Fiitjee phase test paper

FIITJEE - JEE

(Mains)

PHYSICS, CHEMISTRY & MATHEMATICS

JEE – MAINS- 2020 CODE: SET-A Time Allotted: 3 Hours

 

Maximum Marks: 360

Do not open this Test Booklet until you are asked to do so. Please read the instructi ons carefully. You are allotted 5 minutes specifically for this purpose.

Important Instructio I nstructions: ns: 1.

Immediately fill in the particulars particulars on this page of the Test Booklet Booklet with Blue / Black Ball Point Pen. Use of pencil is strictly prohibited.

2.

The Answer Sheet is kept inside this Test Test Booklet. When you are directed to open open the Test Booklet, take take out the Answer Sheet and fill in t he particulars carefully.

3.

The test is of 3 hours duration. hours duration.

4.

The Test Booklet consists of 90 questions. The maximum marks are 360. 360.

5.

There are three three parts in the question paper A, B, C consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each question is allotted 4 (four) marks for correct response.

6.

Candidates will be awarded marks as stated above in instruction No.5 for correct response of each question. ¼ (one fourth) marks will be deducted for indicating incorrect response of each question. No deduction from the total score will be made if no response is indicated for an item in the answer sheet.

7.

There is only only one correct response response for each question. Filling up up more than one response in any question will be treated as wrong response and marks for wrong response will be deducted accordingly as per instruction 6 above.

8.

Use Blue / Black Ball Point Pen only for for writing particulars / marking responses on Side-1 and Side-1 and Side-2 of of the Answer Sheet. Use of pencil is strictly prohibited.

9.

No candidate is allowed to carry any textual material, printed printed or written, written, bits of papers, papers, pager, mobile phone, any electronic device, etc. except the Admit Card inside the examination hall / room.

10. On completion of the test, the candidate must hand over the Answer Sheet to the Invigilator on duty in the Room / Hall. However, the candidates are allowed to take away this Test Booklet with them. 11. Do not fold or make any stray marks on the Answer Sheet.

Name of the Candidate (in Capital Letters) :___________________________ :_____________________________________ __________ Enrolment Number :_________________________________________________________ Batch :__________________ :________________________ ______ Date of Examination : __________________ ________________________ ______

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TWO YEAR CRP C-LOT (1820) MAINS (PCM) SET-A

Useful Data Chemistry: = 8.314 J K1 mol1 = 0.0821 Lit atm K1 mol1 = 1.987  2 Cal K1 mol1 Avogadro's Number Na = 6.023  1023 Planck’s Constant h = 6.626  10 –34 Js = 6.25 x 10-27 erg.s 1 Faraday = 96500 Coulomb 1 calorie = 4.2 Joule 1 amu = 1.66 x 10-27 kg 1 eV = 1.6 x 10-19 J Atomic No : H=1, D=1, Li=3, N a=11, K=19, Rb=37, Cs=55, F=9, Ca=20, He=2, O=8 , Au=79. Atomic Masses: He=4, Mg=24, C=12, O= 16, N=14, P=31, Br= 80, Cu=63.5, Fe= 56, Mn=55, Pb=207, Au=197, Ag=108, F=19, H=2, Cl=35 .5, Sn=118.6 Gas Constant

R

Useful Data Physics: Acceleration due to gravity g = 10 m / s

2



PART –A (PHYSICS) 1.

For a sample of gas with N 0 number of molecules, v

dV v

 av 2 for, O  v  v 0 and zero for

 v 0 a is a constant R.M.S speed of the molecules is

(A) v 0 2 / 5 2.

dN

(B) v 0 3 / 5

(C) v 0 4 / 5

A tube shaped ring, is separated into two parts by a fixed partition at P and movable partition at Q. It carries equal masses of O2 and N 2 in the two parts. Consider the gases to be ideal and tube to the frictionless. The angle θ in equilibrium is – (A) 7  / 8 (B) 15  / 16 (C) 16  / 15 (D) 8  / 7

(D) v 0 P

N2

θ O2

Q

3.

4.

An ideal gas is following a process where P

 P0  aV 2 .Pressure of the gas when

temperature is maximum is (A) P0 / 3

(C) P 0

(D) None of these

A heater is supplying heat at constant rate q J/S to n moles of an ideal monoatomic gas confined in a cylindrical container with movable piston of mass m, such that the piston moves with 2q constant speed, v  ; P is: P  mg mg  P0 A  (A) 1 (C) 5

5.

(B) 2P0 / 3

P2 m

A

(B) 3 (D) 7

One mole of a diatomic gas undergoes undergoes through a process P Molar specific heat of the gas is 3R 5R 7R (A) (B) (C) 2 2 2

 aT 2 where a is a constant. (D) None of these



6.

An ideal gas is taken through a cyclic process ABCDA. The amount of work done by the gas in this process is about (A)



 10 5 J

2 (C) 2   10 5 J

(B)

  10 5 J

(D)

  10 J

P B

3 atm



C

A

3



1atm

D 1m3

5m3

V

7.

A spring is stretched by 2 cm under under a tension of 540 N. Work done done by the force is (A) 2.7 J (B) 5.4 J (C)27 J (D) 54 J

8.

Two wires wires of of same material has lengths in the ratio ratio 1 : 2 and radius in the ratio 2:1. If If equal load is hanged from them, then the ratio of their elongation is – (A) 1 : 2 (B) 1 : 4 (C) 1 : 8 (D) 8 : 1

9.

Two wires, separated by 90 cm, holds a platform horizontally. The radius and Young’s modulus of left and right wires are r and 2r and Y and 2Y respectively. An external force F is applied on the platform at a distance x from the left wire such that the platform still remains horizontal. The value of x is. (A) 40 cm (B) 50 cm (C) 60 cm (D) 80 cm

10.

Two rods of lengths, Young’s modulus and coefficient of linear expansion, l 1 , Y , Y 1 , 1 and l2 ,Y 2 ,  2 ; having same cross sectional area A are connected at one end and placed between two rigid supports. If temperature is now increased by  , then force exerted by the rods on one another is  l   l   Y Y A  l   l  Y Y A (A) 1 1 2 2 1 2 (B) 1 1 2 2 1 2 l1Y1  l1Y2 l1Y2  l 2Y1 (C)

 l12  l2 1  Y1Y2 A  l1Y2

 l 2Y1

(D)

x F

l1 Y 1 1

l 2 Y 2  2

 l12  l2 1  Y1Y2 A  l1Y1

 l 2Y2



11.

