JEE (ADVANCED) 2019 PAPER 2 PART -I PHYSICS...JEE (Advanced) 2019 Paper 2 Page 1 JEE (ADVANCED) 2019...

JEE (Advanced) 2019 Paper 2

JEE (ADVANCED) 2019 PAPER 2 PART-I PHYSICS

SECTION 1 (Maximum Marks:32)

This section contains EIGHT (08) questions. Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct

answer(s). For each question, choose the option(s) corresponding to (all) the correct answer(s). Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If only (all) the correct option(s) is(are) chosen; Partial Marks : +3 If all the four options are correct but ONLY three options are chosen; Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of

which are correct; Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct

option; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : -1 In all other cases.

For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then choosing ONLY (A), (B) and (D) will get +4 marks; choosing ONLY (A) and (B) will get +2 marks; choosing ONLY (A) and (D) will get +2 marks; choosing ONLY (B) and (D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY (B) will get +1 mark; choosing ONLY (D) will get +1 mark; choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get -1 mark.


Q.1. A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod

is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle 600 with vertical?

[g is the acceleration due to gravity]

(A) The angular speed of the rod will be 32

gL

(B) The angular acceleration of the rod will be 2gL

(C) The radial acceleration of the rod’s center of mass will be 34g

(D) The normal reaction force from the floor on the rod will be 16Mg

KEY: (A, C, D)

Sol 2

sin602 3l mlmg

34 3

g l

3 3

4g

l

2

2

2

2

11 cos602 2 3

4 632

l mlmg

g l

gl

radial acceleration 2

2la ; 3

4ga

3 1 3 3 34 2 4 2

3 98 16

1516 16

g gMg N M ll

Mg N Mg

Mg MgN Mg


Q.2. A block of mass 2M is attached to a massless spring with spring-constant k. This block is connected to two other blocks of masses M and 2M using two massless pulleys and strings. The accelerations of the blocks are a1, a2 and a3 as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is x0. Which of the following option(s) is/are correct?

[g is the acceleration due to gravity. Neglect friction]

(A) 0

4mgxk

(B) When spring achieves an extension of 0

2x

for the first time, the speed of the block connected

to the spring is 35Mgk

.

(C) At an extension of 0

4x

of the spring, the magnitude of acceleration of the block connected to

the spring is 310

g.

(D) 2 1 1 3a a a a KEY: (D)

Sol 2 1 1 3a a a a

Effective mass of pulley-block system is 4 2 8

3 3M M M

M .

16

3oMgxK

22

oo

xx A A


Acceleration of block is 2 4 3 22 3 14 7A Mg k g

k M

Speed = 3 8 3

2 14 3 14ox k Mg KA

M K M

Q.3. A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis

undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is 0L L the particle speed is 0v v . The piston is moved

inward at a very low speed V such that V << 0

dL vL

, where dL is the infinitesimal displacement of the

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

(A) The rate at which the particle strikes the piston is /v L (B) After each collision with the piston, the particle speed increases by 2V

(C) If the piston moves inward by dL, the particle speed increases by 2 dLvL

(D) The particle’s kinetic energy increases by a factor of 4 when the piston is moved inward from

0L to 0

12

L .

KEY: (B, D)

Sol Before the collision speed is v.

After the collision speed is v + 2V. Change in speed is 2V. If V1 is the instantaneous speed and ‘x’ remaining distance then V1x = constant. V1 = 2v.

Q.4. An electric dipole with dipole moment 0 ˆ ˆ2

p i j is held fixed at the origin O in the presence of an

uniform electric field of magnitude 0E . If the potential is constant on a circle of radius R centered at

the origin as shown in figure, then the correct statement(s) is/are: ( 0 is permittivity of free space. dipole size)


(A) 1/3

0

0 04pR

E

(B) Total electric field at point A is 0ˆ ˆ2AE E i j

(C) Total electric field at point B is 0BE

(D) The magnitude of total electric field on any two points of the circle will be same. KEY: (A, C) Sol At ‘B’ electric field due to dipole is

3

3

13

42 3

oB

oo

o

o o

A o o o

KPER

KP ER

PRE

E E E E

B A

Q.5. A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially

at pressure 0P , volume 0V , and temperature 0T . If the gas mixture is adiabatically compressed to a volume 0 / 4V , then the correct statement(s) is/are,

(Given 1.2 3.22 2.3; 2 9.2;R is gas constant)

(A) The work W done during the process is 013RT .

(B) The average kinetic energy of the gas mixture after compression is in between 018RT and

019RT .


