P1-16-3-6 PAPER-1 Code : 6100p.s3.amazonaws.com/vidyamandir/JEEAdvanced20… · ·...

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VMC/Paper-1 1 JEE Entrance Exam-2016/Advanced

JEE ENTRANCE EXAM-2016/ADVANCED

P1-16-3-6 PAPER-1 Code : 6 Time : 3 Hours Maximum Marks : 186

Read the following Instructions very carefully before you proceed.

GENERAL

1. This sealed booklet is your Question Paper. Do not break the seal till you are told to do so.

2. The paper CODE is printed on the right hand top corner of this sheet and the right hand top corner of the back cover of this booklet.

3. Use the Optical Response Sheet (ORS) provided separately for answering the questions.

4. The paper CODE is printed on the left part as well as the right part of the ORS. Ensure that both these codes are

identical and same as that on the question paper booklet. If not, contact the invigilator for change of ORS.

5. Blank spaces are provided within this booklet for enough work.

6. Write your name, roll number and sign in the space provided on the back cover of this booklet.

7. After breaking the seal of the booklet at 9.00 am, verify that the booklet contains 36 pages and that all the 54 questions along with the options are legible. If not, contact the invigilator for replacement of the booklet.

8. You are allowed to take away the Question Paper at the end of the examination.

OPTICAL RESPONSE SHEET

9. The ORS (top sheet) will be provided with an attached Candidate’s Sheet (bottom sheet).

The Candidate’s Sheet is a carbon-less copy of the ORS.

10. Darken the appropriate bubbles on the ORS by applying sufficient pressure. This will leave an impression at the corresponding place on the Candidate’s Sheet.

11. The ORS will be collected by the invigilator at the end of the examination.

12. Your will be allowed to take away the Candidate’s Sheet at the end of the examination.

13. Do not tamper with or mutilate the ORS. Do not use the ORS for rough work.

14. Write your name, roll number and code of the examination centre, and sign with pen in the space provided for this purpose on the ORS. Do not write any of these details anywhere else on the ORS. Darken the appropriate bubble under each digit of your roll number.

DARKING THE BUBBLE ON THE ORS

15. Use a BLACK BALL POINT PEN to darken the bubble on the ORS.

16. Darken the bubble COMPLETELY.

17. The correct way of darkening a bubble is as:

18. The ORS is machine-gradable. Ensure that the bubbles are darkened in the correct way.

19. Darken the bubble ONLY IF you are sure of the answer. There is NO WAY to erase or “un-darken” a darkened bubble.

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PART-I PHYSICS

SECTION I (Maximum Marks : 15) This section contains Five questions.

Each question has Four options (A), (B), (C) and (D). ONLY ONE of these four options is correct.

For each question, darken the bubble corresponding to the correct option in the ORS.

For each question, marks will be awarded in one of the following categories : Full Marks : +3 If, only the bubble corresponding to the correct option is darkened. Zero Marks : 0 If none of the bubbles is darkened. Negative Marks : –1 In all other cases.

1. A parallel beam of light is incident from air at an angle on the side

PQ of a right angled triangular prism of refractive index 2.n Light undergoes total internal reflection in the prism at the face PR when has a minimum value of 45 . The angle of the prism is :

(A) 15 (B) 22.5 (C) 30 (D) 45 1.(A)

For face PQ, 01.0sin 45 2 sin

030

Also, 0 090 90 45 180

045

015 Hence (A)

2. In a historical experiment to determine Planck’s constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectron energies were measured by applying a stopping potential. The relevant data for the wavelength ( ) of incident light and the corresponding stopping potential 0( )V are given below:

m 0V (volt)

0.3 0.4 0.5

2.0 1.0 0.4

Given that 8 13 10c ms and 191.6 10 ,e C Planck’s constant (in units of Js) found from such an experiment is :

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(A) 346.0 10 (B) 346.4 10 (C) 346.6 10 (D) 346.8 10

2.(B) 0hceV w

01hc wV

e e

i.e. Slope of 0V us 1

gives .hce

i.e. 6 61 2 0.4 1.010 101 1 1 1

0.4 0.3 0.5 0.4

ymx

This gives : 61.2 10m

hcme

6 19

81.2 10 1.6 10

3 10h

This gives 346.4 10 .h Hence (B) 3. A water cooler of storage capacity 120 litres can cool water at a

constant rate of P watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load). The temperature of water fed into the device cannot exceed 30 C and the entire stored 120 litres of water is initially cooled to 10 .C

The entire system is thermally insulated. The minimum value of P (in watts) for which the device can be

operated for 3 hours is : (Specific heat of water is 4.2 kJ 1 1kg K and the density of water is 1000 3kg m )

(A) 1600 (B) 2067 (C) 2533 (D) 3933 3.(B) Overall water receives heat at the rate of 3000 P watts.

Temperature of water increases from 10° C to 30°C Let time be t. Then, 3000 P t 120 4200 20

t 3 60 60s .

