## JEE Main Session 2 1st February 2023 Shift 2 Question Paper and Answer Key

JEE MAIN 1st February 2023 Shift 2

Physics

SECTION-A

IMPORTANT INSTRUCTIONS:

(1) The test is of 3 hours duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) Section-A: This section contains 20 multiple choice questions which have only one correct answer. each question carries 4 marks for correct answer and 1 mark for wrong answer.

(ii) Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of 10. The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and 1 mark for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. For three low density gases A, B, C pressure versus temperature graphs are plotted while keeping them at constant volume, as shown in the figure.

The temperature corresponding to the point ‘ K ‘ is :

(1)   −273°C

(2)   −100°C

(3)   −40°C

(4)   −373°C

Answer: (1)

2. Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : For measuring the potential difference across a resistance of 600Ω, the voltmeter with resistance 1000Ω will be preferred over voltmeter with resistance 4000Ω.

Reason R : Voltmeter with higher resistance will draw smaller current than voltmeter with lower resistance.

In the light of the above statements, choose the most appropriate answer from the options given below.

(1) Both 𝐀 and 𝐑 are correct and 𝐑 is the correct explanation of 𝐀

(2) Both 𝐀 and 𝐑 are correct but 𝐑 is not the correct explanation of 𝐀

(3) 𝐀 is not correct but 𝐑 is correct

(4) 𝐀 is correct but 𝐑 is not correct

Answer: (3)

3. Figures (a), (b), (c) and (d) show variation of force with time.

The impulse is highest in figure.

(1)   Fig (c)

(2)   Fig (b)

(3)   Fig (d)

(4)   Fig (a)

Answer: (2)

4. An electron of a hydrogen like atom, having Z = 4, jumps from 4th energy state to 2nd  energy state. The energy released in this process, will be :

(Given Rch=13.6eV)

Where R = Rydberg constant

c = Speed of light in vacuum

h = Planck’s constant

(1)   40.8eV

(2)   3.4eV

(3)   10.5eV

(4)   13.6eV

Answer: (1)

5. The ratio of average electric energy density and total average energy density of electromagnetic wave is :

(1)   3

(2)   1/2

(3)   1

(4)   2

Answer: (2)

6. Two objects A and B are placed at 15 cm and 25 cm from the pole in front of a concave mirror having radius of curvature 40 cm. The distance between images formed by the mirror is _______.

(1)   100 cm

(2)   60 cm

(3)   160 cm

(4)   40 cm

Answer: (3)

7. Equivalent resistance between the adjacent corners of a regular n-sided polygon of uniform wire of resistance R would be:

Answer: (3)

8. A Carnot engine operating between two reservoirs has efficiency 1/3. When the temperature of cold reservoir raised by x, its efficiency decreases to 1/6. The value of x, if the temperature of hot reservoir is 99°C, will be:

(1)   66 K

(2)   62 K

(3)   33 K

(4)   16.5 K

Answer: (2)

9. Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Two metallic spheres are charged to the same potential. One of them is hollow and another is solid, and both have the same radii. Solid sphere will have lower charge than the hollow one.

Reason R: Capacitance of metallic spheres depend on the radii of spheres.

In the light of the above statements, choose the correct answer from the options given below.

(1) Both 𝐀 and 𝐑 are true and 𝐑 is the correct explanation of 𝐀

(2) 𝐀 is true but 𝐑 is false

(3) 𝐀 is false but 𝐑 is true 4.

(4) Both 𝐀 and 𝐑 are true but 𝐑 is not the correct explanation of 𝐀

Answer: (3)

10. If the velocity of light c, universal gravitational constant G and Planck’s constant h are chosen as fundamental quantities. The dimensions of mass in the new system is :

(1)   [h1/2c1/2G1]

(2)   [h1/2c1/2G1/2]

(3)   [n1/2c1/2G1/2]

(4)   [h1c1G1]

Answer: (3)

11. Choose the correct statement about Zener diode :

(1) It works as a voltage regulator in forward bias and behaves like simple pn junction diode in reverse bias.

(2) It works as a voltage regulator only in forward bias.

(3) It works as a voltage regulator in both forward and reverse bias.

(4) It works as a voltage regulator in reverse bias and behaves like simple pn junction diode in forward bias.