A wire of length 2l, cross-sectional area A and Young’s modulus Y is light fixed horizontally between two rigid supports. When a block of mass M is hanged from middle, the mid-point gets slightly depressed. The depression is – 1

 Mg  2 (B) l    YA 

 Mg  A) l    YA  1

 Mg  3 (C) l    YA 

M

1

 Mg  4 (D) l    YA 

12.

A stationary wave, s = αsinkx cosωt is produced on a rod of density ρ and cross-sectional area A. The total mechanical energy confined between two adjacent nodes is 2  2 A 2  2 A 2  2 A 2  2 A (A) (B) (C) (D) k 2k 4k 8k

13.

A composite tube has a straight and a circular portion of radius 0.5 m. A source ‘S’ emits all possible audible frequencies which the detector ‘D’ detects at the other end. If speed of sound in the tube is 314 m/s, then which of the following frequency will not be detectable by D (A) 50 Hz (B) 100 Hz (C) 200 Hz (D) 400 Hz

14.

15.

r S

Two coherent loud speakers, A and B are placed at a separation of 2m. A detector ‘D’ moves perpendicular to the line joining the sources starting at P. Speed of sound is 340 m/s. The detector is unable to detect any destructive interference along its path for frequencies below a certain frequency. That frequency is (A) 65 Hz (B) 75 Hz (C) 85 Hz (D) 95 Hz Two coherent sources s1 , and s2 , placed at a separation 2λ, are emitting waves of wavelength λ. A detector is moving along a line perpendicular to the line joining the sources. It detects a maxima at ‘O’ and another at ‘P’. The dis tance between O and P is (d<<λ). (A) d (B) 2d (C) 3d

(D) 2d

D

D A

B





P

2m

 A  s1



P

B

o

d

D

s2

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16.

Two coherent sources, emitting waves of wavelength λ, are placed at a separation of 3λ. A detector is moving around them on a large circle. The number of minima detected by it in one complete rotation is – (A) 2 (B) 4 (C) 8 (D) 12

17.

A string of linear mass density 0.25 kg/m is taut under a tension of 100 N inside a cart. A short pulse is generated at the left end which then propagates in (+)ve x direction. An observer on ground observes the pulse to be at rest. Velocity of the cart is (A) 20 iˆ m / s (B) 20 iˆ m / s (C) 40 iˆ m / s (D) 40 iˆ m / s

18.



If specific heat capacity of a substance in solid and liquid state is proportional to temperature of the substance, then if heat is supplied to the solid initially at – 20°C(having melting point 0°C) at constant rate. Then the temperature dependence of solid with time will be best represented by (A)

(B)

T

T

t

(C)

t

(D)

T

T

t

19.



All the rods have same conductance ‘K’ and same area of cross-section ‘A’. If ends A and C are maintained at temperature 2T 0 and T 0 respectively, then which of the following is/are correct? (A) Rate of heat flow through ABC, AOC and ADC is same (B) Rate of heat flow through BO and OD is not same 3 KAT 0 (C) Total rate of heat flow from A to C is 2a (D) Temperature at junctions B, O and D are same

t

B

a A

a

b b

C

b b

a

a

D



20.

A hollow copper sphere and a hollow copper cube, of same surface area and negligible thickness, are filled with warm water of same temperature and placed in an enclosure of constant temperature, a few degrees below that of the bodies. T hen in the beginning: (A) The rate of energy lost by the sphere is greater than that by the cube (B) The rate of fall of temperature for sphere is greater than that by the c ube (C) The rate of energy lost by the sphere is less than that by the cube (D) The rate of fall of temperature for sphere is less than that f or the cube

21.

There are two thin spheres A and B of the same material and same thickness. They emit like black bodies. Radius of A is double that of B. A and B have same temperature T. When A and B are kept in a room of temperature T 0(
22.

23.

In a cyclic process, a gas is taken from state A to B via path-I as shown in the indicator diagram and taken back to state A from state B via path-II. In the complete cycle: (A) work is done on the gas (B) heat is given to the gas (C) no work is done by the gas (D) nothing can be said about work as data is insufficient

P

A rod of length l is in motion such that its ends A and B are moving along x-axis and y-axis respectively. It is given that

d  dt

 2 rad/s always. P is a fixed point on the rod. Let M be

the projection of P on x-axis. For the time interval in which 



B

A V

y B

P 

l A

, choose the correct statement: 2 (A) The acceleration of M is always directed towards right (B) M executes SHM (C) M moves with constant speed (D) M moves with constant acceleration changes from 0 to



24.

A block of mass ‘m’ is attached to a spring in natural length of spring constant ‘k’. The other end A of the spring is moved with a constant velocity v away from the block, find the amplitude of oscillation of the block in the reference frame of point A of the spring. (A) (C)

25.

1

m 2

4

k m 2

(B)

m 2

2

k

(D) 2

m 2 k

Two light strings, each of length l, are fixed at points A and B on a fixed horizontal rod xy. A small bob is tied by both strings and in equilibrium, the strings are making angle 45° with the rod. If the bob is slightly displaced normal to the plane of the strings and released then time period of the resulting small oscillation will be: (A) 2 (C) 2

26.

k

1

2 2l

l g

(B) 2 (D) 2

x

B 45



2



g l 2 g

A particle of mass m = 2 kg executes SHM in xy-plane between points A and B under action of force  F  F xiˆ  Fy ˆj. Minimum time taken by particle to move

(C) 4 cos  t

45

2l

from A to B is 1 sec. At t = 0 the particle is at x = 2 and y = 2. Then, F x as function of time t is: 2 2 (A) 4 sin  t (B) 4 cos  t

27.

A

A  2, 2  x B

 2,  2 

(D) none of these

An organ pipe of length L is open at one end and closed at other end. The wavelengths of the three lowest resonating frequencies that can be produced by this pipe are: (A) 4L, 2L, L (B) 2L, L, L/2 (C) 2L, L, 2L/3 (D) 4L, 4L/3, 4L/5



28.