Solving gives : P 2067 watt. Hence (B) 4. An infinite line charge of uniform electric charge density lies along the axis of an electrically conducting

infinite cylindrical shell of radius R. At time t = 0, the space inside the cylinder is filled with a material of permittivity and electrical conductivity . The electrical conduction in the material follows Ohm’s law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j(t) at any point in the material?

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

4.(A) Let charge density on wire be ' at time t.

Then, 2kEr

Using j E (differential form of ohm's law)

2k 'j .r

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22

i K Q. .rL r L

(Q is charge on length L of wire)

1 22

dQ . k Qdt

dQ 4 k dtQ

0

Qn 4 k .tQ

0

t

0Q Q e

0t0Q2kj . er L

002 tkj e

r

5. A uniform wooden stick of mass 1.6 kg and length l rests in an inclined manner on a smooth, vertical wall of height h (<l) such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the

stick is perpendicular to the stick. The stick makes an angle of 030 with the wall and the bottom of the stick in on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the

stick. The ratio h/l and the frictional force f at the bottom of the stick are : 2( 10 )g m s

(A) 3 16 3,16 3

h f Nl (B) 3 16 3,

16 3h f Nl

(C) 3 3 8 3,16 3

h f Nl (D) 3 3 16 3,

16 3h f Nl

5.(D)

N sin 30° + N = 1.6 × 10

N = 323

f = N cos 30° = 32 3 16 33 2 3

0 must be zero

mg × cos602 sin 60L hN

1 32 2162 2 3 3L h

3 316

hl

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

Each question has FOUR options options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.

For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS.

For each question, marks will be awarded in one of the following categories: Full Marks : +4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened. Partial Marks : +1 For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened. Zero Marks : 0 If none of the bubbles is darkened. Negative Marks : –2 In all other cases. For example, if (A), (C) and (D) are all the correct options for a question, darkening all these three will result in +4 marks; darkening only (A) and (D) will result in +2 marks; and darkening (A) and (B) will result in -2 marks, as a wrong option is also darkened.

6. The position vector r of a particle of mass m is given by the following equation : 3 2ˆ ˆ( ) ,r t t i t j

where 3 210/ 3 , 5m s m s and 0.1m kg. At 1 ,t s which of the following statement(s) is(are) true

about the particle?

(A) The velocity v is given by 1ˆ ˆ(10 10 )v i j m s

(B) The angular momentum L

with respect to the origin is given by ˆ(5 / 3)L k N m s

(C) The force F

is given by ˆ ˆ( 2 )F i j N

(D) The torque with respect to the origin is given by ˆ(20 / 3)k N m

6.(ABD) 3 2ˆ ˆr t i t j

3 2x t , y t

101 53

ˆ ˆr i j

2dx 3t 10dt

dy 2t 10dt

x10a 6 t 6 1 203

ya 2

(A) ˆ ˆV 10i 10j True

(B) L m r v

10 ˆ ˆ ˆ ˆL 0.1 i 5j 10i 10j3

5 ˆL k3

True

(C) ˆ ˆF 0.1 20i 10 j ˆ ˆ2i j Not true

(D) r F

10 ˆ ˆ ˆ ˆi 50j 2i j3

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20 k̂3

True

7. A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices 1n and 2 1( ),n n as shown in the figure. A

ray of light is incident with angle i from medium 1 and emerges in

medium 2 with refraction angle f with a lateral displacement l.

Which of the following statement(s) is (are) true? (A) l is independent of 2n (B) 1 2sin sini fn n

(C) l is dependent on n (z) (D) 1 2 1sin ( )sini fn n n

7.(ABC) For a slab with variable refractive index, n sin remains constant. Hence (B)

Also, l depends on equation of curve and hence on how refractive index varies with z. Hence (C) As the equation of the curve is independent of n2 hence is independent of n2. Hence (A). 8. A plano-convex lens is made of a material of refractive index n. When a small object is placed 30 cm away in

front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away from the lens. Which of the following statement(s) is(are) true?

(A) The refractive index of the lens is 2.5 (B) The radius of curvature of the convex surface is 45 cm (C) The faint image is erect and real (D) The focal length of the lens is 20 cm 8.(AD) For refraction from lens, 30 u , 60 v

Hence, 1 1 160 30

F

20F cm ................(i)

Hence, (D) For reflection from spherical surface, 30 u , 10 v cm

1 1 210 30

R

30R cm .............(ii)

By lens maker's formula: 1 1 1120 30

( ) 2 5 . Hence, (A)

9. Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the following statement(s) is(are) true ? (A) Relative change in the radii of two consecutive orbitals does not depend on Z (B) Relative change in the radii of two consecutive orbitals varies as 1/n (C) Relative change in the energy of two consecutive orbitals varies as 1/n3 (D) Relative change in the angular momenta of two consecutive orbitals varies as 1/n

9.(ABD) For (AB)

2 2

12

12 1

2n n

n

n nR R nz z

R n

2 1n n

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For (C)

2 2

2 21

2 2

2

2 1( 1)( 1)

n n

n

z zU U nn n

U z nn

1n

For (D) 1 ( 1) ( ) 1n n

n

L L n nL n n

10. A length-scale ( ) depends on the permittivity ( ) of a dielectric material, Botzmann constant Bk , the

absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression(s) for is(are) dimensionally correct ?