Answer: (4)

12. The Young’s modulus of a steel wire of length 6 m and cross-sectional area 3 mm2, is 2 × 1011 N/m2. The wire is suspended from its support on a given planet. A block of mass 4 kg is attached to the free end of the wire. The acceleration due to gravity on the planet is 1/4 of its value on the earth. The elongation of wire is (Take g on the earth =10 m/s2) :

(1)   0.1 cm

(2)   0.1 mm

(3)   1 cm

(4)   1 mm

Answer: (2)

13. In an amplitude modulation, a modulating signal having amplitude of X V is superimposed with a carrier signal of amplitude Y V in first case. Then, in second case, the same modulating signal is superimposed with different carrier signal of amplitude 2Y V. The ratio of modulation index in the two cases respectively will be :

(1)   2 : 1

(2)   1 : 2

(3)   4 : 1

(4)   1 : 1

Answer: (1)

14. The threshold frequency of a metal is f0. When the light of frequency 2f0 is incident on the metal plate, the maximum velocity of photoelectrons is 𝑣1. When the frequency of incident radiation is increased to 5f0, the maximum velocity of photoelectrons emitted is 𝑣2. The ratio of 𝑣1 to 𝑣2 is:

Answer: (4)

15. A coil is placed in magnetic field such that plane of coil is perpendicular to the direction of magnetic field. The magnetic flux through a coil can be changed:

(A) By changing the magnitude of the magnetic field within the coil.

(B) By changing the area of coil within the magnetic field.

(C) By changing the angle between the direction of magnetic field and the plane of the coil.

(D) By reversing the magnetic field direction abruptly without changing its magnitude.

Choose the most appropriate answer from the options given below :

(1)   A and B only

(2)   A, B and D only

(3)   A, B and C only

(4)   A and C only

Answer: (3)

16. Choose the correct length (L) versus square of time period (T2) graph for a simple pendulum executing simple harmonic motion.

Answer: (1)

17. As shown in the figure, a long straight conductor with semicircular arc of radius π/10 m is carrying current I=3 A. The magnitude of the magnetic field. at the center O of the arc is : (The permeability of the vacuum =4π × 10−7NA−2)

(1)   1 μT

(2)   3 μT

(3)   4 μT

(4)   6 μT

Answer: (2)

18. As shown in the figure a block of mass 10 kg lying on a horizontal surface is pulled by a force F acting at an angle 30∘, with horizontal. For μs = 0.25, the block will just start to move for the value of F : [Given g = 10 ms−2]

(1)   20 N

(2)   33.3 N

(3)   25.2 N

(4)   35.7 N

Answer: (3)

19. The escape velocities of two planets A and B are in the ratio 1:2. If the ratio of their radii respectively is 1:3, then the ratio of acceleration due to gravity of planet A to the acceleration of gravity of planet B will be :

(1)   3/2

(2)   2/3

(3)   3/4

(4)   4/3

Answer: (3)

20. For a body projected at an angle with the horizontal from the ground, choose the correct statement.

(1) The vertical component of momentum is maximum at the highest point.

(2) The Kinetic Energy (K.E.) is zero at the highest point of projectile motion.

(3) The horizontal component of velocity is zero at the highest point.

(4) Gravitational potential energy is maximum at the highest point.

Answer: (4)

SECTION-B

21. A block is fastened to a horizontal spring. The block is pulled to a distance x = 10 cm from its equilibrium position (at x = 0 ) on a frictionless surface from rest. The energy of the block at x = 5 cm is 0.25 J. The spring constant of the spring is ________ Nm−1

Answer: (50)

22. A square shaped coil of area 70 cm2 having 600 turns rotates in a magnetic field of 0.4 wbm−2, about an axis which is parallel to one of the side of the coil and perpendicular to the direction of field. If the coil completes 500 revolution in a minute, the instantaneous emf when the plane of the coil is inclined at 60° with the field, will be ________ V. (Take π = 22/7)

Answer: (44)

23. As shown in the figure, in Young’s double slit experiment, a thin plate of thickness t = 10μm and refractive index μ = 1.2 is inserted infront of slit S1. The experiment is conducted in air (μ = 1) and uses a monochromatic light of wavelength λ = 500 nm. Due to the insertion of the plate, central maxima is shifted by a distance of xβ00 is the fringe-width before the insertion of the plate. The value of the x is ________.

Answer: (4)

24. Moment of inertia of a disc of mass 𝑀 and radius ‘R’ about any of its diameter is MR2/4. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, The value of x is ______.