(C) The final pressure of the gas mixture after compression is in between 09P and 010P . (D) Adiabatic constant of the gas mixture is 1.6. KEY: (A, C, D)

Sol 15 o

o

RTPV

2o

o

RTPV

1 2

1 11

mixture 1 11

6

5 15 1

oo

o

P P

V V

RTP P PV

C CC C

5 75 12 2

3 55 12 2

32 1.620

R R

R R

1.61.6

1.6 3.2

constant

P / 4

4 2

9.2

o o o

o o

o

PV

V P V

P P P

P P

2 2 1 19.2

41 1 1.6

2.3 1 1.3 60.6 0.6

13

oo o o

o o o

o

VP PVPV PVW

PV RT

W RT

Q.6. Three glass cylinders of equal height H = 30 cm and same refractive index n = 1.5 are placed on a

horizontal surface as shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radii of curvature of the two curved tops are same (R = 3m). If 1 2,H H and

3H are the apparent depths of a point X on the bottom of the three cylinders, respectively, the correct statement(s) is/are:


(A) 2 1H H (B) 3 1H H

(C) 2 3H H (D) 2 10.8 0.9cm H H cm

KEY: (A, C)

Sol (I)

12 2 30 20

3 3HH cm

(II)

1 1.5 1 1.5

1 1 1.52 3 0.3

V H R

V

1 1 56

1 2966 0.193529

V

V

V


(III)

1 1 56631

V

V

Q.7. In a Young’s double slit experiment, the slit separation d is 0.3 mm and the screen distance D is 1 m. A parallel beam of light of wavelength 600 nm is incident on the slits at angle a as shown in figure. On the screen, the point O is equidistant from the slits and distance PO is 11.0 mm. Which of the following statement(s) is/are correct?

(A) For 0.36

degree, there will be destructive interference at point O.

(B) For 0 , there will be constructive interference at point P.

(C) For 0.36

degree, there will be destructive interference at point P.

(D) Fringe spacing depends on . KEY: (C) Sol sind x

2

3 70.36 36 100.3 10 6 105 180 180

d

Fringe width 7

46 10 1 2

3 10D mmd


Q.8. A free hydrogen atom after absorbing a photon of wavelength a gets excited from the state 1n to

the state 4n . Immediately after that the electron jumps to n m state by emitting a photon of wavelength e . Let the change in momentum of atom due to the absorption and the emission are ap

and ep , respectively. If 1/5a e , which of the option(s) is/are correct?

[Use 91242 ;1 10hc eV nm nm m , h and c are Planck’s constant and speed of light, respectively]

(A) 2m (B) 418e nm

(C) 1/2a ep p

(D) The ratio of kinetic energy of the electron in the state n m to the slate 1n is 14

KEY: (A, D)

Sol 1 11

16a

R

; 2 2

1 1 14e

Rm

;

2 21 1

1415 516

a

e

m

2 21 1 3

164m ; 2

1 1 24

mm

,

Kinetic energy 21n

21

1 1142

1 1 124213.64 163 124213.6

16

m

e

e

KK


SECTION 2 (Maximum Marks:18) This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE. For each question, enter the correct numerical value of the answer using the mouse and the on-screen

virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value of TWO decimal places.

Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct option is entered; Zero Marks : 0 In all other cases.

Q.9. A ball is thrown from ground at an angle with horizontal and with an initial speed 0u . For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is 1V . After hitting the ground, the ball rebounds at the same angle but

with a reduced speed of 0 /u . Its motion continues for a long time as shown in figure. If the

magnitude of average velocity of the ball for entire duration of motion is 0.8 V1, the value of is ______.

KEY: (4)

Sol 20

1sin 2uRg

20 1

2 2 21 sin 2u RR

g

01

12

11 2 1

2 sin

........

1..... 1 11

utg

tt

tT t t t

Average speed = 1

1

Rt

;

Average speed = RT

4


Q.10. A 10 cm long perfectly conducting wire PQ is moving with a velocity 1 cm/s on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor 1L mH and a resistance

1R as shown in figure. The horizontal rails, L and R lie in the same plane with a uniform magnetic field 1B T perpendicular to the plane. If the key S is closed at certain instant, the current in the circuit after 1 millisecond is 310x A , where the value of x is _______.

[Assume the velocity of wire PQ remains constant (1 cm/s) after key S is closed. Given: 1 0.37e , where e is base of the natural logarithm]

KEY: (0.63(0.62-0.64)*)

Sol EMF = BVl = 2 1 31 1 10 10 10 VV

Time constant = 31 10 1 .sec

1L mR

/5

31

3

1

10 11

10 0.63 0.63

toi i e

i e

mA

Q.11. A monochromatic light is incident from air on a refracting surface of a prism of angle 750 and refractive

index 0 3n . The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of 060 . The value of 2n is ________.

KEY: (1.50)


Sol

11 sin 60 3 sin r

1 1

2

2

1sin 302

451 3 sin90 1.52

o

o

o

r r

r

n n

Q.12. A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of

angular frequency such that 24 24 10M mh

with h as Planck’s constant. N photons of

wavelength 68 10 m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 m . If the value of N is 1210x , then the value of x is ________.

[Consider the spring as massless]

KEY: (1)

Sol Km

KM

Momentum change for single photon = 2h

Momentum change = 2Nh


12

12

2

2

452 10 2

10

NhA MV

NhMA

M Nh

N

Q.13. Suppose a 226

88 Ra nucleus at rest and in ground state undergoes -decay to a 22286 Rn nucleus in its

excited state. The kinetic energy of the emitted particle is found to be 4.44 MeV. 22286 Rn nucleus

then goes to its ground state by -decay. The energy of the emitted photon is _______ ke V.