(A) 2

B

nqk T

(B) 2

Bk Tnq

(C) 2

2 3B

qn k T

(D)

2

1 3B

qn k T

10.(BD) 3[ ] [ ]n L [ ] [ ]q AT

1 3 2 4[ ] [ ]M t A T [ ] [ ]T K

[ ] [ ]l L

1 2 2 1[ ] [ ]Bk M L T K

(A) RHS. 3 2 2

1 3 4 2 1 2 2 1[ ]

[ ][ ][ ]L A T

M L T A M L T K K

3 2 2

2 11 2 2

[ ] [ ] [ ][ ]L A T L LL T A

Wrong

(B) RHS 1 3 4 2 1 2 2 1

3 2 2[ ][ ][ ]

[ ][ ]M L T A M L T K K

L A T

1 2 2

3 2 2[ ] [ ][ ]L T A LL T A

Correct

(C) RHS 2 2

31 3 4 2 2 1 2 2 1

[ ][ ][ ][ ][ ]

A T LM L T A L M L T K K Wrong

(D) RHS 2 2

1 3 4 2 1 1 2 2 1[ ]

[ ][ ][ ]A T

M L T A L M L T K

2 2

2 2 2[ ] [ ]

[ ]A T L

L T A Correct

11. Two loudspeakers M and N are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at 60 km/hr along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R , 1800 m away from Q. Let v(t) represent the beat frequency measured by a person sitting in the car at time t. Let

P, Qv v and Rv be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air

is 330 1m s . Which of the following statements(s) is(are) true regarding the sound heard by the person?

(A) The plot below represents schematically the variation of beat frequency with time

(B) 2P R Qv v v

(C) The plot below represents schematically the variation of beat frequency with time

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(D) The rate of change in beat frequency is maximum when the car passes through Q

11.(ABD)

Speed of car, V = 60 km/h

5003

m/s

At a point S, between P & Q

'M M

C V cosv vC

;

'N N

C V cosv vC

cos( ) 1

N MVv v v

C

similarly, between Q & R

cos( ) 1

N MVv v v

C

( ) ( ) sin N M

d v V dv vdt C dt

0º at P & R as they are large distance apart. slope of graph is zero. at , 90º Q

sin is max.

also value of ddt

is max

as

d Vdt r

, where V is its velocity and r is length of line joining P & S.

and r is minimum at Q. slope is maximum at Q. (A) & (B) are correct.

At P, ( ) 1 ( 0º )N MVv v vC

At R, ( ) 1 ( 0º )N MVv v vC

At Q, ( ) ( 90º )N Mv v v

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2( ) 2 P R N M Qv v v V v Hence (B) is correct.

12. A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the 90° vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire.

The current in the triangular loop is in counterclockwise direction and increased at a constant rate of 10 1As . Which of the following statement(s) is (are) true?

(A) The induced current in the wire is in opposite direction to the current along the hypotenuse

(B) The magnitude of induced emf in the wire is 0

volt

(C) There is a repulsive force between the wire and the loop

(D) If the loop is rotated at a constant angular speed about the wire, an additional emf of 0

volt is

induced in the wire 12.(BC) By reciprocity theorem of mutual induction it can be assumed that current in infinite wire is varying at 10 A/S

and EMF is induced in triangular loop. Flux of magnetic field through triangle loop, if current in infinite wire is , can be calculated as follows:

0 22

id y dyy

0id dy

02

i l

EMF 0

2l di

dt

0 0(10cm) 10 A

S

volts.

Hence, (B) is correct. If we assume the current in the wire towards right then as the flux in the loop increases we know that the induced

current in the wire is counter clockwise. Hence the current in the wire is towards right. Field due to triangular loop at the location of infinite wire is into the paper. Hence, force on infinite wire is away

from the loop. Hence (C) By cylindrical symmetry about infinite wire, rotation of triangular loop will not cause any additional EMF. 13. An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric

current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true? (A) The temperature distribution over the filament is uniform (B) The resistance over small sections of the filament decreases with time (C) The filament emits more light at higher band of frequencies before it breaks up (D) The filament consumes less electrical power towards the end of the life of the bulb

13.(CD) (A) Since evaporation is given to be non-uniform, temperature must be non-uniform. (B) Due to evaporation, cross sectional area of wire decreases. Hence resistance of any given section

increases. (C) Consider a small section where area has been reduced.