A, B and C are three tuning forks. Frequency of A is 350Hz. Beats produced by A and B are 5 per second and by B and C are 4 per second. When a wax is put on A, beat frequency between A and B is 2Hz and between A and C is 6Hz. Then frequency of B and C respectively are: (A) 341 Hz, 359 Hz (B) 345 Hz, 341 Hz (C) 359 Hz, 345 Hz (D) 355 Hz, 341 Hz

29.

In resonance tube experiment, if 400 Hz tuning fork is used, the first resonance occurs when length of air column is 19cm. If the 400 Hz tuning fork is replaced by 1600Hz tuning fork then to get resonance, the water level in the tube should be f urther lowered by a minimum value of (take end correction = 1 cm): (A) 5 cm (B) 10 cm (C) 15 cm (D) 20 cm

30.

A car moves towards a hill with speed  c . It blows a horn of frequency f which is heard by an observer following the car with speed  0 . The speed of sound in air is  . (A) The wavelength of sound reaching the hill is  /f. (B) The wavelength of sound reaching the hill is

   c 2 f

.

(C) The wavelength of sound of horn directly reaching the observer is (D) The beat frequency observed by the observed is

   c 2 f

2c    0 

 2   c2



PART –B (CHEMISTRY) 1.

Rank in increasing order of acidity OH

OH

(II) (I) O 2N

NO2

O2N OH

OH NO2

(III)

(IV)

CH3

NO2

(A) III < II < I < IV (C) III < I < II < IV 2.

(B) IV < II < I < III (D) I < II < IV < III

Rank in increasing order of pKa values. O

O II

I

OMe O

O

III

O

O

IV OMe

(A) I < II < IV < III (C) III < I < IV < II

(B) III < IV < I < II (D) IV < I < III < II Space for rough work



3.

Identify the major species in solution for reaction: OH COOH N a OH  o n e e q v.         H 2O

CH 2 OH ONa

OH COOH

(A)

COOH

(B)

CH2OH

CH2ONa

OH

OH COOH

(C)

COONa

(D)

CH2OH

4.

CH2OH

Rank the following carbocations in order of increasing stability (least stable first)

II I IV III

(A) II < IV < I < III < V (C) IV < II < III < V < I

(B) IV < I < V < II < III (D) I < II < III < V < IV Space for rough work



5.

Which of the following is most stable carbocation? +

(A)

(B)

(C) +

(D) +

6.

The most stable diene is (A)

(B)

(C)

7.

(D)

Which of the following is the strongest base?

(A)

(B) N

N H

(C)

(D) N H

8.

O

Arrange the following in the increasing order of their basic strength O

O

O NH 2 (I)

(A) I < II < III < IV (C) IV < III < I < II

N H (II)

NH 2

H 2N

NH

(III)

NH 2 (IV)

(B) IV < III < II < I (D) II < I < III < IV Space for rough work



9.

Which of the following is the resonance form of the following enolate ion O

OH

O

(B) (A) O

O

(C)

10.

(D)

Which of the following compound is most basic? (A)

(B) N H

N H

(C)

(D) N H

11.

N H

What is the major product of the reaction? CH3 D

2  

Ni

D

D CH3

(A)

CH3

(B)

D

D

H

D CH3

CH3 D

(C)

H

(D)

D H

D H



12.

Free radical chlorination of n-hexane produces this number of monochloroderivatives (including stereoisomers)? (A) 5 (B) 7 (C) 8 (D) 3

13.

The major monobromination product in the photochemical bromination of 2,2-dimethyl butane is (A)

(B) Br Br

(C)

(D)

Br

Br

14.

What is the most likely product of the following reaction? Cl

2  

hv

(A)

(B)

Cl

Cl

(C)

(D)

Cl

Cl

15.

An unknown alkene was treated with ozone followed by zinc dust in dilute acetic acid. The reaction products were determined to be acetone and propanol. What is the structure of the unknown? O

O

 i  O ,CH Cl  ii  Zn,CH3 CO2 H

3 2 2   ? 

+ H

(B) (A)

(C)

(D)



16.

Which is the best combination of reagents to carry out the synthesis? OH

?  

OH

(A) m-CPBA,CH2Cl2 (C) Br 2,H2O 17.

(B) (i) O3 (ii) CH3SCH3 (D) OsO4,KHSO3, acetone

What is the product of the reaction? H3C C H

C

CH3 H

OsO4 ,CH3 3 COOH    CH3 3 COH,O H

(A) CH3CHO (B) Racemic (2R,3R) and (2S,3S)-2,3-butanediol (C) Meso-2,3-butanediol (D) Cis-2,3-epoxybutane 18.

What would be the best method to bring about the following conversion? CH3 CH 3 C CH

CH 3 CH 2

CH 3 C

CH3

CHCH 3

CH 3 OH

(A) BH3, THF; then H2O2, OH(C) Conc. H2SO4, then H 2O, heat

(B) HBr; then NaOH/H2O (D) Hg(OAc) 2 /THF-H2O, then NaBH 4, OH-

Space for rough work



19.

Select the reaction(s) that would result in the formation of 2-bromopropane. Peroxides

 I  CH3CH  CH2  HBr  CCl

4  II  CH3CH  CH2  NBS 

hv

 III  C H3CH2CH3  Br2  CCl

4  IV  CH3CH  CH2  Br2 

(A) I and III (C) I, II, III and IV 20.

(B) III only (D) I, II and III

Give the major product of the f ollowing reaction



Catalytic H  Major product Heat

21.

(A)

(B)

(C)

(D)

What is the major organic product of the reaction

C

Hg , H2 SO 4 CH     

(A)

(C)

(B) C

O

H

H H C

H

CCH3

C

C OH

(D) CH2CHO

OH



22.

The product that should be formed in the reaction are

CH 3

i 

H , Lindlar ' s catalyst

 C  C  CH3   2        ii 

H

(I)

Br2 ,CCl 4

Br

H

(II)

CH3

H3C Br

H CH3

Br H

H 3C

(IV)

Br

(III)

CH3

H3C

H

H

Br

Br

Br Br

H

H

H3 C

H 3C

(A) I and III in equal amounts (C) All four in unequal amounts

(B) II and IV in equal amount (D) II and III in equal amount

23.

Choose the best reagent for the following transformation 2  heptyne  cis  2  heptene (A) H2/Pt (B) Na/liq. NH3 + (C) H /H2O (D) H2/Lindlar’s catalyst

24.