[Given : atomic mass of 22688 226.005Ra u , atomic mass of 222

86 222.000Rn u , atomic mass of particle = 4.000 u, 1 u = 931 Me V/c2, c is speed of the light]

KEY: (135(130-140)*)

Sol 226 222 488 86 2Ra Rn He

226.005 222 4 0.005Rnm mRa m m

u

Q-Value = 35 10 931 4.655MeV MeV

K.E. of 22286

4.44 4 0.08222

Rn

Energy of 4.655 4.52 135MeV KeV

Q.14. An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the

focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is _______.

KEY: (1.39(1.35-1.45)*) Sol

1 1 1v u f

2 2 2

1 1 1 2060 30

f cmf

df dv duf v u


SECTION 3 (Maximum Marks:12) This section contains TWO (02) List-Match sets. Each List-Match set has TWO (02) Multiple Choice Questions. Each List-Match set has two lists: List-I and List-II. List-I has Four entries (I), (II), (III) and (IV) and List-II has Six entries (P), (Q), (R), (S), (T) and (U). FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE

of these four options satisfies the condition asked in the Multiple Choice Question. Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the option corresponding to the correct combination is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : -1 In all other cases.

Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph. A musical instrument is made using four different metal strings, 1, 2, 3 and 4 with mass per unit length

,2 ,3 and 4 respectively. The instrument is played by vibrating the strings by varying the free length

in between the range 0L and 02L . It is found that in string-1 at free length 0L and tension 0T the

fundamental mode frequency is 0f . List-I gives the above four strings while List-II lists the magnitude of some quantity. List-I List-II (I) String-1 (P) 1

(II) String-2 2 (Q) 1/2

(III) String-3 3 (R) 1/ 2

(IV) String-4 4 (S) 1/ 3

(T) 3/16 (U) 1/16

Q.15. If the tension in each string is 0T , the correct match for the highest fundamental frequency in 0f units

will be, (A) , , ,I P II Q III T IV S (B) , , ,I P II R III S IV Q

(C) , , ,I Q II S III R IV P (D) , , ,I Q II P III R IV T

KEY: (B)

Sol 1

2o

oo

TfL


1

2

3

4

2

3

2

o

o

o

o

f fff

ff

ff

Q.16. The length of the strings 1, 2, 3 and 4 are kept fixed at 0 00

3 5, ,2 4L LL , and 07

4L

, respectively. Strings

1, 2, 3 and 4 are vibrated at their 1st, 3rd, 5th, and 14th harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of 0T will be,

(A) , , ,I P II R III T IV U (B) , , ,I P II Q III T IV U

(C) , , ,I P II Q III R IV T (D) , , ,I T II Q III R IV U

KEY: (B)

Sol 1

2o

oo

TfL

1

22

33

12

232 2 3

453 2 5

o

o

o

o

TfL

TfL

TfL

Answer Q.17 and Q.18 by appropriately matching the lists based on the information given in the paragraph. In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by T X , where T is temperature of the system and X is the infinitesimal change in a thermodynamic quantity

X of the system. For a mole of monatomic ideal gas 3 ln ln2 A A

T VX R RT V

. Here, R is gas constant,

V is volume of gas, AT and AV are constants. The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities. List-I List-II

(I) Work done by the system in process 1 2 3 (P) 0

1 ln 23

RT


(II) Change in internal energy in process 1 2 3 (Q) 0

13

RT

(III) Heat absorbed by the system in process 1 2 3 (R) 0RT

(IV) Heat absorbed by the system in process 1 2 (S) 0

43

RT

(T) 0

1 3 ln 23

RT

(U) 0

56

RT

Q.17. If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram

with 0 0 0

13

PV RT , the correct match is,

(A) , , ,I Q II R III P IV U (B) , , ,I Q II R III S IV U

(C) , , ,I S II R III Q IV T (D) , , ,I Q II S III R IV U

KEY: (B)

Sol Work done in 12 = ln 23

oRT

2 3 0

U in 1 2 = 0; O in 1 2 = ln 23

oRTW

Q.18. If the process on one mole of monatomic ideal gas is as shown in the TV-diagram with 0 0 0

13

PV RT ,

the correct match is,

(A) , , ,I P II R III T IV S (B) , , ,I P II T III Q IV T

(C) , , ,I P II R III T IV P (D) , , ,I S II T III Q IV U


KEY: (C)

Sol Work done in 1 2 = PoVo

2 3 = 0 Total work in 1 2 3 = P0V0

Q in 1 2 3 = 43 oRT

U in 1 2 3 = RTo


Page 1

JEE (ADVANCED) 2019 PAPER 2 PART-I CHEMISTRY

SECTION 1 (Maximum Marks:32) This section contains EIGHT (08) questions.

Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct

answer(s).

For each question, choose the option(s) corresponding to (all) the correct answer(s).

Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If only (all) the correct option(s) is(are) chosen;

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of

which are correct;

Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct

option;

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks : -1 In all other cases.

For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct

answers, then

choosing ONLY (A), (B) and (D) will get +4 marks;

choosing ONLY (A) and (B) will get +2 marks;

choosing ONLY (A) and (D) will get +2 marks;

choosing ONLY (B) and (D) will get +2 marks;

choosing ONLY (A) will get +1 mark;

choosing ONLY (B) will get +1 mark;

choosing ONLY (D) will get +1 mark;

choosing no option (i.e. the question is unanswered) will get 0 marks; and

choosing any other combination of options will get -1 mark.