Answer: (3)

25. For a train engine moving with speed of 20 ms−1, the driver must apply brakes at a distance of 500 m before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed √x ms−1. The value of x is ________. (Assuming same retardation is produced by brakes)

Answer: (200)

26. A cubical volume is bounded by the surfaces x = 0, x = a, y = 0, y = a, z = 0, z = a. The electric field in the region is given by Where E0 = 4 × 104 NC1 m1. If a = 2 cm, the charge contained in the cubical volume is Q × 1014 The value of Q is _______. (Take ϵ0 = 9 × 1012 C2/Nm2)

Answer: (288)

27. A force F = (5 + 3y2) acts on a particle in the 𝑦-direction, where F is in newton and y is in meter. The work done by the force during a displacement from y = 2 m to y = 5 m is ________ J.

Answer: (132)

28. The surface of water in a water tank of cross section area 750 cm2 on the top of a house is h m above the tap level. The speed of water coming out through the tap of cross section area 500 mm2 is 30 cm/s. At that instant, dh/dt is x × 103 m/s. The value of x will be _______.

Answer: (2)

1. In the given circuit, the value of is ________.

Answer: (2)

30. Nucleus A having Z = 17 and equal number of protons and neutrons has 1.2MeV binding energy per nucleon. Another nucleus B of Z = 12 has total 26 nucleons and 1.8MeV binding energy per nucleons. The difference of binding energy of B and A will be _______ MeV.

Answer: (6)

Chemistry

31. For electron gain enthalpies of the elements denoted as ΔegH, the incorrect option is :

1) Δeg H(Te) < ΔegH(PO)

(2)  2. ΔegH(Se) < ΔegH(S)

(3) ΔegH(Cl) < ΔegH(F)

(4) ΔegH(I) < ΔegH(At)

Answer: (2)

32. All structures given below are of vitamin C. Most stable of them is :

Answer: (1)

33. In figure, a straight line is given for Freundrich Adsorption (y = 3x + 2.505). The value of 1/n and log K are respectively.

(1)   0.3 and 0.7033

(2)   0.3 and log 2.505

(3)   3 and 0.7033

(4)   3 and 2.505

Answer: (4)

34. The correct order of bond enthalpy (kJmol−1) is :

(1) C – C > Si – Si > Sn – Sn > Ge − Ge

(2) C − C > Si − Si > Ge − Ge > Sn − Sn

(3) Si – Si > C – C > Sn – Sn > Ge − Ge

(4) Si – Si > C – C > Ge – Ge > Sn − Sn

Answer: (2)

35. Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : An aqueous solution of KOH when used for volumetric analysis, its concentration should be checked before the use.

Reason (R) : On aging, KOH solution absorbs atmospheric CO2.

In the light of the above statements, choose the correct answer from the options given below :

(1) Both (A) and (R) are correct but (R) is not the correct explanation of (A)

(2) (A) is correct but (R) is not correct

(3) Both (A) and (R) are correct and (R) is the correct explanation of (A)

(4) (A) is not correct but (R) is correct

Answer: (3)

36. O − O bond length in H2O2 is X than the O − O bond length in F2O2. The O − H bond length in H2O2 is Y than that of the O − F bond in F2O2.

Choose the correct option for X and Y from those given below

(1)   X-shorter, Y-longer

(2)   X-shorter, Y-shorter

(3)   X-longer, Y-shorter

(4)   X-longer, Y-longer

Answer: (3)

37. Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): Cu2+ in water is more stable than Cu+.

Reason (R) : Enthalpy of hydration for Cu2+ is much less than that of Cu+.

(1) Both (A) and (R) are correct and (R) is the correct explanation of (A)

(2) (A) is not correct but (R) is correct

(3) (A) is correct but (R) is not correct

(4) Both (A) and (R) are correct but (R) is not the correct explanation of (A)

Answer: (1)

38.

Answer: (4)

39. The complex cation which has two isomers is :

(1) [Co(NH3)5NO2]2+

(2) [Co(H2O)6]3+

(3) [Co(NH3)5Cl]+

(4) [Co(NH3)5Cl]2+

Answer: (1)

40. The graph which represents the following reaction is :

Answer: (3)

41. Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : α-halocarboxylic acid on reaction with dil NH3 gives good yield of 𝛼-amino carboxylic acid whereas the yield of amines is very low when prepared from alkyl halides.

Reason (R) : Amino acids exist in zwitter ion form in aqueous medium.