For the following series of reaction, what can be predicted about stereochemistry of final product? H3C

Na, liq. NH

3  CH3 

Br / CCl

2 4   Product(s)

(A) Meso compound (C) Racemic mixture

(B) A chiral compound (D) A pair of diastereomers Space for rough work



25.

Which of the following compounds would not be considered aromatic in its behaviour? (A)

(B) N

(C)

(D) N H

26.

Which of the following cations is aromatic?



 (I)

(II)





 (IV)

(III)

(A) II, IV (C) I, II, IV 27.

(B) IV (D) II, III, IV

How many of the compounds will give predominantly meta substituted product in electrophilic aromatic substitution? NH3

Cl

CH3

(III)

(IV)

(V)

CN

(I)

OMe (II)

(A) 1 (C) 3

(B) 2 (D) 4 Space for rough work



28.

What product(s) would you expect from the following reaction? O HNO

3  

O

H2 SO4

O

O

(A)

O2N

O

O

(B)

NO2

O

(C)

NO2

O

(D)

O

O

NO2

29.

Predict the major product of the following reaction. Br HNO

3  

H2 SO4

CHO

(A) 4-bromo-2-nitrobenzaldehyde (B) 4-bromo-3-nitrobenzaldehyde (C) 4-bromo-2,6-dinitrobenzaldehyde (D) 4-bromo-3,5- dinitrobenzaldehyde 30.

What unusual characteristic does a halogen substituent exhibit in electrophilic aromatic substitutions? (A) It is activating but m eta directing (B) It is activating and directs to all positins equally (C) It is deactivating but ortho/para directing (D) It is activating but exclusively ortho directing



PART–C (MATHEMATICS) 1.

The domain & range of the function f ( x )  log2 (A) Df

  ,   , Rf  1, 2 

(B) Df

  , 0    0,  , Rf   1,1

(C) Df

 0,   , Rf  1, 2 

sin x  cos x  3 2 2

are given by

(D) None of these 2.

If f(x) be a continuous function defined for 1  x f(1.8) is (where Q is the set of all rational numbers) (A) 1 (B) 5

3.

 3 . If f ( x )  Qx  1, 3 , f (2 )  10 , then (C) 10

(D) 20

Let the function f(x) be defined as follows: f ( x )  x 3  x 2  10 x, 1  x  0 cos x,

0x

  / 2  / 2  x  

1  sin x, Then f(x) has (A) A local minimum at x = π/2 (B) A local maximum at x = π/2 (C) an absolute minimum at x = -1 (D) an absolute maximum at x = π 4.

The slope of the tangent to the curve y  (A) 1/2

5.

x

dx

0 1  x

3

(B) 1

at the point where x = 1 is (C) 1/4

sin x

cos x

cos x

The number of distinct real roots of cos x

sin x

cos x

cos x

cos x

sin x

(D) None

 0 in the interval

 / 4  x   / 4 (A) 0 6.

(B) 2

(C) 1

(D) 3

If ABCDEF is a regular hexagon inscribed in a circle with centre O, then      AB  AC  AD  AE  AF  

(A) 4AO



(B) 5 AO



(C) 6AO



(D) 8AO



7.

The lines 2x  y

 1  0 ,  m  4  x   2m  1 y  0 and 4mx   m  6  y  1  0 are

(A) concurrent for two values of m (B) concurrent for one value of m (C) concurrent for no value of m (D) None of these 8.

A plane meets the coordinate axes in A, B, C and

 ,,   is the centroid of the triangle ABC,

then equation of plane is x y z   3 (A)

1

(C)

9.





3x





(B)



3y





3z



1

x





y





(D) ax  y

z



 z  1

x y   1 , moves in such a way that harmonic mean of a and b is 8. Then the a b least area of triangle made by line and coordinate axes is(A) 8 (B) 16 (C) 32 (D) 64 A variable line

10.

Total no. Of ways of selecting two numbers from set { 1, 2, 3, ...... 3n}, so that their sum is divisible by 3, is equal to2n 2  n 3n 2  n (A) (B) (C) 2n 2  n (D) None 2 2

11.

 z 2  z  i  0 , then (A) z  1

12.

If i z3

(B) z

18

1

(C) z

1

(D) z

 0

 7 3  3.18.7.25  The value of 6 is 3  6.243 .2  15 .81 .4  20 .27 .8  15 .9 .16  6 .3 .32  64 (A) 0

(B) 1

3

(C) 2

(D) None

13.

If a circle passes through the points (1, 0), (5, 0) and touches the y-axis at (0, h), then h = (A) 5 (B) 2 5 (C) 3 5 (D) 4 5

14.

A tangent to the ellipse

x 2 y 2 4   1 , having slope  meets the major and minor axes at A 18 32 3 and B. If O be the origin, then the area of OAB is. (A) 24 sq. units (B) 48 sq. units (C) 36 sq. units (D) Non of these



15.

If the lines (A) -1

16.

x  1 y  1 z  1 x  3 and   2 3 4 1 (B) 2/9



k

y

z 2 1 (C) 9/2

 intersect, then k is equal to: (D) 0

The tangent to the graph of the function y = f(x) at the point with abscissa x = 1 from an angle of π/6 and at the point x = 2 an angle of π/3, at the point x = 3 an angle of π/4. The value of



3

1

f '(x ) f ''( x) dx 



3

1

f ''( x) dx

( f ''( x ) is supposed to be continuous) os; (A) 17.

4 3

1

3 3

3 3 1 2

(B)

The area bounded by the curves y (A) 1/6

(C)

4 3 3

(D) None of these

2

  x  1 , y   x  1 2 2 , the line y = 1/4 an x-axis is:

(B) 2/3

(C) 1/4

(D) 1/3

18.

Let a, b, c be any real numbers. Suppose the three are real numbers x, y, z not all zero such taht x = cy + bz, y = az + cx and z = bx + ay then a2  b 2  c 2  2abc is equal to; (A) 1 (B) 2 (C) -1 (D) 0

19.

If

cox3  1  then θ = 2 cos 2   1 2

(A) n 

20.