Page 2

Q.1. The cyanide process of gold extraction involves leaching out gold from its ore with CN in the presence of Q in water to form R. Subsequently, R is treated with T to obtain Au and Z. Choose the correct option(s)

(A) Q is 2O

(B) T is Zn

(C) Z is 2

4Zn CN

(D) R is 4Au CN

KEY: (A, B, C) Sol Gold extraction:

2 2 2

22 4

4 8 2 4 4

4 2 2 4

R

TR Z

QAu NaCN H O O Na Au CN NaOH

Na Au CN Zn Na Zn CN Au

Q.2. With reference to aqua regia, choose the correct option(s) (A) Aqua regia is prepared by mixing conc. HCl and conc. HNO3 in 3:1 (v/v) ratio (B) Reaction of gold with aqua regia produces an anion having Au in +3 oxidation state (C) Reaction of gold with aqua regia produces NO2 in the absence of air (D) The yellow colour of aqua regia is due to the presence of NOCl and Cl2 KEY: (A, B, D) Sol Aquaregia is a mixture of conc.HCl and conc.HNO3 in 3:1 ratio.

When gold dissolves in aquaregia, the species formed is 4AuCl in which gold is in +3 oxidation state.

In the absence of air the reaction between gold and aquaregia 3 4 24 2Au HNO HCl H AuCl H O NO NO2 will not be produced. Yellow colour is due to the presence of NOCl and Cl2. Q.3. Consider the following reactions (unbalanced) Zn+ hot conc. 2 4H SO G R X

Zn+ conc. NaOH T Q

2 4G H S NH OH Z (a precipitate) X Y

Choose the correct option(s) (A) Z is dirty white in colour (B) The oxidation state of Zn in T is +1 (C) R is a V-shaped molecule (D) Bond order of Q is 1 in its ground state KEY: (A, C, D)


Page 3

Sol 2 4Hot & Conc.

Zn H SO G R X

2 4 4 2 2

2 2 2

4 2 4 4 4 22

2

Conc.G R X

T Q

ZG XY

Zn H SO ZnSO SO H O

Zn NaOH Na ZnO H

ZnSO H S NH OH ZnS NH SO H O

(1) Oxidation state of Zn in Na2ZnO2 is +2 (2) Bond order of Q is one for H2(Q) (3) ZnS is white in colour (4) SO2 is angular in shape Q.4. The ground state energy of hydrogen atom is -13.6 eV. Consider an electronic state of He whose

energy, azimuthal quantum number and magnetic quantum number are -3.4 eV, 2 and 0, respectively. Which of the following statement(s) is (are) true for the state ?

(A) It is a 4d state (B) It has 2 angular nodes (C) It has 3 radial nodes (D) The nuclear charge experienced by the electron in this state is less than 2e, where e is the

magnitude of the electronic charge KEY: (A, B)

Sol 2

2213.6 3.4En

2

2

2 2

213.6 3.4

4 4n

n n

Wave function corresponds to 4,2,0 represents 24z

d – orbital which has only one radial nodes and two angular nodes. It experiences nuclear change of 2e units.

Q.5. Which of the following reactions produce(s) propane as a major product?

(A) (B)

(C)

(D)

KEY: (A, C)


Page 4

Sol

CH3

O

ONa NaOHCaO,Δ

CH3 CH3

CH3

O

ONa H2Oelectrolysis CH3

CH3

Cl

CH3Zn

dil.HCl CH3CH3

Q.6. Choose the correct option(s) that give(s) an aromatic compound as the major product

(A)

(B)

(C)

(D)

KEY: (A, B)

Sol 2

2213.6 3.4En

2

2

2 2

213.6 3.4

4 4n

n n

Wave function corresponds to 4,2,0 represents 24z

d – orbital which has only one radial nodes and

two angular nodes. It experiences nuclear change of 2e units. Q.7. Choose the correct option(s) for the following reaction sequence

Consider Q, R and S as major products


Page 5

(A)

(B)

(C)

(D)

KEY: (C, D) Sol

MeO C C CH2

CHO i)Hg2+,dilH2SO4 MeO C CH2 CH2

CHOOii)AgNO3

NH4OH

MeO C CH2 CH2

COOHO

MeO CH2 CH2 CH2

COOH

MeO CH2 CH2 CH2

COCl MeO

OMeO

iii)Zn-Hg.conc.HCl

i)SOCl2 pyridine

ii)AlCl3 Zn-Hgconc.HCl

'Q'

'R' 'S'

Q.8. Choose the correct option(s) from the following (A) Natural rubber is polyisoprene containing trans alkene units (B) Nylon-6 has amide linkages (C) Teflon is prepared by heating tetrafluoroethene in presence of a persulphate catalyst at high

pressure (D) Cellulose has only -D-glucose units that are joined by glycosidic linkages KEY: (B, C)


Page 6

Sol Nylon-6 has amide linkages.

Cellulose has only β-D-glucose units that are joined by glycosidic linkages. Teflon is prepared by heating tetrafluoromethane in presence of a persulphate

catalyst at high pressure. Natural rubber is polyisoprene containing cis alkene units.

SECTION 2 (Maximum Marks:18) This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE. For each question, enter the correct numerical value of the answer using the mouse and the on-screen



Q.9. The amount of water produced (in g) in the oxidation of 1 mole of rhombic Sulphur by conc. HNO3 to a compound with the highest oxidation state of Sulphur is ________.