In the light of the above statements, choose the correct answer from the options given below :

(1) Both (A) and (R) are correct and (R) is the correct explanation of (A)

(2) (A) is not correct but (R) is correct

(3) Both (A) and (R) are correct but (R) is not the correct explanation of (A)

(4) (A) is correct but (R) is not correct

Answer: (1)

42. The industrial activity held least responsible for global warming is :

(1) Industrial production of urea

(2) Electricity generation in thermal power plants

(3) steel manufacturing

(4) manufacturing of cement

Answer: (1)

43. Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : Gypsum is used for making fireproof wall boards.

Reason (R): Gypsum is unstable at high temperatures.

In the light of the above statements, choose the correct answer from the options given below :

(1) Both (A) and (R) are correct and (R) is the correct explanation of (A)

(2) Both (A) and (R) are correct but (R) is not the correct explanation of (A)

(3) (A) is correct but (R) is not correct

(4) (A) is not correct but (R) is correct

Answer: (2)

44. The starting material for convenient preparation of deuterated hydrogen peroxide (D2O2) in laboratory is :

(1)   BaO

(2)   K2S2O8

(3)   BaO2

(4)   2-ethylanthraquinol

Answer: (2)

45. The effect of addition of helium gas to the following reaction in equilibrium state, is :

PCl5( g) ⇌ PCl3( g) + Cl2( g)

(1) helium will deactivate PCl5 and reaction will stop.

(2) the equilibrium will shift in the forward direction and more of Cl2 and PCl3 gases will be produced.

(3) the equilibrium will go backward due to suppression of dissociation of PCl5.

(4) addition of helium will not affect the equilibrium.

Answer: (2)

46. Which element is not present in Nessler’s reagent ?

(1)   Oxygen

(2)   Potassium

(3)   Mercury

(4)   Iodine

Answer: (1)

47. The structures of major products A,B and C in the following reaction are sequence.

Answer: (4)

48. In a reaction,

Reagents ‘X’ and ‘Y’ respectively are:

(1) (CH3CO)2O/H+ and (CH3CO)2O/H+

(2) CH3OH/H+, Δ and (CH3CO)2O/H+

(3) CH3OH/H+, Δ and CH3OH/H+, Δ

(4) (CH3CO)2O/H+ and CH3OH/H+, Δ

Answer: (4)

49. Which one of the following sets of ions represents a collection of isoelectronic species ? (Given: Atomic Number : F:9, Cl:17, Na=11, Mg=12, Al=13, K=19, Ca=20, Sc=21)

(1)   Ba2+, Sr2+, K+, Ca2+

(2)   Li+, Na+, Mg2+, Ca2+

(3)   N3, O2, F, S2

(4)   K+, Cl, Ca2+, Sc3+

Answer: (4)

50. Given below are two statements :

Statement I : Sulphanilic acid gives esterification test for carboxyl group.

Statement II : Sulphanilic acid gives red colour in Lassigne’s test for extra element detection.

In the light of the above statements, choose the most appropriate answer from the options given below:

(1) Statement I is incorrect but Statement II is correct

(2) Both Statement I and Statement II are incorrect

(3) Statement I is correct but Statement II is incorrect

(4) Both Statement I and Statement II are correct

Answer: (1)

SECTION B

51. 0.3 g of ethane undergoes combustion at 27°C in a bomb calorimeter. The temperature of calorimeter system (including the water) is found to rise by 0.5∘C. The heat evolved during combustion of ethane at constant pressure is_____________ kJmol−1. (Nearest integer)

[Given : The heat capacity of the calorimeter system is 20 kJ K1, R  = 8.3JK1 mol1.

Assume ideal gas behavior.

Atomic mass of C and H are 12 and 1 g mol−1 respectively]

Answer: (1006)

52. Among the following, the number of tranquilizer/s is/are _______

(A) Chloroliazepoxide   (B) Veronal

(C) Valium                     (D) Salvarsan

Answer: (3)

53. Among the following, the number of tranquilizer/s is/are

(A) CuCO3         (B) Cu2S          (C) Cu2O         (D) FeO

Answer: (1)

54. A metal M crystallizes into two lattices :- face centred cubic (fcc) and body centred cubic (bcc) with unit cell edge length of 2.0 and 2.5Å respectively. The ratio of densities of lattices fcc to bcc for the metal M is___________ (Nearest integer)

Answer: (4)

55. The spin only magnetic moment of [Mn(H2O)6]2+ complexes is__________ B.M. (Nearest integer) (Given: Atomic no. of Mn is 25)

Answer: (6)

56. 1 × 10−5M AgNO3 is added to 1 L of saturated solution of AgBr. The conductivity of this solution at   298 K is__________ × 10−8 S m−1

[Given : KSP(AgBr) = 4.9 × 1013 at 298 K

Answer: (14)

57. 20% of acetic acid is dissociated when its 5 g is added to 500 mL of water. The depression in freezing point of such water is___________ × 10−3°C Atomic mass of C,H and O are 12,1 and 16 a.m.u. respectively.