 3

(B) n 

Statement – I (s1 ) : Let

 4

(C) 2n  

 6

(D) None of these

2x    sin1  , 2   1  x 

 1  x 2  du . Then  1 for 0 < x < 1. 2  dv  1  x   2x   2 tan1 x Statement – I  s2  : sin1    1  x 2   1  x 2  For -1 < x < 1 and cos 1   2 tan 1 x 2   1  x  v  cos1 

(A) If both Statement-I and Statement-II explanation of Statement-II. (B) If both Statement-I and Statement-II of Statement-I. (C) If Statement-I is true but Statement-II (D) if Statement-I is false but Statement-II

are true but Statement-II is not the correct are true, and Statement-II is correct explanation is false. is true



21.

The locus of z which statisfies the inequality log0.3 z  1 (A) x + y > 0

22.

23.

(B) x – y < 0

 log0.3 z  i is given by,

(C) x + y < 0

When 5 99 is divided by 13, the remainder is (A) 8 (B) 9 (C) 10

(D) x – y > 0

(D) None

A circle C 1 of radius 2 touches both x-axis and y-axis. Another circle C 2 whose radius is greater than 2 touches circle C 1 and both the axes. Then the radius of circle C 2 is (A) 6  4 2

24.

(B) 6  4 2

The tangent to ellipse 3x 2

(C) 6  4 3

(D) 6  4 3

 3  16y 2  12 . At the point 1,  , intersects the curve y 2  x  0 at  4

(A) no point (C) two distinct points

(B) Exactly one point (D) more than two points

25.

The volume of the tetrahedron included between the plane 3x + 4x – 5z – 60 = 0 and the coordinate planes is (A) 60 (B) 600 (C) 720 (D) None

26.

The tangent to the curve y

 e x drawn at A  c,e c  intersect the line joining the points

P  c  1, ec 1  and Q  c  1, ec 1  : (A) on the left of x = c (C) at no point 27.

(B) on the right of x = c (D) at all points

If l r means (log log log...)x, the log being repeated r times, then

  xl  x  l  x  ...l  x  2

r

1

dx is

equal to: r 1

(A) l

 x   c

(C) l r  x   c 28.

l r 1  x 

 c r 1 (D) None of these (B)

t 2 f ( x )  x 2 f ( t )  1 for t  x t  x

Let f(x) be differentiable in the interval (0, π) such that f(1) = 1 and lim each x > 0. Then f(x) is: 1 2 x 2 (A)  3x 3 1 2 (C)   2 2x x

1  3x 1 (D) x

(B)



4x 2 3



29.

The angle of elevation of the top of a tower standing on a horizontal plane from a point A is α. After walking a distance a towards the foot of the tower the angle of elevation is found to be β. The height of the tower is – a sin  sin  a sin  sin  (A) (B) sin      sin      (C)

30.

a sin      sin  sin 

(D)

Assertion (s1 ) : The equation x 6

a sin      sin  sin 

 5 x 5  3x 3  2 x 2  6  0 has at least two imaginary roots.

Reason (s2 ) : f(x) has two changes in sign and f(-x) also has two changes in sign. You have to select the correct choice. (A) Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 (C) Statement-1 is true, Statement-2 is false (D) Statement-1 is false, Statement-2 is true Space for rough work



FIITJEE -- JEE JEE (Mains) FIITJEE PHYSICS, CHEMISTRY & MATHEMATICS

JEE - MAINS 2020 CODE: SET-A ANSWER KEYS PHYSICS Q. No. 1.

Ans. B

Concept Code

CHEMISTRY Q. No. 1.

Ans. C

Concept Code

MATHEMATICS Q. No. 1.

Ans. A

2.

C

2.

B

2.

C

3.

B

3.

D

3.

B

4.

C

4.

A

4.

A

5.

A

5.

D

5.

C

6.

C

6.

A

6.

C

7.

B

7.

C

7.

C

8.

C

8.

A

8.

A

9.

D

9.

D

9.

C

10.

B

10.

D

10.

B

11.

C

11.

A

11.

C

12.

C

12.

A

12.

B

13.

A

13.

C

13.

A

14.

C

14.

D

14.

A

15.

C

15.

C

15.

C

16.

D

16.

D

16.

D

17.

B

17.

C

17.

D

18.

C

18.

D

18.

A

19.

D

19.

B

19.

C

20.

D

20.

B

20.

B

21.

B

21.

A

21.

D

22.

A

22.

A

22.

A

23.

B

23.

D

23.

B

24.

C

24.

A

24.

B

25.

D

25.

B

25.

B

26.

B

26.

C

26.

A

27.

D

27.

B

27.

A

28.

B

28.

D

28.

A

29.

A

29.

B

29.

A

30.

D

30.

C

30.

A

Concept Code



HINT AND SOLUTIONS Physics 1.



v rms



v0

v 2 dN

0

v0





v / 5 v /3

2.

 



av 2 dN

m RT and M O 2

P  2     rA 



0

av 4 dv

3 v 0 5



P  r A  



0 v 0

dN

0

5 0 3 0

 

v 0

2  

m RT M O 2 M N 2 28 MO 2 32



 2  

7  8  7  2   1     8  16   15



2

3.

Ans.





4.

 nRT  P  P0  aV  P0  a    P  P P0  P dT T  ; for T  max, 0 2

nR

d

P

dP

a

P0  P



dP

P0  P



P0  P 



P0

Ans.

dQ

 nc pdT



qdt

 n R.

Or

qdt



0 P



2 P0  P

 0

P 2

3 2

 PP

2 P0 P

5 PdV 5  mg    P0   Adx 2 nR 2 A  5 dx 2q  P0 A  mg  dx  v   2 dt 5  P0 A  mg 



5.

From first law of thermodynamics, dQ dQ ndT

 nCv dT  PdV

P dV n dT P dV nRT C  Cv  . ; now , P  aT 2  n dT V nR dV nR V     2 aT dT aT 2 5 aT  nR  5 3  RR R C R . 2  2 n  aT  2 2

 Or,

 

 C v 

6.

W   A   . a . b  5  1 m3   3  1  atm      2   2    3 5  2  atm.m  2   10 J 7.

Ans.



8.

9.

W

1 2

1 2

1 2

 kx 2  kx.x  .F. x

1  540  2  10 2  5 .4J 2

F Ans. Y  A l l Fl Fl  l   2 YA r Y 1 2 l 1  1  4r  l2 2l 8 r 2 Ans.

l1 

F1l

r Y



F2 l

 F 1 

4 r .2Y Now F1  F1  F and F1 x  F2 l  x  2

2

F2 8

 F1 x  8F1  l  x   9x  8l  x  10.