(Given data : Molar mass of water = 18 g mol-1) KEY: (288(288.00-288.30)*) Sol S8 + HNO3 H2SO4 + NO2 + H2O Rhombic sulphur Balancing the equation S8 + 48HNO3 8H2SO4 + 48NO2 + 16H2O Rhombic sulphur 1 mole S8 produces 16 x 18 gm of H2O i.e. 288 gms of H2O. Q.10. Total number of cis N Mn Cl bond angles (that is, Mn N and Mn Cl bonds in cis

positions) present in a molecule of cis- 22Mn en Cl complex is ______

2 2 2 2en NH CH CH NH

KEY: (6) Sol

Mn

2H N Cl

Cl2NH

2CH 2CH

2NH

2 2CH CH

2NH

3 1

24

Similarly from bottom N-there will be another (2). Total N – Mn – Cl angle are 6.


Page 7

Q.11. The decomposition reaction 2 5 2 4 22 2N O g N O g O g

is started in a closed cylinder under

isothermal isochoric condition at an initial pressure of 1 atm. After 310Y s , the pressure inside the

cylinder is found to be 1.45 atm. If the rate constant of the reaction is 4 15 10 s , assuming ideal gas behavior, the value of Y is ______

KEY: (2.3) Sol 2 2 4 252 2gN O N O g O g

Given rate constant of the reaction = 4 15 10 sec

2 5

2 5

2 5

2 5

overall

overall

12

2

N ON O

N ON O

dPK P

dtdp

K Pdt

2 5 2 4 21 /2

2 2xx x

N O g N O g O g

3

4

1 1.4520.90 atm

2.303 110 log0.12 5 10

23.03 2.3010

x

x

y

y

Q.12. The mole fraction of urea in an aqueous urea solution containing 900 g of water is 0.05. If the density

of the solution is 1.2 g cm-3, the molarity of urea solution is ______ (Given data : Molar masses of urea and water are 60 g mol-1 and 18 g mol-1, respectively) KEY: (2.98(2.80-3.05)*) Sol Xurea = 0.05 Mass of water = 900 gm

Number of moles of water = 2

900 5018H On

Let moles of urea = Un

0.05505019

U

U

U

nn

n

Mass of urea = 50 6019

Given density (d) of solution = 1.2 g/ml.


Page 8

50 60900massof solution 19density / 1.2solutionV

g cc

50191000 2.98

50 60900 1000191.2

U

ml

nMV

Q.13. Total number of hydroxyl groups present in a molecule of the major product P is ____

KEY: (6) Sol

i)H2,Pd-BaSO4,quinoline

ii)dil.KMnO4(excess),273K

OH

OH

OH

OH

OHOH

Q.14. Total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the

molecular formula 4 8C H O is ______ KEY: (10) Sol

OO

CH3

O

CH3

O

CH3CH3

O

CH3 CH3

O

CH3

* * * *


Page 9

SECTION 3 (Maximum Marks:12) This section contains TWO (02) List-Match sets. Each List-Match set has TWO (02) Multiple Choice Questions. Each List-Match set has two lists: List-I and List-II. List-I has Four entries (I), (II), (III) and (IV) and List-II has Six entries (P), (Q), (R), (S), (T) and (U). FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE

of these four options satisfies the condition asked in the Multiple Choice Question. Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the option corresponding to the correct combination is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : -1 In all other cases.

Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph. Consider the Bohr’s model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the thn orbit of the atom and List-II contains options showing how they depend on n. List-I List-II (I) Radius of the thn orbit (P) 2n

(II) Angular momentum of the electron in the thn orbit (Q) 1n

(III) Kinetic energy of the electron in the thn orbit (R) 0n (IV) Potential energy of the electron in the thn orbit (S) 1n

(T) 2n

(U) 1/2n

Q.15. Which of the following options has the correct combination considering List-I and List-II? (A) (I), (P) (B) (I), (T) (C) (II), (Q) (D) (II), (R) KEY: (B)

Sol 2nr

Z

2

2

2

2

L n

ZKEn

ZPEn


Page 10

Q.16. Which of the following options has the correct combination considering List-I and List-II ? (A) (III), (P) (B) (III), (S) (C) (IV), (Q) (D) (IV), (U) KEY: (A)

Sol 2nr

Z

2

2

2

2

L n

ZKEn

ZPEn

Answer Q.17 and Q.18 by appropriately matching the lists based on the information given in the paragraph. List-I includes starting materials and reagents of selected chemical reactions. List-II gives structures of compounds that may be formed as intermediate products and/or final products from the reactions of List-I.