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Power generated per unit volume will determine the temperature of this section:

2dP i dR

2dP dRidV dV

2

( )i dx

A dx A

2

2dP idV A

Power radiated = 4 AT 43

iTA

Assuming that effect of 3A dominates i, (C) will be correct.

(D) 2

VPR

. Overall R decreases, V remains constant. Hence (D)

SECTION III (Maximum Marks : 15) This section contains FIVE questions.

The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.

For each question, darken the bubble corresponding to the correct integer in the ORS.

For each question, marks will be awarded in one of the following categories: Full Marks : +3 If only the bubble corresponding to the correct answer is darkened. Zero Marks : 0 In all other cases.

14. Two inductors 1L (inductance 1 mH, internal resistance 3 ) and 2L (inductance 2 mH, internal resistance 4

), and a resistor R (resistance 12 ) are all connected in parallel across a 5 V battery. The circuit is switched on at time 0t . The ratio of the maximum to the minimum current max minI / I drawn from the battery is

________. 14.(8)

At 0t

minI at t = 0 as current in both inductors = 0

Using KVL, 5 – 12 I = 0

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min5

12i A

I max is at t as current in both inductors will be max. t

Inductors will behave like resistors only 1 1 1 1 23 4 12 3eqR

max5 10

3/ 2 3I max

min

10 / 3 85 /12

II

15. A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P) by the metal. The sensor has a scale that displays 2log 0P / P , where 0P is a constant. When the metal surface is

at a temperature of 487°C, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to 2767 °C?

15.(9) Power radiated 4P A

1st reading, 4

1 20

760log

AR

P

; 2nd reading, 4

2 20

3040log

AR

P

4 40

2 1 2 240

3040 3040log log760760

A PR RP A

24log 4 8

2 18 9R R

16. A hydrogen atom in its ground state is irradiated by light of wavelength 970 Å . Taking 61 237 10hc / e . eV m and the ground state energy of hydrogen atom as 13 6. eV , the number of lines

present in the emission spectrum is________.

16.(6) Incident energy 6

101.237 10 12.75970 10

ihcE eV

e jumps to state of E = –13.6+12.75= –0.85 eV 4n Number of lines in emission spectrum is 6.

17. Consider two solid spheres P and Q each of density 8 gm 3cm and diameters 1 cm and 0.5 cm, respectively.

Sphere P is dropped into a liquid of density 0.8gm 3cm and viscosity 3 poiseulles. Sphere Q is dropped into

a liquid of density 1.6 gm 3cm and viscosity 2 poiseulles. The ratio of the terminal velocities of P

and Q is________.

17.(3) Terminal velocity 22 ( )

9TrV d p g

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2

2 ( )p Q PP

Q P QQ

r dVV dr

2 8 – 0.81 2

0.5 3 (8 –1.6)

2 7.243 6.4

=3

18. The isotope 125 B having a mass 12.014 u undergoes -decay to 12

6 C. 126 C. has an excited state of the nucleus

126( C*) at 4.041 MeV above its ground state. If 12

5 B decays to 126 C* , the maximum kinetic energy of the

-particle in units of MeV is________. ( 21u 931.5 MeV / c , where c is the speed of light in vacuum)

18.(9) 12 12 06 –5 iB C e v

Mass of 126 C = 12. 000 u (by definition if 1 a.m.u.)

Q – value of reaction, Q = (MB – MC) × c2 = (12.014 – 12.000) × 931.5 = 13.041 MeV

4.041 MeV of energy is taken by 126 C* max. K.E. of -particle is (13.041 – 4.041) = 9 MeV

PART-II CHEMISTRY





19. P is the probability of finding the 1s electron of hydrogen atom in a spherical shell of infinitesimal thickness, dr,

at a distance r from the nucleus. The volume of this shell is 24 r dr . The qualitative sketch of the dependence of P on r is :

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

19.(C) r = 0 at nucleus For 1s orbital probability of finding electron can’t be zero for any value of r. 20. One mole of an ideal gas at 300 K in thermal contact with surroundings expands isothermally from 1.0 L to 2.0

L against a constant pressure of 3.0 atm. In this process, the change in entropy of surroundings surrS in

–1J K is (1 L atm = 101.3 J)

(A) 5.763 (B) 1.013 (C) –1.013 (D) –5.763 20.(C) For isothermal expansion : irrev. irrev.U 0 q w ( P V) 3(2 1) 3 L atm

1irrev.Surrounding

q 3 101 3S 1 013 J KT 300

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21. Among 24 4 3 4 2 3 6 2 2 2[Ni(CO) ], [NiCl ] , [Co(NH ) Cl ]Cl, Na [CoF ], Na O and CsO , the total number of

paramagnetic compounds is (A) 2 (B) 3 (C) 4 (D) 5

21.(B) 24 3 6 2[NiCl ] , Na [CoF ],CsO are paramagnetic.