[Given : Molal depression constant and density of water are 1.86 K kg mol−1 and 1 g cm−3 respectively.

Answer: (372)

58. A → B

20% of acetic acid is dissociated when its 5 g is added to 500 mL of water. The depression in freezing point of such water is___________ × 10−3°C   Atomic mass of C,H and O are 12, 1 and 16 a.m.u. respectively.  [Given : Molal depression constant and density of water are 1.86 K kg mol−1 and 1 g cm−3 respectively.

Answer: (75)

59. Testosterone, which is a steroidal hormone, has the following structure.

The total number of asymmetric carbon atom /s in testosterone is___________

Answer: (6)

60. The molality of a 10%(v/v) solution of di-bromine solution in CCl4 (carbon tetrachloride) is ‘x’. x =_________ × 10−2 (Nearest integer)

[Given : molar mass of Br2 = 160 g mol1

atomic mass of C = 12 g mol1

atomic mass of Cl = 35.5 g mol1

density of dibromine = 3.2 g cm3

density of CCl4 = 1.6 g cm3]

Answer: (139)

Mathematics

SECTION-A

61. Let αx = exp(xβyγ) be the solution of the differential equation 2x2y dy – (1 – xy2) dx = 0, x > 0, y(2) = √loge 2. Then α + β + γ equals :

(1)   1

(2)   −1

(3)   3

(4)   0

Answer: (1)

62. The sum

Answer: (4)

63. Let be two vectors. Then which one of the following statements is TRUE?

(1)   Projection of and the direction of the projection vector is same as of

(2)   Projection of and the direction of the projection vector is opposite to the direction of

(3)   Projection of and the direction of the projection vector is same as of

(4)   Projection of and the direction of the projection vector is opposite to the direction of

Answer: (*)

64. Let and be three given vectors. If is a vector such that then is equal to :

(1)

(2)   √914/7

(3)

(4)   11/7

Answer: (3)

65. Let f : ℝ − 0, 1 → ℝ be a function such that Then f(2) is equal to

(1)   9/2

(2)   7/4

(3)   9/4

(4)   7/3

Answer: (3)

66. Let P(S) denote the power set of S = {1, 2, 3, ………., 10}. Define the relations R1 and R2 on P(S) as AR1B if (A ∩ BC) ∪ (B ∩ AC) = ∅ and AR2B if A ∪ BC = B ∪ AC, ∀ A, B ∈ P(S). Then :

(1) only R1 is an equivalence relation

(2) only R2 is an equivalence relation

(3) both R1  and R2 are equivalence relations

(4) both R1  and R2 are not equivalence relations

Answer: (3)

67. The area of the region given by {(x, y) : xy ≤ 8, 1 ≤ y ≤ x2} is :

Answer: (2)

68. If then :

(1)   A30 + A25 + A = I

(2)   A30 = A25

(3)   A30 + A25 –A = I

(4)   A30 – A25 = 2I

Answer: (3)

69. Which of the following statements is a tautology ?

(1)   p ⋁ (p ⋀ q)

(2)   (p ⋀ (p → q)) → ~q

(3)   (p ⋀ q) → (~(p) → q)

(4)   p → (p ⋀ (p → q))

Answer: (3)

70. The sum of the absolute maximum and minimum values of the function f (x) = |x2 – 5x + 6| − 3x + 2 in the interval [–1,3] is equal to :

(1)   12

(2)   13

(3)   10

(4)   24

Answer: (4)

71. Let the plane P pass through the intersection of the planes 2x + 3y – z = 2 and x + 2y + 3z = 6 and be perpendicular to the plane 2x + y – z = 0. If d is the distance of P form the point (–7, 1, 1,) then d2 is equal to :

(1)   250/83

(2)   250/82

(3)   15/53

(4)   25/83

Answer: (1)

