2018-jee-advanced · sol: 2 gh r 2 cos r h r r g 2 cos h r g (a)for given material , constant 1 h r...

2018-Jee-Advanced Question Paper-1_Key & Solutions

Sri Chaitanya IIT Academy # 304, Kasetty Heights, Ayyappa Society, Madhapur, Hyderabad – 500081

www. srichaitanya.net, [email protected]

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JEE (ADVANCED) 2018 PAPER 1

PART-I PHYSICS SECTION 1 (Maximum Marks: 24)

This section contains SIX (06) questions. Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of

these four option(s) is (are) correct option(s). For each question, choose the correct option(s) to answer the question. 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, both of

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

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : −2 In all other cases. For Example: If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks.Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option),without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks. Q.1 The potential energy of a particle of mass m at a distance r from a fixed point O is given by

2

2rV r k .where is a positive constant of appropriate dimensions. This particle is

moving in a circular orbit of radius R about the point O. If is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

A) 2kv Rm

B) kv Rm

C) 2L mk R

D) 2

2mkL R

Key. (B,C) Sol:




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O

r

v

2

2krV , F kr At r=R,

2mvkRR

(Centripetal force) 2kR kv R

m m , 2kL m R

m

Q.2 Consider a body of mass 1.0 at rest at the origin at time 0t . A force F t i j

is applied on the body, where 1.0 /N s and 1.0N . The torque acting on the body

about the origin at time 1.0t s is . Which of the following statements is (are) true?

A) 13

Nm

B) The torque is in the direction of the unit vector k

C) The velocity of the body at 1t s is 1 2 /2

v i j m s

D) The magnitude of displacement of the body at 1t s is 16

m

Key. (A, C) Sol:

F t i j F ti j ,

dvm ti jdt

On integrating

2

2tmv i t j

2

2dr t i tjdt

On integrating 3 2

6 2t tr i j

At t=1s, 1 1 16 2 3

r F i j i j k

At t=1, 1 1 2 / sec2 2

v i j i j m

At t=1, 1 0s r r , 1 16 2

s i j

2 21 16 2

s

106

m




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Q.3 A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water.

The water rises to a height ℎ in the capillary tube above the water surface in the beaker. The

surface tension of water is . The angle of contact between water and the wall of the capillary

tube is . Ignore the mass of water in the meniscus. Which of the following statements is(are)

true?

A) For a given material of the capillary tube, ℎ decreases with increase in r B) For a given material of the capillary tube, ℎ is independent of C) If this experiment is performed in a lift going up with a constant acceleration, then ℎ decreases D) ℎ is proportional to contact angle Key. (A, C)

Sol:

2 ghR

2

cosrh R

R g

2 coshr g

(A)For given material , constant

1hr

(B) h depend on (C)If lift is going up with constant acceleration, ( )effg g a

2 cosh

r g a

,It means h decreases.

(D) h is proportional to cos not Q.4 In the figure below, the switches 1S and 2S are closed simultaneously at t =0 and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude MaxI at time

t . Which of the following statements is (are) true?

A) 2MaxVIR

B) 4MaxVIR

C) ln 2LR

D) 2 ln 2LR




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Key. (B, D) Sol:

1i 2ii

R L 2LR

max 2 1 maxi i i

2 2

2 1 1 1R R R Rt t t tL L L LV V Vi i i e e e e

R R R

For max

0d i

idt

2 02

R Rt tL lV R Re e

R L L

21

2

R Rt tL le e

, 2 1

2

R tLe

ln 22R tL

, 2 ln 2LtR

time, when i is maximum

2 2ln 2 ln 22

max

R L R LL R L RVi e e

R

max

1 1 14 2 4

V ViR R

Q.5 Two infinitely long straight wires lie in the xy-plane along the lines x R . The wire located at

x R carries a constant current 1I and the wire located at carries a constant current

2I . A circular loop of radius R is suspended with its centre at 0,0, 3R and in a plane

parallel to the xy-plane. This loop carries a constant current I in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the j direction.

Which of the following statements regarding the magnetic field B is (are) true? A) If 1 2I I , then B cannot be equal to zero at the origin (0, 0, 0)

B) If 1 0I and 2 0I , then B can be equal to zero at the origin (0, 0, 0)

C) If 1 0I and 2 0I , then B can be equal to zero at the origin (0, 0, 0)

D) If 1 2I I , then the z-component of the magnetic field at the centre of the loop is 02

IR

Key. A, B, D

Sol:

Z

C 0,0, 3R1

x R O x R

Y

Y

x R




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A)At origin , B =0 due to two wires if 1 2I I ,hence netB at origin is equal to B due to ring , which is non –zero.

B)If 1 0I and 2 0I , B at origin due to wires will be along + k direction and B due to ring is along - k direction and hence B can be zero at origin.

C)If 1 20 0I and I , at origin due to wires along - k and also along - k due to ring , hence B cannot be zero .

D)

X

Z 1B

2BxB

2I1I

At centre of ring, B due to wires is along x-axis. Q.6 One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure

(where V is the volume and T is the temperature). Which of the statements below is (are)

true?