Page 11

Q.17. Which of the following options has correct combination considering List-I and List-II ? (A) (I), (Q), (T), (U) (B) (II), (P), (S), (T) (C) (II), (P), (S), (U) (D) (I), (S), (Q), (R) KEY: (C) Sol

i)O3

ii)Zn,H2O

iv)conc.H2SO4

iii)NaBH4CH2

COOH

O

COOH

OH

COOH

O

O

CN

O

O

i)DIBAL-Hii)dilHCl

CHO

CHO

NaBH4

iii)conc.H2SO4

OH

OH

O

COOMe

COOMe

i)LiAlH4 iI)conc.H2SO4OH

OH

O=

O


Page 12

Cl

COOCH3

i)KCN ii)H3O+,Δ iii)LiAlH4CN

COOCH3

COOH

COOH OH

OH

O

iv)conc.H2SO4

Q.18. Which of the following options has correct combination considering List-I and List-II? (A) (III), (S), (R) (B) (IV), (Q), (U) (C) (III), (T), (U) (D) (IV), (Q), (R) KEY: (D) Sol Sol

i)O3

ii)Zn,H2O

iv)conc.H2SO4

iii)NaBH4CH2

COOH

O

COOH

OH

COOH

O

O

CN

O

O

i)DIBAL-Hii)dilHCl

CHO

CHO

NaBH4

iii)conc.H2SO4

OH

OH

O

COOMe

COOMe

i)LiAlH4 iI)conc.H2SO4OH

OH

O=

O


Page 1

JEE (ADVANCED) 2019 PAPER 2 PART-I MATHS


This section contains EIGHT (08) questions. Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct

answer(s). For each question, choose the option(s) corresponding to (all) the correct answer(s). Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If only (all) the correct option(s) is(are) chosen; Partial Marks : +3 If all the four options are correct but ONLY three options are chosen; Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of

which are correct; Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct

option; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : -1 In all other cases.

For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then choosing ONLY (A), (B) and (D) will get +4 marks; choosing ONLY (A) and (B) will get +2 marks; choosing ONLY (A) and (D) will get +2 marks; choosing ONLY (B) and (D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY (B) will get +1 mark; choosing ONLY (D) will get +1 mark; choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get -1 mark.


Page 2

1. For , 1a a , let

3 3

7/32 2 2

1 2 ......lim 541 1 1....

1 2

n

n

nan an an n

Then the possible value(s) of a is/are 1. – 9 2. – 6 3. 7 4. 8 2. Let :f be a function. We say that f has

PROPERTY 1 if

0

0limh

f h fh

exists and is finite, and

PROPERTY 2 if

20

0limh

f h fh

exists and is finite

Then which of the following options is/are correct ? 1. f x x has PROPERTY 1 2. sinf x x has PROPERTY 2

3. f x x x has PROPERTY 2 4. 2/3f x x has PROPERTY 1 3. For non – negative integers n, let

0

2

0

1 2sin sin2 2

1sin2

n

kn

k

k kn nf n

kn

Assuming cos-1x takes values in 0, which of the following options is/are correct?

1. If 1tan cos 6f ,then 2 2 1 0

2. 1sin 7cos 5 0f

3. 342

f

4. 1lim2n

f n

4. Let :f be given by 1 2 5f x x x x . Define 0

, 0x

F x f t dt x

Then when of the following options is/are correct? 1. F has a local minimum at x = 1 2. 0F x for all 0,5x

3. F has a local maximum at x = 2 4. F has two local maxima and one local minimum in 0,


Page 3

5. Let

1 2 3

1 0 0 1 0 0 0 1 00 1 0 , 0 0 1 , 1 0 00 0 1 0 1 0 0 0 1

P I P P

4 5 6

0 1 0 0 0 1 0 0 10 0 1 , 1 0 0 , 0 1 01 0 0 0 1 0 1 0 0

P P P

and 6

1

2 1 31 0 23 2 1

Tk k

k

X P P

Where TkP denotes the transpose of the matrix kP . Then which of the following options is/ are

correct? 1. X is a symmetric matrix 2. The sum of diagonal entries of X is 18

3. X – 30I is an invertible matrix 4. If

1 11 1 ,1 1

X

then 30

6. Let 2sin , 0xf x x

x

Let x1 < x2 < x3 < …….. < xn <……. be all the points of local maximum of f and y1 < y2 < y3 < ……. < yn < ….. be all the points of local minimum of f . Then which of the following options is/are correct ? 1. 1 1x y 2. 1 2n nx x

3. 12 ,22nx n n

for every n 4. 1n nx y for every n

7. Let x and let

1 1 1 20 2 2 , 0 4 00 0 3 6

x xP Q

x x

and 1R PQP

Then which of the following options is/are correct?

1. det

2det 0 4 0 8

5

x xR

x x

, for all x

2. For x = 1, there exists a unit vector i j k for which

000

R

3. There exists a real number x such that PQ = QP

4. For x = 0, if

1 16R a a

b b

, then a + b = 5


Page 4

8. Three lines

1 : ,L r i

2 : ,L r k j

3 : ,L r i j vk v

are given . For which point(s) Q on L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear?