Cl is a weak field ligand with 2Ni in 24[NiCl ] hence no pairing of electrons

F is a weak field ligand with 3Co in 3 6Na [CoF ] hence no pairing of electrons

2O is superoxide anion in 2CsO .

22. The increasing order of atomic radii of the following Group 13 elements is

(A) Al < Ga < In < Tl (B) Ga < Al < In < Tl (C) Al < In < Ga < Tl (D) Al < Ga < Tl < In

22.(B) Increasing order of atomic radii: Ga < Al < In < Tl Ga is smaller due to poor shielding of d-orbital 23. On complete hydrogenation, natural rubber produces

(A) ethylene-propylene copolymer (B) vulcanised rubber (C) polypropylene (D) polybutylene




For each question, marks will be awarded in one of the following categories: Full Marks : +4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened.

Partial Marks : +1 For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened. Zero Marks : 0 If none of the bubbles is darkened. Negative Marks : –2 In all other cases. For example, if (A), (C) and (D) are all the correct options for a question, darkening all these three will result in +4 marks; darkening only (A) and (D) will result in +2 marks; and darkening (A) and (B) will result in -2 marks, as a

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wrong option is also darkened.

24. The product(s) of the following reaction sequence is(are)

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

25. The correct statement(s) about the following reaction sequence is(are)

2 3

3

i O CHCl / NaOH9 12 ii H O

Cumene C H major minor P Q R

2

NaOHPhCH Br

Q S

(A) R is steam volatile (B) Q gives dark violet coloration with 1% aqueous FeCl3 solution (C) S gives yellow precipitate with 2,4-dinitrophenylhydrazine (D) S gives dark violet coloration with 1% aqueous FeCl3 solution.

26. The crystalline form of borax has

(A) tetranuclear 2

4 5 4B O OH

unit

(B) all boron atoms in the same plane

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(C) equal number of sp2 and sp3 hybridized boron atoms (D) one terminal hydroxide per boron atom.

27. The reagent(s) that can selectively precipitate 2S from a mixture of 2S and 2

4SO in aqueous solution is (are)

(A) CuCl2 (B) BaCl2

(C) Pb(OOCCH3)2 (D) 2 5Na Fe CN NO

27.(A) By using CuCl2, 2S can be precipitated selectively from a mixture of 2S and 24SO

28. A plot of the number of neutrons (N) against the number of protons (P) of stable nuclei exhibits upward deviation

from linearity for atomic number, Z > 20. For an unstable nucleus having N/P ratio less than 1, the possible mode(s) of decay is(are) :

(A) decay emission (B) orbital or K-electron capture

(C) neutron emission (D) decay positron emission

28.(BD) N/P can be increased by decay and K electron capture. In both processes number of protons decreases and

number of protons increases. 29. Positive Tollen’s test is observed for

(A) (B) (C) (D) 29.(ABC) Aldehyde gives positive Tollen's test.

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-Hydroxy ketones are also oxidized by mild oxidizing agent including Tollen's reagent. 30. The compound(s) with TWO lone pairs of electrons on the central atom is(are) : (A) 5BrF (B) 3ClF (C) 4XeF (D) 4SF

31. According to the Arrhenius equation, (A) a high activation energy usually implies a fast reaction. (B) rate constant increases with increase in temperature. This is due to a greater `

number of collisions whose energy exceeds the activation energy. (C) higher the magnitude of activation energy, stronger is the temperature dependence of the rate

constant. (D) the pre-exponential factor is a measure of the rate at which collisions occur, irrespective of

their energy

31.(BCD) Ea

RTk Ae

High activation energy means lower value of k and slow reaction. On increasing temperature energy of particles increases hence greater number of collisions occur whose energy exceeds the activation energy. Rate of increase of k with temperature is higher when Ea has a large value Pre-exponential factor is a measure of frequency of collisions.




For each question, marks will be awarded in one of the following categories:

Full Marks : +3 If only the bubble corresponding to the correct answer is darkened. Zero Marks : 0 In all other cases.

32. In the following monobromination reaction, the number of possible chiral products is _________.

32.(5) Total five products are formed.

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33. The mole fraction of a solute in a solution is 0.1. At 298 K, molarity of this solution is the same as its

molality. Density of this solution at 298 K is 2.0 g cm–3. The ratio of the molecular weights of the solute

and solvent,

solute

solvent

MWMW

is_________.

33.(9) B

A A

1000m

M

10009 A

mM

. . . .(i)

Also using,

10001000 B

Mmd MM

1000BMM . . . .(ii)

B

10001

1000 2 MM

Using (i) and (ii) : 9B

A

MM

(Molality = molarity)

34. The number of geometric isomers possible for the complex 2 2 2 2 2CoL Cl L H NCH CH O

is_________. 34.(5) Total five isomers are possible

35. In neutral or faintly alkaline solution, 8 moles of permanganate anion quantitatively oxidize thiosulphate

anions to produce X moles of a sulphur containing product. The magnitude of X is_________.

