (4k 2) 2bc 3 3 [16k2 4 (4k) 2] = 8k (4k 2) 48k2 96k = 32k2 16k 16k2 = 80k k = 5 So, sides are 22, 20, 18

*45.

3 i and P = {wn : n = 1, 2, 3, ….. }. Further H1 = 2 1 z C : Re z , where C is the set of all complex numbers. If z1 2 represents the origin, then z1 Oz2 = (A) (B) 2 6 2 5 (C) (D) 3 6

Let w =

1 z C : Re z and H2 = 2

P H1, z2 P H2 and O

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JEE(ADVANCED)2013-Paper 2-PCM-25

Sol. (C), (D) i 3 i = e 6 , so w n e 6 2 n 1 n 1 Now, for z1, cos and for z2, cos 6 2 6 2

i

n

B1 B2 B3 /6 O

A3 A2 A1

w=

x = 1/2 x = 1/2 Possible position of z1 are A1, A2, A3 whereas of z2 are B1, B2, B3 (as shown in the figure) 2 5 So, possible value of z1Oz2 according to the given options is or . 3 6

*46.

If 3x = 4x1, then x = 2 log3 2 (A) 2 log3 2 1 (C)

1 1 log 4 3

(B) (D)

2 2 log 2 3 2 log 2 3 2 log 2 3 1

Sol.

(A, B, C) log23x = (x 1) log24 = 2 (x 1) x log23 = 2x – 2 2 x= 2 log 2 3 Rearranging, we get 2 log 3 2 2 x= 1 2 log 3 2 1 2 log 3 2 Rearranging again, 1 log3 4 log 4 3 1 x= = . 1 log3 4 1 1 log 4 3 1 log 4 3 Let be a complex cube root of unity with 1 and P = [pij] be a n n matrix with pij = i+j. Then P2 0, when n = (A) 57 (B) 55 (C) 58 (D) 56 (B, C, D) 2 3 4 3 4 5 P n 2 n 3 .....

4 6 ..... 5 7 9 ..... 2 P n 4 n 6 .....

47.

Sol.

..... n 2 ..... n 3 ..... 2n 4 5 7 9 ..... ..... ..... ..... ..... ..... ..... 2n 4 2n 6 ..... .....

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JEE(ADVANCED)2013-Paper 2-PCM-26

P2 = Null matrix if n is a multiple of 3 48. The function f(x) = 2 |x| + |x + 2| ||x + 2| 2 |x|| has a local minimum or a local maximum at x = 2 (A) 2 (B) 3 2 (C) 2 (D) 3 (A), (B) f x gx f x gx As, = Min (f (x), g (x)) 2 2 x x2 x 2 2 x = Min (|2x|, |x + 2|) 2

Sol.

2x + 4 2x 4 2x + 4 2 2/3 8/3 4x 2 4x

According to the figure shown, points of local minima/maxima are x = 2,

2 , 0. 3

SECTION 2 : (Paragraph Type)

This section contains 6 multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. Paragraph for Questions 49 and 50 Let f : [0, 1] R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0) = f(1) = 0 and satisfies f (x) 2f (x) + f(x) ex, x [0, 1]. 49. Which of the following is true for 0 < x < 1 ? (A) 0 < f(x) <

1 (C) < f(x) < 1 4

(B)

1 1 < f(x) < 2 2

(D) < f(x) < 0

Sol.

(D) Let g (x) = e–x f (x) and g (x) > 1 > 0 So, g (x) is concave upward and g (0) = g (1) = 0 Hence, g (x) < 0 x (0, 1) e–x f (x) < 0 f (x) < 0 x (0, 1) Alternate Solution f (x) – 2f(x) + f (x) ex

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JEE(ADVANCED)2013-Paper 2-PCM-27

x2 f x ex 2

0

Let g (x) = f (x) e–x – g (0) = 0, g (1) =

1 2

x2 2

Since g is concave up so it will always lie below the chord joining the extremities which is y = f x e x f (x) <

x 2

x2 x 2 2

2 0 x (0, 1)

x2 x ex

50.

If the function ex f(x) assumes its minimum in the interval [0, 1] at x =

1 3 x 4 4 1 (C) f x f x , 0 x 4 C Let, g (x) = e–x f (x) As g (x) > 0 so g (x) is increasing. So, for x < 1/4, g (x) < g (1/4) = 0 (f (x) – f (x))e–x < 0 f (x) < f (x) in (0, 1/4).

(A) f x f x ,

1 , which of the following is true ? 4 1 (B) f x f x , 0 x 4 3 (D) f x f x , x 1 4

Sol.