8l 9

8 9

  90cm  80cm

Ans.



Thermal expansion = Elastic compression Fl Fl  1  2 l11   l 2  2  Y1 A Y2 A

 F 

 l11  l 2 2  

 l1 l 2  1 Y  Y  A  1 2   l   l   Y Y A  F  1 1 2 2 1 2 l1Y2  l 2Y1 Faridabad (Delhi-NCR) FIITJEE Ltd., Sec-15A Market, Near Vidya Mandir Public School, Ajrounda Chowk, Faridabad, Haryana-121007: Ph-0129-4174571 Website: www.fiitjee.com


11.

Ans.

Elongation,

l  l sec   l  l  sec   1

At equilibrium, 2T sinθ = Mg  2YA  sec   1 sin   Mg Or,  tan   sin   

Mg

2YA Mg x  tan   sin   2YA 1



x

 l

2

 x

2

    x 2  2  x 3   1  1  2    3 1 l  2l    1

x

1

Mg  3  x  l    YA  12.

Ans. dx

Kinetic energy of the elemental mass 2

 ds   1 Adx [ sin kx cos t ] dK  Adx   2  dt  2 1

1 2

 2  2 Adx sin2 kx cos2  t dK max

1 2

 2 2 Adx sin2 kx dx 



1 sin 2kx  2 2 2 A  x  4 2k  0 

1  2  2 A  8 13.

14.

Ans.

Here  x







v



2  2 A

  2 r   2n  1 for destructive interference 2

4 r 2n  1 v  2n  1 v





4k

4 r

  2n  1 50Hz

Ans.

For destructive interference v   x   2n  1   2n  1 2 2v  2n  1 v v   v min  2 xmax 2 xmax



340m / s 2  2m

 85Hz



15.

 x0  d   d  2    2 , So, for maxima at P,  x p  

Ans.

P





s1



θ  2λ s2  xP

16.

O  S1 P  S2 P  2  cos   



cos  





Now, x



 x 

1 2

 3 d tan

 d tan60 o  3d

Along the axis, path difference is 3λ and along the bisector, path difference is zero. In between, there are path differences of 2λ and λ which all satify the condition of maxima. In this way, there are 12 maxima on whole circle. Between every two maxima, there is a minima. So, the number of minima is also 12

O





2λ

2λ B

o



3λ



3λ

3λ 2λ

2λ

 17.

Ans.

Speed of the pulse, v 



λ

F



100

 20ms 0.25   For the pulse to remain stationary, v PG  v PC 

v CG

18.

O





 vCG  0

 v PC  20iˆ m / s

As dQ = msdT

dQ dt

 ms

dT

From question : S

dt

T S

Or

 ms

dT dt

 K2

 m  K1T 

dT dt

 K2

 K1  T 2 m   t  K 2  2

 K1T

(K1 being proportionality constant) Also,

dQ dt

 constant  Given   K 2  say   T 

t

Hence, the graph will be: T

t



19.

By summetyry,

I AB  I BC and I AD  No current in BO and OD

 I DC

 T B  TO  T D 20.

dQ dt

  KA 1   2 

[Newton’s law of cooling]  Rate of loss of heat is same.

dQ

From

dt d

 ms

d 

dt KA      S  dt ms



Since, ‘m’ is larger for s phere, rate of cooling will be less for sphere. 21.

The rate of heat loss by a thin hollow sphere of thickness ‘  x,, mean radius ‘r’ and made of density '  ' is given by:

ms

dT dt

    A T 4  T04  dT

  4 r x  s dt     4 r T 2

2

4

 T04 

  T 4  T 04   dt s x

dT



is independent of radius. Hence rate of cooling is same f or both spheres. 22.

As work done in state (II) is more than in state(I). P B

A V

d  23.

 2;

  2t

dt Let BP  a ;

 x  OM  a sin   a sin  2t 

Hence, M executes SHM within the given time

y B

a 

P

l A o

M

x

Period and its acceleration is opposite to ‘x’ that means towards left.



24.

A = xmax

25.

Resulting torque on the bob = mg

MI of bob about axix xy  



2 g

I

l

 

T  2

26.

1 2

sin 

ml 2 2

 l

 .

2

l 2 g

mg

Let the line joining AB represents axis ‘r’. By the conditions given’r’ coordinate of the particle at time t is



r

2





2 cos  t

2

T



2 2

 

r  2 2 cos  t  r r



A 2 , 2 

0

x 2

B x  r cos 45 

  27.

r 2

 2 cos  t

  2 x   2 2cos  t 2 F x  max  4 cos  t a x

D



28. 355 Hz

5 B A  350 Hz 

wax

2

29.

351Hz 349Hz

4 345 Hz  C common 6

A '  347 *

359Hz

4 C 

A  B

 5,

A '  A,

B C

 4,

A ' C

A ' B

6

341Hz common 341Hz

C

A  350 / 347  2   B  345  C  341

For first resonance with 400 Hz tuning fork leq.



v 4f0



v 4  400 

400Hz

 19  1  20cm

If we use 1600 Hz tuning fork v 4f0





v 4  1600 

20 cm

20  5cm 4

For Resonance l eq.



v 3v 5v 7v , , , ,..... 4f0 4f0 4f0 4f0

1cm  l l

 5cm,15cm, 25cm, 35cm, 45cm.....

 4cm,14cm, 24cm, 34cm, 44cm.....

Water level should be further lowered by 24 – 19 = 5cm Or 34 – 19 = 15 cm

30.

Frequency of horn directly heard by observer

 v  v 0    f  v  v c  Frequency of echo



v v

 v c

f

Frequency of echo of horn as heard by observer v v

 v c

 v  v 0    v 

f . 

Frequency of beats:

 1 1  2v  v  v 0   v  v 0  f     2 2 f  Beat f v  v v  v c c    v  v  0



Hint and Solutions Chemistry 1.

- NO2 group increases acid strength of phenol greatly when present at ortho/para position due to resonance effect. Therefore, IV is the s trongest acid. When – NO 2 is present at meta position, it increases acid strength by only inductive effect. Therefore,II is stronger acid than I. III is weakest acid due to +I eff ect of – CH3.