A) Process I is an isochoric process

B) In process II, gas absorbs heat

C) In process IV, gas releases heat

D) Processes I and III are not isobaric

Key. (B, C, D) Sol:

A)Process-I is not isochoric,V is decreasing .

B)Process-II is isothermal expansion 0, 0U W 0Q

C)Process-IV is isothermal compression ,

0, 0U W 0Q

D)Process-I and III are NOT isobaric because in isobaric process T V hence isobaric T-V

graph will be linear.




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SECTION 2 (Maximum Marks: 24)

This section contains EIGHT (08) questions. The answer to each question is a NUMERICAL VALUE. For each question, enter the correct numerical value (in decimal notation, truncated/rounded-off to

the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer.

Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct numerical value is entered as answer.

Zero Marks : 0 In all other cases.

Q.7 Two vectors A and B are defined as A ai and cos sinB a t i t j , where is a

constant and /6

rad s . If 3A B A B at time t for the first time, the value of

, in seconds, is Key. (2.00) Sol:

B

Aata

2 cos

2tA B a

2 sin2tA B a

So 2 s 3 2 sin2 2t ta co a

1tan2 3t

2 6 3t t

2.00sec

6 3t t

Q.8 Two men are walking along a horizontal straight line in the same direction. The man in front

walks at a speed 1.0 /m s and the man behind walks at a speed 2.0 /m s . A third man is standing at a height 12m above the same horizontal line such that all three men are in a

vertical plane. The two walking men are blowing identical whistles which emit a sound of

frequency 1430Hz . The speed of sound in air is 330 /m s . At the instant, when the moving men are 10m apart, the stationary man is equidistant from them. The frequency of beats in

Hz, heard by the stationary man at this instant, is .

Key. (5.00) Sol:

Observer(stationary)

13m12m

A5m2m/s 1m/s5m

13m

B




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330 1 21430 1430 1430 12cos330 2cos 3301330

Acosf

3301430 1430 1330 cos 330B

cosf

3 51430 13cos 13 5.00330 13A Bcosf f f Hz

Q.9 A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is 2 3 / 10s , then the height of the top of the inclined plane, in meters, is .

Take 210 /g m s . Key. (0.75m)

Sol:

h

060

2

sin

1c

C

ga IMR

sin2ring

ga , 2 sin

3discga

21 1 2

1 sin 4 16sin 2 2 sin 3

h g h ht tg g

22 2 2

1 2 sin 3 4sin 2 3 sin

h g h ht tg g

16 4 2 33 10

h hg g

4 2 2 33

h

2 3 3 3 0.75

24 2 3h m

Q.10 A spring-block system is resting on a frictionless floor as shown in the figure. The spring

constant is 2.0 /N m and the mass of the block is 2.0 g. Ignore the mass of the spring. Initially

the spring is in an unstretched condition. Another block of mass 1.0 moving with

a speed of 2.0 /m s collides elastically with the first block. The collision is such that the

2.0 block does not hit the wall. The distance, in meters, between the two blocks when

the spring returns to its unstretched position for the first time after the collision is .




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Key. (2.09 m) Sol:

2 2 secmTk

Block returns to original position in sec2T

2kg1kg0

2m/s

Just after collision

2/3m/s 4/3/m/s

1kg 2kg Just after collision

2 2 3.14 2.09333 3

d m

2.09d m

Q.11 Three identical capacitors 1 2 3,C C and C have a capacitance of 1.0 F each and they are

uncharged initially. They are connected in a circuit as shown in the figure and 1C is then

filled completely with a dielectric material of relative permittivity r . The cell electromotive

force (emf) 0 8V V . First the switch 1S is closed while the switch 2S is kept open. When the

capacitor 3C is fully charged, 1S is opened and 2S is closed simultaneously. When all the

capacitors reach equilibrium, the charge on 3C is found to be 5 C .The value of r

Key. 1.50 Sol:

0 8V V8 C

8 C1 F

1 Fr

3 C3 C

5 C

5 C

1 F

3 C3 C

Applying loop rule5 3 3 01 1r

3 2 1.50rre




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Q.12 In the xy-plane, the region y > 0 has a uniform magnetic field 1B k and the region y < 0 has

another uniform magnetic field 2B k A positively charged particle is projected from the origin

along the positive y-axis with speed 0 /v m s at 0t , as shown in the figure. Neglect

gravity in this problem. Let t T be the time when the particle crosses the x-axis from below

for the first time. If 2 14B B , the average speed of the particle, in /m s , along the x-axis in the

time interval T is .

Key. 2.00 Sol: 1)Average speed along x-axis

1r

2r

1 2

1 2

xx

V dt d dVt tdt

2)We have 1 21 2

,mv mvr rqB qB

Since 21 4

BB 1 24r r Time in 1 11

mB tqB

Time in 2 22

mB tqB

Total distance along x-axis 1 2 1 2 1 2 22 2 2 2 5d d r r r r r

Total time 1 2 25T t t t Average speed 2 2

2 2

10 2 25

r mv qBt qB m

Q.13 Sunlight of intensity 21.3kWm is incident normally on a thin convex lens of focal length

20cm . Ignore the energy loss of light due to the lens and assume that the lens aperture size

is much smaller than its focal length. The average intensity of light, in 2kWm , at a distance

22cm from the lens on the other side is

Key. 130




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Sol: 20cm 2cm

R r

2 120 10

rR Ratio of area

1100

Let energy incident on lens be E. Given 1.3EA

So final , ??Ea 1.30E A

1100

aAlsoA

Average intensity of light at 22cm 21.3 100 1.3 130 /E A kW ma a

Q.14 Two conducting cylinders of equal length but different radii are connected in series between

two heat baths kept at temperatures 1 300T K and 2 100T K , as shown in the figure.