Paragraph for Questions 51 and 52 Let PQ be a focal chord of the parabola y2 = 4ax. The tangents to the parabola at P and Q meet at a point lying on the line y = 2x + a, a > 0. *51. Length of chord PQ is (A) 7a (C) 2a (B)

(B) 5a (D) 3a

Sol.

2a a Let P at 2 , 2at , Q 2 , as PQ is focal chord t t Point of intersection of tangents at P and Q 1 a, a t t as point of intersection lies on y = 2x + a 1 a t 2a a t

1 1 t 1 t 5 t t

1 length of focal chord = a t = 5a t

2 2

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JEE(ADVANCED)2013-Paper 2-PCM-28

*52.

If chord PQ subtends an angle at the vertex of y2 = 4ax, then tan = 2 2 (A) 7 (B) 7 3 3 2 2 (C) 5 (D) 5 3 3 (D) Angle made by chord PQ at vertex (0, 0) is given by 1 2 2 t t 2t t = 2 5 tan = 1 4 3 3 Paragraph for Questions 53 and 54

Sol.

Let S = S1 S2 S3, where

z 1 3 i S1 = {z C : |z| < 4}, S2 = z C : Im 0 and S3 = {z C : Re Z > 0}. 1 3 i

*53.

Area of S = 10 (A) 3 16 (C) 3 (B) Area of region 42 42 4 6 1 1 = 42 4 6 20 = 3

20 3 32 (D) 3

(B)

Sol.

y 3x 0

S1

S2

S3

=

shaded area

60º 60º x2 + y2 < 16

*54.

min 1 3i z z S

2 3 2 3 3 (C) 2

(A)

2 3 2 3 3 (D) 2

(B)

Sol.

(C) Distance of (1, –3) from y 3x 0 >

3 3 1 2

3 3 2

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JEE(ADVANCED)2013-Paper 2-PCM-29

Paragraph for Questions 55 and 56 A box B1 contains 1 white ball, 3 red balls and 2 black balls. Another box B2 contains 2 white balls, 3 red balls and 4 black balls. A third box B3 contains 3 white balls, 4 red balls and 5 black balls. 55. If 1 ball is drawn from each of the boxes B1, B2 and B3, the probability that all 3 drawn balls are of the same colour is 82 90 (A) (B) 648 648 558 566 (C) (D) 648 648 (A) P (required) = P (all are white) + P (all are red) + P (all are black) 1 2 3 3 3 4 2 4 5 = 6 9 12 6 9 12 6 9 12 6 36 40 82 = . 648 648 648 648 If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box B2 is 116 126 (A) (B) 181 181 65 55 (C) (D) 181 181 (D) Let A : one ball is white and other is red E1 : both balls are from box B1 E2 : both balls are from box B2 E3 : both balls are from box B3 E Here, P (required) = P 2 A

A P P E2 E2 = A A A P P E1 P P E2 P P E3 E1 E2 E3

2

Sol.

56.

Sol.

1 55 6 = 1 = . 3 2 3 3 4 1 1 2 181 C1 C1 1 C1 C1 1 C1 C1 1 9 12 6 3 3 3 5 6 11 C2 C2 C2

9

C1 3 C1 C2

1 3

SECTION 3 : (Matching list Type)

This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct. *57. Match List I with List II and select the correct answer using the code given below the lists :

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JEE(ADVANCED)2013-Paper 2-PCM-30

List I P.

cos tan 1 y y sin tan 1 y 1 y 4 takes value y 2 cot sin 1 y tan sin 1 y If cosx + cosy + cosz = 0 = sinx + siny + sinz then x y possible value of cos is 2 If cos x cos2x + sinx sin2x secx = cosx sin2x secx 4 + cos x cos2x then possible value of secx is 4

List II

1/2

1.

1 5 2 3

Q.

2.

2

R.

3.

1 2

S.

If cot sin 1 1 x 2 sin tan 1 x 6 possible value of x is

,

x 0, then

4.

1

Codes : (A) (B) (C) (D) (B) P P 4 4 3 3 Q 3 3 4 4 R 1 2 2 1 S 2 1 1 2

Sol.

cos tan 1 y ysin tan 1 y cot sin 1 y tan sin 1 y

1 y2 1 y2

=

1 y2

1 y2 y y 1 y2

=

1 y2 y 1 y4 1 y 1 y2

2

1 1 1 cos tan y ysin tan y 2 y4 cot sin 1 y tan sin 1 y y 1 2 4 4 = 2 y 1 y 4 y 4 = 1 y y 1 y Q cos x + cos y + cos z = 0 sin x + sin y + sin z = 0 cos x + cos y = – cos z sin x + sin y = – sin z (1)2 + (2)2 1 + 1 + 2(cos x cos y + sin x sin y) = 1 2 + 2 cos (x – y) = 1 2 cos (x – y) = –1 1 cos (x – y) = 2 1 xy 2 cos 2 1 2 2 xy 1 2 cos 2 2 2