2.

III is a 1,3-diketo, most acidic followed by IV. I is more acidic than II.

3.

The most acidic group (- COOH) will be neutralized.

4.

V is most stable as it is 3° as well as benzylic. III is 2nd most stable and more stable than I as both the resonance structures of III 2°. II is least stable as it is vinylic.

5.

It has resonance stabilization

6.

It is conjugated as well as trans at both double bonds

7.

Lone pair of nitrogen is not involed in resonance as well as it has less s-character than the nitrogen of pyridine; hence the former is better electron donor

8.

IV is most basic as its conjugate acid has two equivalent resonance structures. amides I, II and III, III is most basic and I is least basic.

Among the

9. O

O

10.

Lone pair is not delocalized

11.

Syn addition of D2 will take place. In chair conformation of cyclohexane, axial-equatorial position on adjacent ring carbons are cis.

12.

(racemic),

Cl, Cl

Cl (racemic): Total 5 isomers

13. +

Br 2

hv  

2,2- dimethyl butane

Br



Bromination occur selectively at carbon where more stable free radical is formed.



14.

Free radical chlorination occur selectively at benzylic carbon

15. + O3

H3C

Zn / H O

2  

O CH3

C

CH3 + CH3CH2CHO

16.

OsO4/KHSO3 in acetone gives syn vicinal diol.

17.

Syn hydroxylation of alkene. Here the alkene is symmetrical hence meso diol will be produced.

18.

Oxymercuration-demercuration reaction brings about Markownikoff’s hydration of alkene without rearrangement.

19.

CH3CH2CH3

hv  Br2   CH3  CH  CH3

Br

20. H

+

+

-H

21.

+

OH |

C6H5

 C  C  H  H   C6 H5  C  CH2 Markownikoff's additionof H2O O

tautomerisation

||

  C6H5  C  CH3 22.

H3C

C

C

CH3 Lindlar's   C catalyst H

CH3+ H2 CH3 C H

Br

2   anti additiongives racemic mixture of dibromides.



23. 24.

Lindlar’s catalyst brings about cis hydrogenation of alkyne to alkene. CH3

Na, liq.NH

3  C  C  CH3  

H

H3C C H

C

Br

CH3

2  

Anti  addition of Br2 to trans  2 butene gives meso dibromide 25.

Lack of planarity prevent it to be aromatic.

26.

I, II and IV have Huckel number of π electrons, delocalized in cyclic loop, hence aromatic

27.

I and III

28.

Nitration occur in phenoxy ring and at para position mainly, due to steric reason.

29.

Nitration occur at ortho position to less deactivating bromin

30.

Halogens are ortho/para directing but deactivating groups due to dominance of effect over resonance effect

inductive



Hint and Solutions Mathematics A

1.

u



sin x  cos x

3 

sin  x   / 4 

3:2 

1  ....  1 2 u  4 :log2 u is defined for x   ,   Rf : y

2.

 Df

 log2 u, 2  u  4 : log2 2  y  log2 4 1  y  2 : y  1, 2   Rf

C y 10

x

0

1

3

2

A slight increase or decrease in the value of the function from the value at f(2) will result in the inclusion of infinite irrational numbers and it is given that f(x) ϵ θ only possibility is that f(x) is constant in the given interval. 3.

B 2

1  x  0, f '( x )  3x 2  2 x  10  2 x 2   x  1  11  0  f(x) is monotonically decreasing in the interval 1  x  0

In

-1 In 0  x

+ + 0 π/2 π   / 2, f '( x )   sin x  0,

 f ( x) is m.d. In  / 2  x  , f '( x )  cos x  0,  f ( x) is m.d. Displaying the trend of values of the function in different intervals, we get t he adjoining graph.  f (  / 2  h )  2  f   / 2 , f   / 2  h   f   / 2 

 f  x  has a local maximum at x = π/2 4.

A

dy dx



1 1  x

3



dy 

1

  dx  x 1 2



5.

C Applying C

 C1  C2  C3 , we get

sin x  2 cos x

cos x cos x

sin x  2 cos x

sin x

cos x

sin x  2 cos x

cos x

sin x

1 cos x

 0 cos x

  sin x  2 cos x  1 sin x cos x  0 1 cos x 1

sin x

cos x

cos x

  sin x  2 cos x  0 sin x  cos x 0

0

 0

sin x  cos x

0

2

  sin x  2 cos x  sin x  cos x   0  sin x  2 cos x  0, sin x  cos x  0  tan x  2. tan x  1 . 6.

C

AB  AC  AD  AE  AF  ?

 AC  AB  BC  AB  AO  AE  AF  FE  AF  AO

...(2)

By (1) + (2)  AC  AE  AB  AF

...(3)

 2 AO

E

...(1)

F

D

C

O

Also, AD  2AO AB  AF

 AB  BO  AO  AB  AC  AD  AE  AF   AB  AF    AC  AE   AD  AO  3AO  2AO 7.

A

B

 6AO

C

1 1 m  4 2m  1 0  0 4m m 6 1 2  2m  1   1  m  4    m  4 m  6   4m  2m  1  0  4m  2  m  4  m 2  2m  24  2m 2  4m  0  7m 2  7m  22  0 2 D   7   4 7  22   49  616 2

D<0



8.

A

G  ,,   is the centroid of  ABC Equation of the plane

(0, q, 0) B

which is passing through the points A, B, and C is x y z   1 ....(1) p q r

  p  0  0  p  3   3  0  q  0  G   q  3 3  0  0  r   r  3   3  x

y

y

(0, 0, r) C

o

x A (p, 0, 0)

z

z

 By (1)    3    9.

C

2ab  8 ab  2ab  8  a  b   ab

 4 a  b 

A.M .  G.M .

a  b  2 ab ab  2 ab  ab  8 4 1   ab 2 1  min  64   32 2 10.

 ab  64

B Group A : 1, 4, 7 ..... : (n) Group B : 2, 5, 8 ..... : (n) Group C : 3, 6, 9 ..... : (n)

Select C

nos. [3n+1] from nos. [3n+2] from nos. [3n] from

Select C  2 B 1

 2

 nC2  nC1. nC 1 n  n  1 n 2  n  2n 2 3n 2  n   n.n   2

11.