The radius of the bigger cylinder is twice that of the smaller one and the thermal

conductivities of the materials of the smaller and the larger cylinders are 1K and 2K

respectively. If the temperature at the junction of the two cylinders in the steady state is

200K , then 1

2

KK

Key. 4.00

Sol:

k2100k

200k

2r

L

k1r300k

L

We have in steady state,

2 21 2

200 300 200 100 0

2L L

k r k r

2 21 2 1

2

100 100 4 4k r k r kL L k




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This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) questions. Each question has FOUR options. ONLY ONE of these four options corresponds to the correct answer. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct option is chosen.

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

Negative Marks : −1 In all other cases. PARAGRAPH “X”

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each

other. In the questions below, E and B stand for dimensions of electric and magnetic fields respectively, while 0 0and stand for dimensions of the permittivity and permeability of free space respectively. L and T are dimensions of length and time

respectively. All the quantities are given in SI units. (There are two questions based on PARAGRAPH “X”, the question given below is one of them)

Q.15 The relation between E and B is A) E B L T

B) 1E B L T

C) 1E B L T

D) 1 1E B L T Key. C

Sol: We have 1 1 1E B E L T E B L TC

PARAGRAPH “X” In electromagnetic theory, the electric and magnetic phenomena are related to each other.

Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, E and B stand for dimensions of electric and magnetic

fields respectively, while 0 0and stand for dimensions of the permittivity and permeability of free space respectively. L and T are dimensions of length and time

respectively. All the quantities are given in SI units. (There are two questions based on PARAGRAPH “X”, the question given below is one of them)

Q.16 The relation between 0 0and is

A) 2 20 0 L T

B) 2 20 0 L T

C) 1 2 20 0 L T

D) 1 2 20 0 L T




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Key. D

Sol: We have 1 2 22 2 2

0 00 0 0 0

1 1C L T L T

PARAGRAPH “A”

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation

/z x y . If the errors in x,yand z are ,x y and z , respectively, then 1

1 1x x x x yz zy y y x y

The series expansion for 1

1 yy

, to first power in /y y , is 1 /y y . The relative

errors in independent variables are always added. So the error in z will be x yz

x y

The above derivation makes the assumption that / 1, / 1x x y y . Therefore, the higher power of these quantities are neglected.

(There are two questions based on PARAGRAPH “A”, the question given below is one of them)

17. Consider the ratio 11

ar

a

to be determined by measuring a dimensionless quantity a. If the

error in the measurement of a is / 1a a a , then what is the error r in determining r?

A) 21

aa

B) 2

21

aa

C) 2

21

aa

D) 221

a aa

Key. B

Sol: 11

ara

1 1 1 11 1 1 1

a a a a arr a a a a

212 2

1 1 1 1

aa aa a a a




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PARAGRAPH “A” If the measurement errors in all the independent quantities are known, then it is possible to

determine the error in any dependent quantity. This is done by the use of series expansion and

truncating the expansion at the first power of the error. For example, consider the relation

/z x y . If the errors in x,yand z are ,x y and z , respectively, then

1

1 1x x x x yz zy y y x y

The series expansion for 1

1 yy

,to first power in /y y , is 1 /y y . The relative

errors in independent variables are always added. So the error in z will be x yz

x y

The above derivation makes the assumption that / 1, / 1x x y y . Therefore, the higher

power of these quantities are neglected. (There are two questions based on PARAGRAPH “A”, the question given below is one of them)

Q.18 In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40

nuclei decayed in the first 1.0 s. For |x| ≪ 1, ln(1 +x) = x up to first power in x. The error ,

in the determination of the decay constant , in 1S , is

A) 0.04

B) 0.03

C) 0.02

D) 0.01

Key. C

Sol: 0tN N e 0ln lnN N t

N t

N

40 0.02

2000N is number ofnuclei left undecayed

L




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PART-I CHEMISTRY SECTION 1 (Maximum Marks: 24)

This section contains SIX (06) questions. Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of

these four option(s) is (are) correct option(s). For each question, choose the correct option(s) to answer the question. 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, both of

which are correct options. Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct options. Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : −2 In all other cases. For Example: If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option),without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks. 1. The compound(s) which generate(s) 2N gas upon thermal decomposition below 0300 C is (are)

A) 4 3NH NO

B) 4 2 72NH Cr O

C) 3 2( )Ba N D) 3 2Mg N Key. BC

Sol. A) 04 3 2 23002

below CNH NO N O H O

B) 4 2 7 2 2 3 224NH Cr O N Cr O H O

C) 3 2 2( ) 3Ba N Ba N

D) 3 2Mg N (it does not decompose into 2N )

2. The correct statement(s) regarding the binary transition metal carbonyl compounds is

(are)(Atomic number: 26, 28Fe Ni )

A) Total number of valence shell electrons at metal centre in 5 4( ) ( )Fe CO or Ni CO is 16

B) These are predominantly low spin in nature

C) Metal-carbon bond strengthens when the oxidation state of the metal is lowered

D) The carbonyl C-O bond weakens when the oxidation state of the metal is increased




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Key. BC

Sol. A) [Fe(CO)5] & [Ni(CO)4] complexes have 18-electrons in their valence shell.