..... (1) ..... (2)

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JEE(ADVANCED)2013-Paper 2-PCM-31

xy 1 cos 2 2 4 xy 1 cos 2 2 R cos x cos 2x sin x sin 2x sec x 4 = cos x sin 2x sec x cos x cos 2x 4 cos 4 x cos 4 x cos 2x = (cos x sin 2x – sin x sin 2x) sec x 2 sin x cos 2x cos x sin x sin 2x sec x 2 2 sin x cos 2x cos x sin x 2sin x

1 x 4 2 cos x sin x sec x sec 2 4 1

S cot sin 1 1 x 2 x cot 1 x2 tan 1 x 6

sin

x 6 6x 2 1

2

1 x 6x 2 1 2 6x + 1 = 6 – 6x 12x2 = 5

2

x

x 6

x

5 1 5 = 12 2 3

*58.

A line L : y = mx + 3 meets y-axis at E(0, 3) and the arc of the parabola y2 = 16x, 0 y 6 at the point F(x0, y0). The tangent to the parabola at F(x0, y0) intersects the y-axis at G(0, y1). The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum. Match List I with List II and select the correct answer using the code given below the lists : List I P. m= 1. 2. 3. 4. S 3 2 4 2

1 2 4 2 1

List II

Q. Maximum area of EFG is R. y0 = S. y1 = Codes : P Q R (A) 4 1 2 (B) 3 4 1 (C) 1 3 2 (D) 1 3 4

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JEE(ADVANCED)2013-Paper 2-PCM-32

Sol.

(A) A(t) = 2t2(3 4t) For max. A(t), t = m=1

1 sq. units 2 y0 = 4 and y1 = 2 1 2

E G

4t 2 , 8t

F

A t

max .

59.

Match List I with List II and select the correct answer using the code given below the lists : P. List I Volume of parallelepiped determined by vectors a, b and c is 2. Then the volume of the parallelepiped determined by vectors 2 a b , 3 b c and c a is Volume of parallelepiped determined by vectors a, b and c is 5. Then the volume of the parallelepiped determined by vectors 3 a b , b c and 2 c a is List II 1. 100

Q.

2.

30

R.

Area of a triangle with adjacent sides determined by vectors a and b is 20. Then the area of the triangle with adjacent sides determined by vectors 2a 3b and a b is

3.

24

S.

Area of a parallelogram with adjacent sides determined by vectors a and b is 30. Then the area of the parallelogram with adjacent sides determined by vectors a b and a is

4.

60

Codes : (A) (B) (C) (D) Sol. (C) P 4 2 3 1 Q 2 3 4 4 R 3 1 1 3 S 1 4 2 2

P a b c 2 2a b 3b c c a = 6 a Q a b c 5 6 a b b c c a = 12 a 1 R a b 20 2 1 2a 3b a b 2

2 b c 6 4 24

b c 60

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JEE(ADVANCED)2013-Paper 2-PCM-33

1 2 a b 3 a b 2 5 40 100 2 S a b 30 a b a = b a 30

60.

x 1 y z 3 x4 y 3 z 3 , L2 : and the planes P1 : 7x + y + 2z = 3, P2 2 1 1 1 1 2 : 3x + 5y 6z = 4. Let ax + by + cz = d be the equation of the plane passing through the point of intersection of lines L1 and L2, and perpendicular to planes P1 and P2. Match List I with List II and select the correct answer using the code given below the lists :

Consider the lines L1 :

List I P. Q. R. S. Codes : (A) (B) (C) (D) Sol. P 3 1 3 2 Q 2 3 2 4 R 4 4 1 1 S 1 2 4 3 a= b= c= d= 1. 2. 3. 4. 13 3 1 2

List II

(A) Plane perpendicular to P1 and P2 has Direction Ratios of normal ˆ ˆ k i j ˆ ˆ ˆ ˆ 7 1 2 = 16i 48j 32k … (1)

3 5 6

For point of intersection of lines (21 + 1, 1, 1 3) (2 + 4, 2 3, 22 3) 21 + 1 = 2 + 4 or 21 2 = 3 1 = 2 3 or 1 + 2 = 3 1 = 2, 2 = 1 Point is (5, 2, 1) … (2) From (1) and (2), required plane is 1 (x 5) + 3 (y + 2) + 2 (z + 1) = 0 or x + 3y + 2z = 13 x 3y 2z = 13 a = 1, b = 3, c = 2, d = 13.

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