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36. The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The

absolute temperature of an ideal gas is increased 4 times and its pressure is increased 2 times. As a result, the diffusion coefficient of this gas increases x times. The value of x is _________.

36.(4) Diffusion co-efficient . meanC

TP

meanC T

Diffusion co-efficient T

Diffusion co-efficient T TP

Diffusion co-efficient3/ 2TP

If T is increased four time and pressure is increased two times diffusion co-efficient should become 4 times.

PART-III MATHEMATICS





37. A computer producing factory has only two plants 1T and 2T . Plant 1T produces 20% and plant 2T produces

80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P (computer turns out to be defective given that it is produced in plant 1T ) = 10 P (computer turns

out to be defective given that it is produced in plant 2T ).

Where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant 2T is

(A) 3673

(B) 4779

(C) 7893

(D) 7583

Vidyamandir Classes


37.(C)

P [Defective] = 0.07

[Non – defective] = 0.93 Let P[Defective from T1] = x

P [Defective from T2] = 10x

Figure

A.T.P. 0.2(1 ) 0.8 1 0.9310xx

0.07 = 0.28 x 14

x

P [Non-defective from T2] =

8 390.8 110 10 40

930.93100

x = 78

93

38. A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the

selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

(A) 380 (B) 320 (C) 260 (D) 95 38.(A) 6 Girls 4 Boys

Case-I 4 Girls

Select 4 Girls from 6 in 64C ways.

Select captain in 41C ways

by FPC, 6 44 1 60C C

Case-II 3 Girls, 1 Boy

Select 3 Girls from 6 in 63C & 1 Boy from 4 in 4

1C

Select captain in 41C

by FPC, 6 4 43 1 1 320C C C

Total = 320 + 60 = 380.

39. The least value of R for which 2 14 1xx

, for all x > 0, is :

(A) 164

(B) 132

(C) 127

(D) 125

39.(C) 34 1 0 0h h x

212 1 0f x n

Vidyamandir Classes


2 112

x

112

x

3

21

21 14 1 012 12

3 1 1 1

2 2 22

4 4 4 1 012 1212

3 1

2 2

3 32 2

4 12 12 012

3 1

2 28 12 0

3 1

2 212 0

1

23

2

8

12

64 12144

49 12

127

40. Let 6 12

. Suppose 1 1and are the roots of the equation 2 2 sec 1 0x x and 2 2and are the

roots of the equation 2 2 tan 1 0x x . If 1 1 2 2> and > , then 1 2+ equals :

(A) 2(sec tan ) (B) 2sec

(C) 2 tan (D) 0

40.(C) For 1 & 1

Roots are 2sec 2 tan2

sec tan

6 12

sec 0

tan 0 1 1

1 sec tan ; 1 sec tan

For 2 & 2

Roots are 2 tan 2sec2

sec tan & sec tan

2 2 2 sec tan ; 2 sec tan

1 2 2 tan

41. Let , : 0,2

S x x

. The sum of all distinct solutions of the equation

3 2 0sec x cosec x tan x cot x in the set S is equal to :

(A) 79

(B) 29

(C) 0 (D) 59

Vidyamandir Classes


41.(C) 2 cos 23 1 0cos sin sin cos

xx x x x

3 sin cos 2cos2x x x cos cos 23

x x

2 23

x n x 2 2

3x n x

23

x n 2

3 9nx

Possible values 3

7 5, ,9 9 9

3 7 5sum 09




For each question, marks will be awarded in one of the following categories: Full Marks : +4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened. Partial Marks : +1 For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened. Zero Marks : 0 If none of the bubbles is darkened. Negative Marks : –2 In all other cases.

For example, if (A), (C) and (D) are all the correct options for a question, darkening all these three will result in +4 marks; darkening only (A) and (D) will result in +2 marks; and darkening (A) and (B) will result in -2 marks, as a wrong option is also darkened.

42. Let : 0, f R be a differential function such that 2

f xf x

x for all 0,x and 1 1f .

Then:

(A) 0

1lim 1x

fx

(B) 0

1lim 2x

xfx

(C) 20

lim 0x

x f x

(D) 2f x for all 0, 2x

42.(A) : 0,f R

' 2 0, , 1 1f x

f x x fx

1 2dy ydx x

If 1 dxxe x

2xy x dx c 2x c

cy xx

1, 1 0x y c

Vidyamandir Classes


(i) 2' 1 cf xx

; 21' 1f cxx

0

1lim ' 1x

fx

(ii) 21 1 1xf x cx cxx x

0

1lim 1x

xfx

(iii) 2 2'x f x x c

2

0lim ' 0

xx f x c

(iv) | ( ) | , 0cf x x xx

cxx

Let c = 2 2( )f x xx

Let x = 1 ( ) 3f x contradicts 2f x

43. The circle 2 21 : 3C x y , with centre at O, intersects the parabola 2 2x y at the point P in the first quadrant.

Let the tangent to the circle 1C at P touches other two circles 2 3andC C at 2 3andR R , respectively. Suppose