72. The number of integral values of k, for which one root of the equation 2x2 – 8x + k = 0 lies in the interval (1,2) and its other root lies in the interval (2, 3), is :

(1)   3

(2)   0

(3)   2

(4)   1

Answer: (4)

73. Let P(x0, y0) be the point on the hyperbola 3x2 – 4y2 = 36, which is nearest to the line 3x + 2y = 1. Then √2(y0 – x0) is equal to:

(1)   −9

(2)   −3

(3)   3

(4)   9

Answer: (1)

74. Two dice are thrown independently. Let A be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, B be the event that the number appeared on the 1st die is even and that on the second die is odd, and C be the event that the number appeared on the 1st die is odd and that on the 2nd is even. Then :

(1) the number of favourable cases of the events A,B and C are 15,6 and 6 respectively

(2) the number of favourable cases of the event (A ∪ B) ∩ C is 6

(3) B and C are independent

(4) A and B are mutually exclusive

Answer: (2)

75. If y(x) = xx, x > 0, then y”(2) – 2yʹ(2) is equal to :

(1)   4loge 2 + 2

(2)   8loge 2 – 2

(3)   4(loge 2)2 + 2

(4)   4(loge 2)2 – 2

Answer: (4)

76. Let

If n(S) denotes the number of elements in S then :

(1)   n(S) = 2 and only one element in S is less then 1/2.

(2)   n(S) = 1 and the element in S is more than 1/2.

(3)   n(S) = 0

(4)   n(S) = 1 and the element in S is less than 1/2.

Answer: (1)

77. The value of the integral is :

(1)   π2/12√3

(2)   π2/6√3

(3)   π2/6

(4)   π2/3√3

Answer: (2)

78. For the system of linear equations αx + y + z = 1, x + αy + z = 1, x + y + αz = β, which one of the following statements is NOT correct?

(1)   It has infinitely many solutions if α = 2 and β = −1

(2)   It has no solution if α = −2 and β = 1

(3)   if α = 2 and β = 1

(4)   It has infinitely many solutions if α = 1 and β = 1

Answer: (1)

79. Let 9 = x1< x2 < ….. < x7 …….., x7 be in an A.P. with common difference d. If the standard deviation of x1 ∙ x2 ………, x7 is 4 and mean is is equal to

(1)

(2)

(3)   25

(4)   34

Answer: (4)

80. Let a, b be two real numbers such that ab < 0. IF the complex number is of unit modulus and a + ib lies on the circle |z – 1| = |2z|, then a possible value of where [t] is greatest integer function, is :

(1)   −1/2

(2)   −1

(3)   1

(4)   1/2

Answer: (*)

SECTION-B

81. Let αx + βy + yz = 1 be the equation of a plane through the point (3, –2, 5)and perpendicular to the line joning the points (1, 2, 3) and (–2, 3, 5). Then the value of αβy is equal to

Answer: (6)

82. If the term without x in the expansion of is 7315, then |α| is equal to

Answer: (1)

83. If the x – intercept of a focal chord of the parabola y2 = 8x + 4y + 4 is 3, then the length of this chord is equal to

Answer: (16)

84. Let the sixth term in the binomial expansion of in the increasing powers of be 21. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of A.P., then the sum of the squares of all possible values of x is

Answer: (4)

85. The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the point A(–3, –61) and B(2, –4, –3) divides the line segment AB internally in the ratio k:. If a, b, c (|a|, |b|, |c|) are coprime are the direction ratios of the perpendicular form the point C on the line then |a + b + c| is equal to

Answer: (10)

86. The line x = 8 is the directrix of the ellipse with the corresponding focus (2, 0). If the tangent to E at the point P in the first quadrant passes through the point (0, 4√3) and intersects that x-axis at Q then (3PQ)2 equal to

Answer: (39)

87. The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6 , is

Answer: (81)

88. Number of integral solutions to the equation x + y + z = 21, where x ≥ 1, y ≥ 3,  z ≥ 4, is equal to

Answer: (105)

89. The sum of the common terms of the following three arithmetic progressions.

3, 7, 11, 15, ……., 399

2, 5, 8, 11, ……., 359 and

2, 7, 12, 17, …….., 197

is equal to

Answer: (321)

90. If

Then k is equal to

Answer: (13)

## JEE Main Session 2 31st January 2023 Shift 2 Question Paper and Answer Key

JEE MAIN 31st January 2023 Shift 2

Physics

SECTION-A

IMPORTANT INSTRUCTIONS:

(1) The test is of 3 hours duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) Section-A: This section contains 20 multiple choice questions which have only one correct answer. each question carries 4 marks for correct answer and 1 mark for wrong answer.