B) Carbonyl complexes are predominantly low spin complexes due to strong ligand field. C) As electron density increases on metals (with lowering oxidation state on metals), the extent

of synergic bonding increases. Hence M–C bond strength increases D) While positive charge on metals increases and the extent of synergic bond decreases and

hence C–O bond becomes stronger. 3. Based on the compounds of group 15 elements, the correct statement(s) is (are) A) 2 5Bi O is more basic than 2 5N O B) 3NF is more covalent than 3BiF C) 3PH boils at lower temperature than 3NH D) The N-N single bond is stronger than the P-P single bond Key. ABC Sol.

A) Bi2O5 is metallic oxide but N2O5 is non metallic oxide therefore Bi2O5 is basic but N2O5 is acidic. B) In NF3 , N and F are non metals but BiF3 , Bi is metal but F is non metal therefore NF3 is more covalent than BiF3. C) In PH3 hydrogen bonding is absent but in NH3 hydrogen bonding is present therefore PH3 boils at lower temperature than NH3. D)Due to small size in N–N single bond l.p. – l.p. repulsion is more than P–P single bond therefore N–N single bond is weaker than the P–P single bond.

4. In the following reaction sequence, the correct structure(s) of X is (are)

X 3 21) ,PBr Et O

2

3 2

2) ,3) ,

Nal MeCONaN HCONMe

Me 3N

enantiomerically pure

A)

B)

C)

D) Key. B Sol.

X 3 21) ,PBr Et O

2

3 2

2) ,3) ,

Nal MeCONaN HCONMe

Me 3N

all the three reaction are 2NS so X is




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5. The reaction(s) leading to the formation of 1,3,5-trimethylbenzene is (are)

A)

B)

heated iron tube

873 K

C)

D)

Key. ABD

Sol. A)

B) Fe

C) COOHHOOC

COOH

21 BrNaOH

32 HO(3) soda lime

D) COOHHOOC

COOH

ZnHg,HCl

6. A reversible cyclic process for an ideal gas is shown below. Here, P,V, and T are pressure, volume and temperature, respectively. The thermodynamic parameters q, w, H and U are heat, work, enthalpy and internal energy, respectively.

Temperature(T)

Vol

ume(

V)

1, 1, 1( )A PVT

2, 2, 1( )B PV T

2, 1, 2( )C PVT

The correct option(s) is (are) A) 2 2 1AC BC ABq U and w P V V

B) 2 2 1BC BC ACw P V V and q H

C) CA CA AC BCH U and q U D) BC AC CA CAq H and H U




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Key. BC Sol.

AC Isochoric AB Isothermal BC Isobaric # qAC = UBC = nCV (T2 – T1)

WAB = nRT1 ln 2

1

VV

A (wrong)

# qBC = HAC = nCP (T2 – T1) WBC = – P2(V1 – V2) B (correct) # nCP (T1–T2) < nCV(T1 – T2) C (correct) HCA < UCA # D (wrong)



the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer.

Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct numerical value is entered as answer.

Zero Marks : 0 In all other cases. 7. Among the species given below, the total number of diamagnetic species is .

H atom, NO2 monomer, O2− (superoxide), dimeric sulphur in vapour phase, Mn3O4,

(NH4)2[FeCl4], (NH4)2[NiCl4], K2MnO4, K2CrO4

Key. 1

Sol. *H-atom = 11S Paramagnetic

* NO2 =

N

OO odd electron species Paramagnetic

* 2O (superoxide) = One unpaired electrons in * M.O. Paramagnetic * S2(in vapour phase) = same as 2O , two unpaired e– s are present in * M.O. Paramagnetic

* 2 4

23 4 2 .Mn O MnO MnO

Paramagnetic * 4 42

NH FeCl 2 6 03 4Fe d s




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ClCl Cl Cl

3sp -hybridisation Paramagnetic * (NH4)2[NiCl4] =Ni = 3d8 4s2 Ni+2 = 3d8 4s0

ClCl Cl Cl

3sp -hybridisation Paramagnetic

* K2MnO4 =2K

Mn

O

O

OO , 6 13Mn Ar d Paramagnetic

*

2KCr

O

O

OO

2 4KCrO

, 6 03Cr Ar d Diamagnetic 8. The ammonia prepared by treating ammonium sulphate with calcium hydroxide is

completely used by NiCl2.6H2O to form a stable coordination compound. Assume that both the reactions are 100% complete. If 1584 g of ammonium sulphate and 952 g of NiCl2.6H2O are used in the preparation, the combined weight (in grams) of gypsum and the nickel-

ammonia coordination compound thus produced is .