2 3andC C have equal radii 2 3 and centres 2 3andQ Q respectively. If 2 3andQ Q lie on the y-axis, then :

(A) 2 3 12Q Q (B) 2 3 4 6R R

(C) Area of the triangle 2 3OR R is 6 2 (D) Area of the triangle 2 3PQ Q is 4 2 43.(ABC)

22 3y y

2 2 3 0y y

( 3)( 1) 0y y 1, 3y

( 2, 1)P

Tangent at P is 2 3x y

Let 2 3orQ Q (0, k)

Vidyamandir Classes


3 2 33

k

3 6x 9 or 3

2 30, 3 , 0,9Q Q

2 3 12Q Q 222 3 12 4 3 144 48 96R R

Area of 2 31 34 6 6 22 3

OR R squints

Area of 2 31 12 2 6 22

PQ Q

44. A solution curve of the differential equation 2 24 2 4 0 0dyx xy x y y xdx

, passes through the point

(1, 3). Then the solution curve : (A) intersects y = x + 2 exactly at one point (B) intersects y= x + 2 exactly at two points (C) Intersects 22y x (D) does NOT intersect 23y x 44.(AD) Let x + 2 = X x > 0 y = Y X > 2

2

2( 2) ( 2)x y x dx

dyy

2

2X YX dX

dYY

2X X dX

Y Y dY

Let dX dtX tY t YdY dY

2 dtt t t YdY

22

Ydt dt dXtdY Yt

1 log Y ct

logY Y cX

…… (i)

(i) passes through (3, 3)

1 log 3 c 1log log3ce

1log3

ce

curve is

1log log3

Y YX e

or log3

Y YX e

…… (ii)

Now option (A) Y = X using in (ii)

1 log 33Y Ye

x = 3

at one point. Option (C) is y = x2 Replace in (ii)

2 2

log3

X XX e

2 3 . XX e e

But x > 2 LHS > 4

RHS 3e

No solution.

Vidyamandir Classes


Option (D) is 2( 1)Y X

Replace in (ii)

2 2( 1) ( 1)log

3X X

X e

1 2

2( 1) 3 .X

XX e e

X > 2 LHS > 9

RHS < 7 / 23 . e

Not intersects. 45. In a triangle XYZ , let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and

2s x y z . If 4 3 2

s x s y s z and area of incircle of the triangle XYZ is 8

3 , then :

(A) area of the triangle XYZ is 6 6 (B) the radius of circumcircle of the triangle XYZ is 35 66

(C) 4sin sin sin2 2 2 35X Y Z

(D) 2 3sin2 5

X Y

45.(ACD) 2 83

r

34 3 2 9 9

s x y zs x s y s z s

9 9 4 ,3 3 ,9 9 2s x s s y s s z s 5 9 , 2 3 ,7 9s x s y s z

5 2 79 3 9s s ss s s s

4 29 3 9s s ss 22 2

9 3s

2 2 2 23 9 3

r s

9 5, 6, 7s x y z

2 2 81 6 69 3

5 6 7 35 35 64 244 6 6 4 6xyzR

4 sin sin sin2 2 2x y zr R

2 22 2 6 43sin sin sin

2 2 2 35 3535 6 36

X Y Z

22sin 1 cos2

X Y X Y

1 cos Z 2 2 2

12

x y zxy

25 36 49160

12 6160 5

2 3sin2 5

X Y

46. Let RS be the diameter of the circle 2 2 1x y , where S is the point (1, 0). Let P be a variable point (other than

R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point(s).

(A) 1 13 3

(B) 1 1

4 2

(C) 1 13 3

(D) 1 1

4 2

Vidyamandir Classes


46.(AC) 1 0 1 0S R

cos sinP

Tangent at P: cos sin 1x y

Tangent at S : x = 1

Solving 1 tan2

Q

Normal at P : sin cos 0x y

Line through Q and to RS

tan2

y

Solving 22

11 sec tan2 2

E h k

Locus : 211 12

h k

or 211 12

x y or 2

21 2 12 2yx x y

Options (A) and (C) satisfy the locus

47. Let 3 1 22 03 5 0

P

, where . Suppose ijQ q is a matrix such that PQ = kI, where 0k k and

I the identity matrix of order 3. If 2

23 and det8 2k kq Q

, then :

(A) 0 8k (B) 4 8 0k (C) 9det 2P adj Q (D) 13det 2Q adj P

47.(BC) 1Q KP

1

5 103 6 (3 4)10 12 2

20 12P

23 8kq

.( 3 4)20 12 8

k k

1

2

| |2

kQ

2

1| |2

kkP | | 8 | | 8P Q

3 2

| | 2k kP

3 2

20 12 2k k

k = 4 4 8 0k Ans. B

| |adjP Q 2| || |P Q 3 62 2 92 Ans. C

Vidyamandir Classes


48. Let andf g h be differentiable functions such that 3 3 2f x x x ,

g f x x and h g g x x for all x . Then :