(ii) Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of 10. The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and 1 mark for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. Given below are two statements:

Statement I: In a typical transistor, all three regions emitter, base and collector have same doping level.

Statement II: In a transistor, collector is the thickest and base is the thinnest segment.  In the light of the above statements, choose the most appropriate answer from the options given below.

(1) Both Statement I and Statement II are correct

(2) Statement I is incorrect but Statement II is correct

(3) Statement I is correct but Statement II is incorrect

(4) Both Statement I and Statement II are incorrect

Answer: (2)

2. If the two metals A and B are exposed to radiation of wavelength 350 nm. The work functions of metals A and B are 4.8eV and 2.2eV. Then choose the correct option.

(1) Both metals A and B will emit photo-electrons

(2) Metal A will not emit photo-electrons

(3) Metal B will not emit photo-electrons

(4) Both metals A and B will not emit photo-electrons

Answer: (2)

3. Heat energy of 735 J is given to a diatomic gas allowing the gas to expand at constant pressure. Each gas molecule rotates around an internal axis but do not oscillate. The increase in the internal energy of the gas will be :

(1)   525 J

(2)   441 J

(3)   572 J

(4)   735 J

Answer: (1)

4. Match List I with List II

Choose the correct answer from the options given below:

(1) A – III, B – I, C – IV, D – II

(2) A – II, B – III, C – IV, D – I

(3) A – IV, B – II, C – I, D – III

(4) A – I, B – IV, C – III, D – II

Answer: (1)

5. A stone of mass 1 kg is tied to end of a massless string of length 1 m. If the breaking tension of the string is 400 N, then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is :

(1)   40 ms1

(2)   400 ms1

(3)   20 ms1

(4)   10 ms1

Answer: (3)

6. A microscope is focused on an object at the bottom of a bucket. If liquid with refractive index 5/3 is poured inside the bucket, then microscope have to be raised by 30 cm to focus the object again. The height of the liquid in the bucket is :

(1)   12 cm

(2)   50 cm

(3)   18 cm

(4)   75 cm

Answer: (4)

7. The number of turns of the coil of a moving coil galvanometer is increased in order to increase current sensitivity by 50%. The percentage change in voltage sensitivity of the galvanometer will be :

(1)   0%

(2)   75%

(3)   50%

(4)   100%

Answer: (1)

8. A body is moving with constant speed, in a circle of radius 10 m. The body completes one revolution in 4s. At the end of 3rd second, the displacement of body (in m) from its starting point is:

(1)   15π

(2)   10√2

(3)   30

(4)   5π

Answer: (2)

9. The H amount of thermal energy is developed by a resistor in 10 s when a current of 4 A is passed through it. If the current is increased to 16 A, the thermal energy developed by the resistor in 10 s will be :

(1)   H/4

(2)   16H

(3)   4H

(4)   H

Answer: (2)

10. A long conducting wire having a current I flowing through it, is bent into a circular coil of N turns. Then it is bent into a circular coil of n turns. The magnetic field is calculated at the centre of coils in both the cases. The ratio of the magnetic field in first case to that of second case is:

(1)   n: N

(2)   n2 : N2

(3)   N2 : n2

(4)   N : n

Answer: (3)

11. A body weight W, is projected vertically upwards from earth’s surface to reach a height above the earth which is equal to nine times the radius of earth. The weight of the body at that height will be :

(1)   W/100

(2)   W/91

(3)   W/3

(4)   W/9

Answer: (1)

12. The radius of electron’s second stationary orbit in Bohr’s atom is R. The radius of 3rd orbit will be

(1)   R/3

(2)   3R

(3)   2.25R

(4)   9R

Answer: (3)

13. A hypothetical gas expands adiabatically such that its volume changes from 08 litres to 27 litres. If the ratio of final pressure of the gas to initial pressure of the gas is 16/81. Then ratio of cp/cv will be

(1)   1/2

(2)   4/3

(3)   3/2

(4)   3/1

Answer: (2)

14. For a solid rod, the Young’s modulus of elasticity is 3.2 × 1011Nm−2 and density is 8 × 103 kg m−3. The velocity of longitudinal wave in the rod will be.