(Atomic weights in g mol-1: H = 1, N = 14, O = 16, S = 32, Cl = 35.5, Ca = 40, Ni = 59) Key. 2992 Sol.

(NH 4)2 SO 4 + Ca(OH)2 CaSO 4.2H 2O+2NH3 1584 g gypsum ( M =172) 24mole =12 mol 12 mol

2 2 3 3 2 26952 4 24 232

4

.6 6 6g mol mol M

mol

NiCl H O NH Ni NH Cl H O

Total mass 12 172 4 232 2992g 9. Consider an ionic solid MX with NaCl structure. Construct a new structure (Z) whose unit cell

is constructed from the unit cell of MX following the sequential instructions given below. Neglect the charge balance.

(i) Remove all the anions (X) except the central one

(ii) Replace all the face centered cations (M) by anions (X)

(iii) Remove all the corner cations (M)

(iv) Replace the central anion (X) with cation (M)

The value of number of anionsnumber of cations

in Z is .

Key. 3




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Sol. . .X OV

M FCC

M+ X–

(i) 4 1 (ii) 4–3 3+1 (iii) 4 – 3 – 1 3+1 (iv) 1 3

3 31

Z

10. For the electrochemical cell, Mg(s) Mg2+ (aq, 1 M) Cu2+ (aq, 1 M) Cu(s) the standard emf of the cell is 2.70 V at 300 K. When the concentration of Mg2+ is changed to M, the cell potential changes to 2.67 V at 300 K. The value of is .

(given, 111500F KVR

,where F is the Faraday constant and R is the gas constant,

ln(10)=2.30) Key. 10

Sol. 0 2

2

2 2

2.672.701

CellCell EE Mg x MCu M

Mg S Cu aq Mg aq Cu s

0

lnCell Cell

RTE E xnF 2.67 2.70 ln

RTxnF

300

0.03 ln2R

xF

0.03 2ln

300FxR

0.03 2 11500300 1

ln 2.30 ln 10x X=10

11. A closed tank has two compartments A and B, both filled with oxygen (assumed to be ideal

gas). The partition separating the two compartments is fixed and is a perfect heat insulator

(Figure 1). If the old partition is replaced by a new partition which can slide and conduct heat but

does NOT allow the gas to leak across (Figure 2), the volume (in m3) of the compartment A

after the system attains equilibrium

is .




Page

-21

Key. 2.22 Sol. P1= 5 P2= 1

v1= 1 v2= 3 T1= 400 T2= 300

15

400n

R 2

3300

nR

Let volume be(v+x) v=(3-x) 15-5x=4+4x

A B

A B

P PT T

1 2

1 2

b b

b b

n R n Rv v

5 3400 4 300 3

5 3 4 4119

11 201 1 2.229 9

x R x

x x

x

v x

12. Liquids A and B form ideal solution over the entire range of composition. At temperature T, equimolar binary solution of liquids A and B has vapour pressure 45 Torr. At the same temperature, a new solution of A and B having mole fractions Ax and Bx

, respectively, has

vapour pressure of 22.5 Torr. The value of /A Bx x in the new solution is . (given that the vapour pressure of pure liquid A is 20 Torr at temperature T) Key. 19 Sol.

0 0

0 0

0

0

1 1452 290....... 1

2070 22.5 20 70 1

70 5070 22.5 0.95

500.05

0.95 190.05

A B

A B

A

B A A

A

A

B

A

B

P P

P P

given P torrP torr torr x x

x

x

xxSox

13. The solubility of a salt of weak acid (AB) at pH 3 is 310Y mol 1L . The value of Y is _.

(Given that the value of solubility product of AB 102 10spK and the value of ionization

constant of HB 81 10aK )

Key. 4.47

Sol. 3

10 5 38

101 2 10 1 2 10 4.47 1010sp

a

HS K M

K




Page

-22

14. The plot given below shows P-T curves(where P is the pressure and T is the temperature) for

two solvents X and Y and isomolal solutions of NaCl in these solvents. NaCl completely

dissociates in both the solvents.

760

1 2 3 4

360

362

367

368

Temperature(K)Pr

essu

re(m

mH

g)

1.solvent X2.solution of NaCl in solvent X3.solvent Y4.solution of NaCl in solvent Y

On addition of equal number of moles of a non-volatile solute S in equal amount (in kg) of

these solvents, the elevation of boiling point of solvent X is three times that of solvent Y. Solute S is known to undergo dimerization in these solvents. If the degree of dimerization is

0.7 in solvent Y, the degree of dimerization in solvent X is . Key. 0.05 Sol. From graph

For solvent X’ 2bxT ( )bx NaCl b xT m K ……….(1) For solvent ‘Y’ 1byT NaClb y b yT m K ……….(2) Equation (1) /(2)

2b x

b y

K

K For solute S

22 S S

11 / 2

1 / 2i

11

2b x s b xT K

212b x s b yT K

Given 3b x s b y sT T

1 21 3 1

2 2b x b yK K

1 22 1 3 12 2

2 0.7 So 1 0.05

SECTION 3 (Maximum Marks: 12) This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) questions. Each question has FOUR options. ONLY ONE of these four options corresponds to the correct answer. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct option is chosen.