1. k 2. 12

k j 3. k j 4. 12

k j


This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE. For each question, enter the correct numerical value of the answer using the mouse and the on-screen



1. Suppose

2

0 0

0 0

det 03

n nn

kk k

n nn n k

k kk k

k C k

C k C

Holds for some positive integer n. The 0 1

nnk

k

Ck equals____

2. The value of

10

1

0

11 7 7sec sec sec4 12 2 12 2k

kk

In the interval 3,

4 4

equals ___

3. The value of the integral

/2

50

3 cos

cos sind

equals ____

4. Let X denote the number of elements in a set X. Let S= {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that 1 B A , equals ____


Page 5

5. Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ___

6. Let 2a i j k and 2b i j k

be two vectors. Consider a vector

, ,c a b

. If the projection of c

on the vector a b

is 3 2 , then the minimum

value of .c a b c

equals_____


This section contains TWO (02) List-Match sets. Each List-Match set has TWO (02) Multiple Choice Questions. Each List-Match set has two lists: List-I and List-II. List-I has Four entries (I), (II), (III) and (IV) and List-II has Six entries (P), (Q), (R), (S), (T) and (U). FOUR options are given in each Multiple 1. Answer the following by appropriately matching the

lists based on the information given in the paragraph Let sin cosf x x and cos 2 sing x x be two functions defined for x > 0. Define

the following sets whose elements are written in the increasing order : : 0 ,X x f x 1: 0Y x f x

: 0 ,Z x g x 1: 0W x g x

List – I contains the sets X, Y, Z and W. List – II contains some information regarding these sets. List – I List – II

(I) X (P) 3, ,4 ,7

2 2

(II) Y (Q) an arithmetic progression (III) Z (R) NOT an arithmetic progression

(IV) W (S) 7 13, ,

6 6 6

(T) 2, ,

3 3

(U) 3,

6 4

Which of the following is the only CORRECT combination? 1. (IV), (P), (R), (S) 2. (III), (P), (Q), (U) 3. (III), (R), (U) 4. (IV), (Q), (T) 2. Answer the following by appropriately matching the lists based on the information given in the

paragraph Let sin cosf x x and cos 2 sing x x be two functions defined for x > 0. Define

the following sets whose elements are written in the increasing order : : 0 ,X x f x 1: 0Y x f x


Page 6

: 0 ,Z x g x 1: 0W x g x

List – I contains the sets X, Y, Z and W. List – II contains some information regarding these sets. List – I List – II

(I) X (P) 3, ,4 ,7

2 2

(II) Y (Q) an arithmetic progression (III) Z (R) NOT an arithmetic progression

(IV) W (S) 7 13, ,

6 6 6

(T) 2, ,

3 3

(U) 3,

6 4

Which of the following is the only CORRECT combination? 1. (I), (P), (R) 2. (I), (Q), (U) 3. (II), (R), (S) 4. (II), (Q), (T) 3. Answer the following by appropriately matching the lists based on the information given in the

paragraph.

Let the circles 2 21 : 9C x y and 2 2

2 : 3 4 16C x y , intersect at the points X and

Y. Suppose that another circle 2 2 23 :C x h y k r satisfies the following conditions :

(i) Centre of 3C is collinear with the centres of 1C and 2C

(ii) 1C and 2C both lie inside 3C , and

(iii) 3C touches 1C at M and 2C at N Let the line through X and Y intersect C3 at Z and W, and let a common tangent of C1 and C3 be a

tangent to the parabola 2 8x y . There are some expressions given in the List – I whose values are given in List – II below List – I List – II (I) 2h k (P) 6

(II) Length of ZWLength of XY

(Q) 6

(III) Area of triangle MZNArea of triangle ZMW

(R) 54

(IV) (S) 215

(T) 2 6

(U) 103

Which of the following is the only CORRECT combination? 1. (I), (U) 2. (II), (T) 3. (II), (Q) 4. (I), (S)


Page 7

4. Answer the following by appropriately matching the lists based on the information given in the paragraph

Let the circles 2 21 : 9C x y and 2 2

2 : 3 4 16C x y , intersect at the points X and

Y. Suppose that another circle 2 2 23 :C x h y k r satisfies the following conditions :

(i) Centre of 3C is collinear with the centres of 1C and 2C

(ii) 1C and 2C both lie inside 3C , and

(iii) 3C touches 1C at M and 2C at N Let the line through X and Y intersect C3 at Z and W, and let a common tangent of C1 and C3 be a

tangent to the parabola 2 8x y . There are some expressions given in the List – I whose values are given in List – II below List – I List – II (I) 2h k (P) 6

(II) Length of ZWLength of XY

(Q) 6

(III) Area of triangle MZNArea of triangle ZMW

(R) 54

(IV) (S) 215

(T) 2 6

(U) 103

Which of the following is the only INCORRECT combination? 1. (III), (R) 2. (IV), (U) 3. (IV), (S) 4. (I), (P)

Mathematics Key and Solutions

Section-I: 1) 1,4 2) 1,4 3) 1,2,3 4) 1,2,3 5) 1,2,4 6) 2,3,4 7) 1,4 8) 2,4

Section-II: 1) 6.20 2) 0.00 3) 0.50 4) 1523.00 5) 30.00 6) 18.00

Section-III: 1) 1 2) 4 3) 3 4) 3


Page 8

SECTION – I

1.

1/33

11

7/32 2

1 1

1

lim lim1 1 1

nn

rrn nn n

r r

rr n n

nn ran r a

n

=

11/3 4/3 1

00

1102

0

3 34 4 541 11

1

x dx x l

ldx a x a aa x

1 8 9 9 8a a or 8 9a or

2. If 0 0

0lim lim 0h h

hf x x h

h

If 20

sinh 0sin lim .h

f x x D Eh

If 20

0lim .h

h hf x x x D E

h

If 2/3

1/32/31/30 0

0lim lim 0h h

hf x x hh

3. 0

0

2 3cos cos2 2

2 21 cos2

n

k

n

k

kn nf n

kn

1sin

2 31 cos cos2 2 2sin

21

sin2 21 cos

2 2sin2

nn nn

n n nn

nn nn

n nn

2 cos

2 cos2 2

nn

n n

.