(A) 1215

g (B) 1 666h (C) 0 16h (D) 3 36h g

48.(BC) 23 3 0f x x x

1g x f x

1 1 1 1g g x f f x f of

1 1

0 3g x

f

h x fof x 33 33 2 3 3 2 2x x x x

23 2 23 3 2 3 3 3 3 3h x x x x x

1 3 36 6 18h 666

0 8 6 2 16h

13 3h g f f f 3f 27 9 2 38

49. Consider a pyramid OPQRS located in the first octant ( 0, 0, 0)x y z with O as origin, and OP and OR

along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3. Then :

(A) the acute angle between OQ and OS is 3

(B) the equation of the plane containing the triangle OQS is 0x y

(C) the length of the perpendicular from P to the plane containing the triangle OQS is 32

(D) the perpendicular distance from O to the straight line containing RS is 152

49.(BCD)

(A) cos OQ OSOQ OS

1cos3

1 1cos3

Vidyamandir Classes


(B) 3 3 0 03 3 32 2

x y z 0x y

(C) 3 0 32 2

(D)

RSRM ORRS

32

RM …… (i)

3OR …… (ii)

2 2OM OR RM

152

OM




For each question, marks will be awarded in one of the following categories: Full Marks : +3 If only the bubble corresponding to the correct answer is darkened. Zero Marks : 0 In all other cases.

50. Let 1 3 , where 1, and , 1, 2, 32

iz i r s . Let

2

2

( )r s

s rz z

Pz z

and I be the identity matrix of order

2. Then the total number of ordered pairs (r, s) for which 2P I is __________.

50.(1) 1 32

iz

z

2

2

sr

s rP

2P I

Vidyamandir Classes


2 2

2 2

1 00 1

s sr r

s sr r

4 2

2 4 2

1 1 1 00 11 1

rr s r s

rr s s r

2 4 1r s and 2 1 1 0rr s r can be 1 or 3

If 1r 2 4 1s 4 21s

1s s If 3r

6 4 1s 4 2s

No “s” possible Ordered pair of ,r s is 1, 1

51. Let m be the smallest positive integer such that the coefficient of x2 in the expansion of 2 3 49 50(1 ) (1 ) ... (1 ) (1 )x x x mx 51

3is (3 1)n C for some positive integer n. Then the value of n is __________.

51.(5) S = (1 + x)2 + (1 + x)3 + ….. + (1 + x)49 + (1 + mx)50

2 4850

1 (1 ) 1(1 )

x xS mx

x

50 2501 1

(1 )x x

S mxx x

Coefficient of x2

3 50 2 50. of in (1 ) 0 . of in (1 )Coeff x x coeff x mx

50 50 2 513 2 3(3 1)C C m n C .

50 5023 2

51 513 3

3 1C C m nC C

250 49 48 50 49 6 3 151 50 49 2 51 50 49

m n

216 1 3 117 17

m n

2 1317 17m n

2 1 51m n 2 51 1m n

5n

52. The total number of distinct 0, 1x for which 2

40

2 11

xt dt x

t

is __________.

Vidyamandir Classes


52.(1) Let 2

40

2 11

xtf x dt x

t

. . . . .(i)

2

4 21

xf xx

Now as 22

1 2xx

2

2

1 11 2xx

0f x f x is decreasing function

Now 0 1f . . . . (ii)

and 1 2

40

1 11

tft

. . . . (iii)

2

4121

tt

1 12

40 0

121

tt

1 2

40

11 121

tt

. . . . .(iv)

1 0f . . . . . (v)

Therefore f x will cross X-axis exactly at one point between [0, 1]

53. The total number of distinct x R for which

2 3

2 3

2 3

1

2 4 1 8 10

3 9 1 27

x x x

x x x

x x x

is __________.

53.(2)

2 3

2 3

3

1

2 4 1 8 10

3 9 1 27

x x x

x x x

x x x

2 2 3

2 2 3

2 2 3

1

2 4 1 2 4 8 10

3 9 1 3 9 27

x x x x x

x x x x x

x x x x x

2 2

2 2

2 2

1 1

1 2 4 2 3 1 2 4 10

1 3 9 1 3 9

x x x x

x x x x x x x

x x x x

32 1 6 10x x x x

3 32 1 6 10x x

3 31 6 5x x

3x t

26 5t t

26 5 0t t

26 6 5 5 0t t t

Vidyamandir Classes


6 1 5 1 0t t t

1 5/ 6t t

3 31 5 / 6x x

1/ 31 and 5/ 6x x

54. Let , be such that 2

0

sin( )lim 1sinx

x xx x

. Then 6( ) equals __________.

54.(7) 2

0

sinlim 1

sinx

x xx x

3

sin0lim 1x xx

x

1 6 1

16

16 6 1 76

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