2

2

C

 z 2  z  i  0  i z 2  z  i    z  i   0

Given, i z

3

1

  z  i   i z2  1  0  z  i or z2    i i

Now, z  i  z And z 2

 i 1

 i  z2   i  z2  1  z  1

Thus, in both cases z

1.



12.

B The numerator is of the form

 b 3  3ab  a  b    a  b 

a3

3

Where a = 18 and b = 7 3

Numerator  18  7   25 3 For denominator, 31  3, 3 2  9, 3 3  27 , 3 4  81, 3 5  243 6 C1  6, 6C2  15, 6 C3  20 6 C4  6C2  15, 6C5  6C1  6 , 6C6  1 denominator 3 6  6C1 3 5.21  6C2 3 4.2 2  6C3 3 3.2 3  6C4 3 2.2 4  6C5 3.2 5  6C6 2 6

This is clearly the expansion of 6

 3  2   5 6  25 3 3

 25  Numerator  Denominator  25 3

 13.

1

A Let the equation of the given circle be

x 2

Y

 y 2  2gx  2fy  c  0

Since it passes through (1, 0) and (5, 0) therefore 1 + 2g + c = 0 and 25 + 10g C = 0. On solving the two equations, we get g = -3 and C = 5.

 y 2  6x  2fy  5  0 Since it touches yaxis at (0, h), therefore h 2  2fh  5  0 So, the circle is x

2

(0,h)

Discriminant of this quadratic must be zero, i.e.

O

4f 2  20

 0 f   5  h2  2 5h  5  0

X (1,0)

(5,0)

2

  h  5   0  h  5 14.

A The coordinates of any point on the given ellipse are

x 2



3 2

2



y 2

 

2



4 2

 1 are  3 2 cos , 4 2 sin 

The slope of tangent



b2 x 1 2

a y 1



b2 .  3 2 cos  



a2 . 4 2 cos 



4 3

  cot 

Given : Slope of the tangent



4 3

4 4  3 3 4 Thus, the equation of the tangent is 1 1 x. y . x cos  y sin  2  2  1  1  3 2 4 2 3 2 4 2

 cot    or tan q  1   

Or

x 6



y 1 8

Therefore, the coordinates of A and B are (6, 0) and (0, 8) respectively.

1 2

 Area of OAB   6  8  24sq. units



15.

C

x  1 y  1 z  1   2 3 4 x  3 y  k z   1 2 1   a 1, 1,1 ; b  2, 3, 4 











 a  b    r  c  d r

c  3, k, 0  ; d 1, 2,1

These lilnes qwill intersect if the lines are coplanar     a  c , b and d are coplanar



    16.



  

2

k

[a  c, b, d ]

 1 1

2

3

4

1

2

1

2  5    k 2 k

0

 1 2   1(1)  0

 1  11

 9 / 2

k

D

f '(1)  tan  / 6  1 / 3 , f '( 2 )  tan  / 3  3 and f '(3 )  tan  / 4  1 Now

  17.

3

1

f '( x) f ''( x) dx 

1

2

f '  x  2

1 3

3

1

1  3 

3

2 f ''( x) dx

3 1  1  f '  x  2  . 1    1  3  2  3

1 43 3 3





D Required area

y=(x+1)2

1 1 1 2  2    1/2  x  1 dx   4 2  1 1  1  2      8 24  3

18.

A

19.

C

y=(x-1)2 y=1/4 O1/2 1

cos3   3 cos  1  2 4 cos2   3 3 1 cos2   or cos   4 2 4

  2n   20.

B

21.

D

 3

By the given condition, z  1

 zi

 x  1  iy  x  i  y  1 2

2

  x  1  y 2  x 2   y  1  2 x  2y  x  y  x  y  0 Faridabad (Delhi-NCR) FIITJEE Ltd., Sec-15A Market, Near Vidya Mandir Public School, Ajrounda Chowk, Faridabad, Haryana-121007: Ph-0129-4174571 Website: www.fiitjee.com


22.

A We have

5 99

24

 5 3.5 96  125 625 

 13  9  8  1  48 13 

24

2 24  13  9  8  1  24C1  48 13   24C2  48  13   ...   48  13  

24

= 8 + terms containing power of 13. Hence remainder = 8 23.

B First circle touches both axes and radius is 2 unit. Hence, centre of circle is (2, 2). Let radius of other circle be a and this circle also touches both the axes. Hence, centre of circle is (a, a). This circle touches first cirle. Hence,

2

2

a  2   a  2   a  2

On squaring both the sides, we get a2  12a  4 12  12 2  4  4  1 2  6  4 2

a 

But a > 2, therefore a = 6



12

0

 128 2

4 2 is neglected.

Hence, a = 6 + 4 2 24.

B Tangent to 3 x

2

 3  16y 2  12 at 1,  is  4

 3   12  x  4y  4  4  Solving with y 2  x  0 , we get y 2  4  4y  0 3x(1)  16 

2

 y  2  0  y  2  tangent touches y 2  x  0 at (-4, 2) 25.

B The plane intersects the axes at (20, 0, 0), (0, 15, 0) and (0, 0, 12) So the volume of tetrahedron OABC

0



0

0

1

1 20 0 6 0 15

0

1

0

1

0

0

12 1



1 6

20

0

0

15

0

0

Z C

0 0

 600

O

B

12 A X


Y


26.

A

dy  e x dx Tangent at A  c,e c  is] y

 e x 

 ec  e c  x  c  i.e., y  xec  e c  cec y

C ....(1)

Line PQ is

ec 1

 ec 1 y e   x   c  1   c  1  c  1  c  e e1  e ec y  .  x  c  1  c 1

C-1

C C+1

....(2) 2 e Solving (1) and (2) e2  2e  1  c x  c  e2  2e  1 Tangent meets the line PQ on the left of x = c. Aliter: From the graph 27.

A

 l r  loglog...log( x) l     r times

Putting t

1 2

xl ( x ) l ( x )....l r ( x )

dx

x ) . Then  l r ( x )  loglog...log(   r times

dt 

1

.

1

..

1

.

dx

loglog...log( x ) loglog...log( x) log x x     r -1



r -2

1 2

l ( x ) l ( x )....l

 r 1

1 . dx . ( x) x

dt  log t  c t  log loglog...log  ( x )  c

I  

 

 l r 1( x)  c

r times


Fiitjee phase test paper

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