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

Negative Marks : −1 In all other cases.




Page

-23

PARAGRAPH “X” Treatment of benzene with CO/HCl in the presence of anhydrous AlCl3/CuCl

followed by reaction with Ac2O/NaOAc gives compound X as the major product. Compound X upon reaction with Br2/Na2CO3, followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H2/Pd-C, followed by H3PO4 treatment gives Z as the major product.

15. The compound Y is

A)

B)

C)

D) Key. C Sol.

3/CO HCl

AlCl CuCl

PARAGRAPH “X”

Treatment of benzene with CO/HCl in the presence of anhydrous AlCl3/CuCl followed by reaction with Ac2O/NaOAc gives compound X as the major product. Compound X upon reaction with Br2/Na2CO3, followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H2/Pd-C, followed by H3PO4 treatment gives Z as the major product.

16. The compound Z is

A)

B)

C)

D)




Page

-24

Key. A Sol.

PARAGRAPH “A”

An organic acid P (C11H12O2) can easily be oxidized to a dibasic acid which reacts with ethyleneglycol to produce a polymer dacron. Upon ozonolysis, P gives an aliphatic ketone as one of the products. P undergoes the following reaction sequences to furnish R via Q. The compound P also undergoes another set of reactions to produce S.

17. The compound R is

A)

B)

C)

D) Key. A

11 12 2C H O O P

COOH

COOH

OHH dacron

OH




Page

-25

Sol.

PARAGRAPH “A”

An organic acid P (C11H12O2) can easily be oxidized to a dibasic acid which reacts with ethyleneglycol to produce a polymer dacron. Upon ozonolysis, P gives an aliphatic ketone as one of the products. P undergoes the following reaction sequences to furnish R via Q. The compound P also undergoes another set of reactions to produce S.

18. The compound S is

A)

B)

C)

D)




Page

-26

Key. B

11 12 2C H O O P

COOH

COOH

OHH dacron

OH

Sol.




Page

-27

PART-1:MATHEMATICS SECTION -1 (Maximum Marks: 24)

This section contains SIX (06) questions Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is

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

both of which are correct options Partial Marks: +1If 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: −2 In all other cases. For Example: If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting

only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option) ,without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks.

Q1. For a non-zero complex number , let arg ( ) denote the principal argument with arg( )z . Then, which of the following statement(s) is (are) FALSE?

A) arg( 1 ) 4

i , where 1i

B) the function : ,f , defined by arg 1f t it for all t , is continuous at all

points of , where 1i

C) For any two non-zero complex numbers 1z and 2z , 11 2

2

arg arg argz Z Zz

is an integer

multiple of 2 D) For any three given distinct complex numbers 1z , 2z and 3z , the locus of the point satisfying the

condition

1 2 3

3 2 1

argz z z zz z z z

, lies on a straight line

Key: A,B,D

Sol: a) 3arg 14

i

b)

1

1

tan , 0arg 1

tan 0t t

f t itt t

Discontinuous at 0t

c) 11 2

2

arg arg argz z zz

1 2 1 2arg arg 2 arg arg 2z z n z z n




Page

-28

d)

1 2 3

3 2 1

argz z z zz z z z

1 2 3

3 2 1

z z z zz z z z

is real. 1 2 3, , ,z z z z are concyclic

Q2. In a triangle , 30let PQR and the sides PQ and QR have lengths 10 3 and 10, respectively. Then , which of the following statement(s) is (are) TRUE?

A) 45QPR

B) The area of the triangle PQR is 25 3 and 120QRP

C) The area of the incircle of the triangle PQR is 10 3 15 D) The area of the circumcircle of the triangle is 100 Key: B,C,D

Sol:

2 2 210 3 10cos30

2 10 3 10

PR

10PR QR PR PQR QPR 30QPR

Area of 1 1 110 3 10 sin30 10 10 32 2 2

PQR

25 3 180 30 30 120QRP

25 3 25 3

10 5 310 10 10 32

rS

5 3. 2 3 10 3 15

10 10

2sin 2sin30aR

A

2 100Area R

Q3. Let 1 : 2 3P x y z and 2 : 2 2P x y z be two planes. Then, which of the following

statements(s) is (are) TRUE? A) The line of intersection of 1P and 2P has direction ratios 1,2,−1

B) The line 3 4 1 3

9 9 3x y z

is perpendicular to the line of intersection of 1P and 2P

C) The acute angle between 1P and 2P is 60

D) If 3P is the plane passing through the point (4,2,−2)and perpendicular to the line

of intersection of 1P and 2P , then the distance of the point (2,1,1) from the plane 3P is23




Page

-29

Key: C,D Sol: D.C. of line of intersection (a,b,c) 2 0a b c 2 0a b c

1 2 1 2 4 1

a b c

D.C is 1, 1,1 3 4 1 3

9 9 3x y z

4 / 3 1 / 33 3 3

x y z

lines are parallel

Acute angle between 1P and 2P 1 2 1 1 2 1 1cos6 6

1 13 1cos cos 606 2

Plane is given by 4 2 2 0x y z 0x y z

Distance of 2,1,1 from plane 2 1 1 2

3 3

Q4. For every twice differentiable function : 2,2f with 22 10 0 85f f which of the