26 cos 2 1 2 1 08 8

f Tan

1sin 7 cos 5 sin 7. 07

f

34 cos

6 2f

; lim 1n

f n


Page 9

4. 1 1 2 5F x f x x x x At 1x and 5x F x has local minima At 2x , F x has local maxima and 0 0,5F x x

1 2 5

5. Let 6

1

2 1 31 0 2 ,3 2 1

Tk k

k

A X P AP

Here T Tk k k kP P P P I for all k =1,2,3,4,5,6.

1 &T Tk k kP P P A A

6 6

1 1

TT T Tk k k k

k k

X P AP P AP X

Let 6 6 6

1 1 1

1 2 2 2 30 11 2 2 2 30 30 1 301 2 2 2 30 1

TK K K K

K K KB XB P AP B P AB P AB AB B

30 and 30X I is singular

6

1

18Tk k

k

Tr P AP

6. 2

14

cos 2 sin , 0x x x xf x xx

4

2 cos tan2 0

xx x xcos x

x

y

0 1/23/2 3

1y 1x2y

7. 1 2 2

24 12 4 10 8 0 4 0 8

5

x xR PQP Q x x

x x

If 1 44 0 0x R


Page 10

For 1 1 1 2 0 0 6 3 0

10 0 2 2 0 4 0 0 3 2

60 0 3 0 0 6 0 0 2

x R

12 6 4

1 0 24 86

0 0 36

24 1

346 0 23

0 0 0

R I

2 44 0, 2 03 3b ba a

3, 2 5b a a b If PQ = QP no value of x . 8. , 0,0 , 0, ,1 , 1,1,P Q R

If P, Q, R are collinear 11 1

1 11 1,01

Section – II

1. 2 1

1

11 2 2 02

2 4

n n

n n

n nn n n

n

2 1 2 3 21 2 1 2 2 0n n nn n n n 4 1 1 2 0n n n n 2 3 4 0 4 0,n n n n n N

4 54

0

2 1 31 6.201 5 5k

k

Ck

2. 10

0

7 7sec sec12 2 12 2 2k

k kS

10

0

72cos6k

ec k

10

1

0

2 1 cos6

k

k

ec

4 1 4

1 15sec sec 1 04

3.

/2 /2

5 50 0

3 cos 3 sin

cos sin sin cosI d d


Page 11

/2 /2 2

4 40 0

3 3sec2sin cos 1

I d dTan

Put, 2 2sec 2Tan t d tdt

4 3 4

0 0

6 1 16

1 1 1t dt dtt t t

3 2

0

1 1 1 1 13 3 0 1 0 1 0.503 2 3 2 2t t

I l

4. If n(B) = 1, n(A) = 6 no. of ordered pairs (A, B) = 6 If n(B) = 2, n(A) = 3 no. of ordered pairs (A, B) = 180 If n(B) = 2, n(A) = 6 no. of ordered pairs (A, B) = 15 If n(B) = 3, n(A) = 4 no. of ordered pairs (A, B) = 180 If n(B) = 3, n(A) = 6 no. of ordered pairs (A, B) = 20 If n(B) = 4, n(A) = 6 no. of ordered pairs (A, B) = 15 If n(B) = 5, n(A) = 6 no. of ordered pairs (A, B) = 6 Total number of ordered pairs (A, B) = 422. 5. Total number of ways = 2 3 5 30 6. 3 3a b i j

.3 2 2

c a b

a b

2

18 . 18c c c a b

SECTION III 1, 2 For x , 0 sin cos 0 cosf x x x n cos 0 ( )1( ) 1x or or

,2

nx n N

For Y, 1 0 cos sin 0f x cos x x sin 0 ( ) cos cos 0x or x

, ( ) cos 2 1 ,2

x n n N or x n n N

1 1cos ,2 2

x

22 ,2 , , 03 3

x n n n I x

For Z, 0 cos 2 sin 0 2 sin 2 1 ,2

g x x n n N

2 1 1 1 3 3sin , , ,4 4 4 4 4

nx

For W, 1 0 2 sin 2 sin cos 0g x x


Page 12

sin 2 sin 0 cos 0x or x

2 1 , 2 sin2

x n n N or x n

1sin 1, ,02 2nx

2 1 , ,2

x n n n N

, 1 , 16 6

n nn n n

3,4

WN

M ZO(0,0)

Y

X

(3,4)

2 21

2 2 2 22 2

2 2 23

9 0,0

3 4 16 6 8 9 0 3,4

C x y O

C x y x y x y C

C x h y k r

Let the centre of C3 be C.

29 12: 3: 2 ,5 5

OC CC C

2 6, 2 3 5 4 12 6h k r r Eq. of common chord of C1 and C2 is 3x+4y-9=0. Let P be the point of intersection of OC1 and common chord.

6 12 6 24, ,5 5 5

81 242 925 5

6

54

CP ZP WP MP

XY

ZWXYMZNZMW

Equation of the common chord of C1 and C3 is 103 4 15 03

x y .

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