Following statement(s) is (are) TRUE? A) There exist ,r s , where r s , such that f is one-one on the open interval .r s

B) There exists 0 4,0x such that 10 1f x

C) lim 1x

f x

D) The exists 4,4 such that 11 0f f and 1 0f Key: A,B,D Sol: ƒ(x) can't be constant throughout the domain. Hence we can find x (r, s) such that ƒ(x) is one-one

option (A) is true

Option (B) : 10

0 41

4f f

f x

(LMVT)

Option (C): sin 85f x x satisfies given condition

But limsin 85x

x

D.N.E Incorrect

Option (D): 22 1g x f x f x

11 1f x (by LMVT) 1 2f x (given)

1 15 4,0g x x

Similarly 2 25 0,4g x x

0 85g g x has maxima in 1 2,x x say at .

1 0 & 85g g 1 112 0f f f

If 1 0f 2 85g f Not possible

11 0f f 1 2, 4,4x x




Page

-30

Option (D) correct. Q5. Let :f and :g be two non-constant differentiable functions. If

1 1f x g xf x e g x for all x , and 1 2 1f g , then which of the following

statement (s) is (are) TRUE? A) 2 1 log 2ef

B) 2 1 log 2ef

C) 1 1 log 2eg

D) 1 1 log 2eg Key: B,C

Sol: 1 1f x g xf x e g x x

1. 0f x g xe f x e g x

1 1.f x g xe f x e g x dx C

f x g xe e C

1 1 2 2f g f ge e e e

1 21 1g fe ee e

ln 2 1f xe and 1 ln 2 1g

2 1 ln 2f and 1 1 ln 2g

Q6. Let : 0,f be a continuous function such that 0

1 2x x tf x x e f t dt for all

0,x . Then, which of the following statement(s) is (are) TRUE ?

A) The curve y ƒ x passes through the point (1, 2)

B) The curve y ƒ x passes through the point (2, –1)

C) The area of the region 2x, y 0, 1 : 1f x y x is 2

4

D) The area of the region 2x, y 0, 1 : x y 1f x is 1

4

Key: B,C

Sol: 0

1 2x

x tf x x e f t dt 0

1 2x

x x te f x e x e f t dt

Differentiate w.r.t.w 1 1 2 2x x x x xe f x e f x e x e e f x

1 1 2 2f x f x x f x 1 2 2 3f x f x x

Integrating factor 2xe




Page

-31

2 2. 2 3x xf x e e x dx

2 22 3 2x xx e dx e dx dx 2 22 3

2 2

x xx e e c

22 3 12 2

xxf x ce

3 10 1 02 2

f c c 1f x x

Area 1 2

4 2 4



the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer.

Answer to each question will be evaluated according to the following marking scheme: Full Mark: +3 If ONLY the correct numerical value is entered as answer.

Zero Marks : 0 In all other cases.

Q7. The value of 2 2 4

1 12 log log 9 log 7

2log 9 7 is ______________________

Key: 8

Sol: 2 2 4

2 1/2log log 9 log 7

2log 9 7 2 7log 92

1 log 42 log 22log 9 7 4 2 8

Q8. The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is .

Key: 625 Sol: Option for last two digits are (12), (24), (32), (44) are (52). Total No. of digits = 5 × 5 × 5 × 5 = 625 Q9. Let X be the set consisting of the first 2018 terms of the arithmetic progression 1,6,11, …….. and Y be

the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, ………… then, the

number of elements in the set X U Y is ____

Key: 3748 Sol: X : 1, 6, 11, ........, 10086 Y : 9, 16, 23, ......, 14128 X Y : 16, 51, 86, ...... Let m n X Y

16 m –1 35 10086 m 288.71 m = 288

– n X Y n X n Y n X Y




Page

-32

= 2018 + 2018 – 288 = 3748 Q10. The number of real solutions of the equation

1 1 1

1 1 1 1sin cos

2 2 2

i iii

i i i i

x xx x x

lying in the interval

1 1,2 2

is __________

(Here, the inverse trigonometric functions 1sin x and 1cos x assume value in ,2 2

and 0,

respectively.) Key: 2

Sol: 2

1

1 1i

i

XXX

1 2 2

i

i

x xx

1 2 2

i

i

x xx

1 1i

i

xxx

To have real solutions

1

1 1 1 12 2

i iii

i i i i

X XX X X

2 2

1 2 2 1X X X X

X X X X

3 22 5 2 0X X X X

0x and let 3 22 5 2f X X X X

1 1. 02 2

f f

Hence two solutions exist

Q11. For each positive integer n, let 11 1 2 ... n

nY n n n nn

For x ,let x be the greatest

integer less than or equal to x. If lim nny L

, then the value of L is________

Key: 1

Sol:

1

1 21 1 ...... 1n

nny

n n n

1/

1

1nn

nr

ryn

1

1log 1n

nr

ry l nn n

1

1limlog lim 1n

nn x r

ry lnn n

1

0

log 1L ln x dx 4log logLe

4Le

1L




Page

-33

Q12. Let a and b

be two unit vectors such that . 0a b

. For some ,x y , let c xa yb a b

.

If 2c

and the vector c is inclined at the same angle to both a and b

, then the value of 28cos is ___________

Key: 3 Sol: c xa yb a b

. 2cosc a x and x

. 2cosc a y and y

Also, 1a b

2cosc a b a b

2 22 24cos 2cos .c a b a b a b a b

24 8cos 1 28cos 3 Q13. Let a, b, c be three non-zero real numbers such that the equation

3 cos 2 sin , ,2 2

a x b x c x has two distinct real roots and with

3

. Then,

the value of ba

is ………

Key: 0.5

Sol: 23 cos sinb cx xa a

Now, 23 cos sinb ca a

-----------(1)

23 cos sinb ca a

------------ (2)

23 cos cos sin sin 0ba

23 2sin sin 2cos sin 0

2 2 2 2ba

3 2 3. 0ba

1 0.52

ba

Q14. A farmer 1F has a land in the shape of a triangle with vertices at (0,0), (1,1) and (2,0). From this land, a neighbouring farmer 2F takes away the region which lies between the side and a curve of the

form 1ny x n . If the area of the region taken away by the farmer 2F is exactly 30% of the area ofPQR , then the value of is

Key: 4

Sol: Area 1

0

310

nX X dX

12 1

0

32 1 10

nX Xn

1 1 32 1 10n

1 5n 4n




Page

-34

SECTION 3 (Maximum Marks: 12) This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) questions. Each question has FOUR options. ONLY ONE of these four options corresponds to the correct answer. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct option 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.

************************************************************************************ Paragraph "X"

Let S be the circle in the xy-plane defined by the equation 2 2 4x y .

(There are two question based on Paragraph "X", the question given below is one of them)

Q15. Let 1 2E E and 1 2F F be the chord of S passing through the point 0 1, 1P and parallel to the x-axis and the y-axis, respectively. Let 1 2G G be the chord of S passing through 0P and having slope –1. Let the tangents to S at 1E and 2E meet a 3E , the tangents of S at 1F an 2F meet at 3F , and the tangents to S at 1G and 2G meet at 3G . Then, the points 3E , 3F and 3G lie on the curve

A) 4x y

B) 2 2 – 4 – 4 16x y C) – 4 – 4 4x y D) 4xy Key: A

Sol: co-ordinates of 1E and 2E are obtained by solving 1y and 2 2 4x y

1 3,1E and 2 3,1E

co-ordinates of 1F and 2F are obtained by solving x=1 and 2 2x 4y

1 1, 3F and 2 1, 3F Tangent at 1 : 3 4E x y

Tangent at 2 : 3 4E x y 3 0,4E

Tangent at 1 : 3 4F x y Tangent at 2 : 3 4F x y 3 4,0F

And similarly 3 2,2G 0,4 , 4,0 and 2,2 lies on 4x y




Page

-35

Paragraph "X"

Let S be the circle in the xy-plane defined by the equation 2 2 4x y .

(There are two question based on Paragraph "X", the question given below is one of them) Q16. Let be a point on the circle with both coordinates being positive. Let the tangent to at intersect

the coordinate axes at the points and . Then, the mid-point of the line segment must lie on the curve

A) 2x y 3xy

B) 2/3 2/3 4/32x y

C) 2 2 2x y xy

D) 2 2 2 2x y x y Key: D

Sol: Tangent at 2cos ,2sinP is cos sin 2x y

2sec ,0M and 0,2cosN ec

Let midpoint be ,h k sech , cosk ec

2 2

1 1 1h k

2 2

1 1 1x y

PARAGRAPH “A” There are five students 1 2 3 4, , ,S S S S and 5S in a music class and for them there are five seats 1R , 2 3 4, , ,R R R and 5R arranged in a row, where initially the seat iR is allotted to the student iS , =1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats.

(There are two questions based on PARAGRAPH “A”, the question given below is one of them) Q17. The probability that, on the examination day, the student 1S gets the previously allotted seat 1R , and NONE

of the remaining students gets the seat previously allotted to him/her is

A) 340

B) 18

C) 740

D) 15

Key: A




Page

-36

Sol: Required probability

1 1 14!9 32! 3! 4!

5! 120 40

PARAGRAPH “A”

There are five students 1 2 3 4, , ,S S S S and 5S in a music class and for them there are five seats 1R , 2 3 4, , ,R R R and 5R arranged in a row, where initially the seat iR is allotted to the student iS , =1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on PARAGRAPH “A”, the question given below is one of them) Q18. For =1,2,3,4, let iT denote the event that the students iS and 1iS do NOT sit adjacent to each other on the

day of the examination. Then, the probability of the event 1 2 3 4T T T T is

A) 1

15

B) 1

10

C) 760

D) 15

Key: C Sol: 1 2 3 4n T T T T Total 1 2 3 4n T T T T

4 3 3 2 41 1 1 1 15! 4!2! .3!2! 3!2!2! 2!2! .2.2! 2C C C C C

14

Probability 14 75! 60

2018-jee-advanced · sol: 2 gh r 2 cos r h r r g 2 cos h r g (a)for given material , constant 1 h r...

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