(1)   145.75 × 103 ms1

(2)   18.96 × 103 ms1

(3)   3.65 × 103 ms1

(4)   6.32 × 103 ms1

Answer: (4)

15. A body of mass 10 kg is moving with an initial speed of 20 m/s. The body stops after 5 s due to friction between body and the floor. The value of the coefficient of friction is: (Take acceleration due to gravity g = 10 ms−2)

(1)   0.3

(2)   0.5

(3)   0.2

(4)   0.4

Answer: (4)

16. Given below are two statements :

Statement I : For transmitting a signal, size of antenna (l) should be comparable to wavelength of signal (at least l = λ/4 in dimension)

Statement II : In amplitude modulation, amplitude of carrier wave remains constant (unchanged).

In the light of the above statements, choose the most appropriate answer from the options given below.

(1) Statement 𝐈 is correct but Statement II is incorrect

(2) Both Statement I and Statement II are correct

(3) Statement I is incorrect but Statement II is correct

(4) Both Statement I and Statement II are incorrect

Answer: (1)

17. An alternating voltage source V=260sin⁡(628t) is connected across a pure inductor of 5mH. Inductive reactance in the circuit is :

(1)   0.318Ω

(2)   6.28Ω

(3)   3.14Ω

(4)   0.5Ω

Answer: (3)

18. Under the same load, wire A having length 5.0 m and cross section 2.5 × 10−5 m2 stretches uniformly by the same amount as another wire B of length 6.0 m and a cross section of 3.0 × 10−5 m2 The ratio of the Young’s modulus of wire A to that of wire B will be :

(1)   1 : 1

(2)   1 : 10

(3)   1 : 2

(4)   1 : 4

Answer: (1)

19. Match List I with List II

Choose the correct answer from the options given below:

(1) A−IV,B – III, C – I, D – II

(2) A−IV,B−I,C – II, D – III

(3) A – III, B – II, C – I, D – IV

(4) A – II, B – IV, C – III, D – I

Answer: (1)

20. Considering a group of positive charges, which of the following statements is correct?

(1) Both the net potential and the net electric field cannot be zero at a point.

(2) Net potential of the system at a point can be zero but net electric field can’t be zero at that point.

(3) Net potential of the system cannot be zero at a point but net electric field can be zero at that point.

(4) Both the net potential and the net field can be zero at a point.

Answer: (3)

SECTION-B

21. A series LCR circuit consists of R = 80Ω, XL = 100Ω, and XC = 40Ω. The input voltage is 2500 cos⁡(100πt)V. The amplitude of current, in the circuit, is _____A.

Answer: (25)

22. Two bodies are projected from ground with same speeds 40 ms−1 at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of 60°, with horizontal then sum of the maximum heights, attained by the two projectiles, is _____m. (Given g = 10 ms−2)

Answer: (80)

23. For the given circuit, in the steady state, |VB – VD| = ________ V.

Answer: (1)

24. Two parallel plate capacitors C1 and C2 each having capacitance of 10μF are individually charged by a 100 V D.C. source. Capacitor C1 is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor C2 is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor C1 is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be ______V. (Assuming Dielectric constant =10)

Answer: (55)

25. Two light waves of wavelengths 800 and 600 nm are used in Young’s double slit experiment to obtain interference fringes on a screen placed 7 m away from plane of slits. If the two slits are separated by 0.35 mm, then shortest distance from the central bright maximum to the point where the bright fringes of the two wavelength coincide will be ______ mm.

Answer: (48)

26. A ball is dropped from a height of 20 m. If the coefficient of restitution for the collision between ball and floor is 0.5, after hitting the floor, the ball rebounds to a height of _____ m

Answer: (5)

27. If the binding energy of ground state electron in a hydrogen atom is 13.6eV, then, the energy required to remove the electron from the second excited state of Li2+ will be : x × 10−1 The value of x is ____.

Answer: (136)

28. A water heater of power 2000 W is used to heat water. The specific heat capacity of water is 4200 J kg−1 K−1. The efficiency of heater is 70%. Time required to heat 2 kg of water from 10∘C to 60°C is _____s. (Assume that the specific heat capacity of water remains constant over the temperature range of the water).

Answer: (300)

29. Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of materials are in the ratio 3:5. The moment of inertia of these discs respectively about their diameters will be in the ratio of x/6. The value of x is ______.

Answer: (5)

30. The displacement equations of two interfering waves are given by  y2 = 5[sin ωt + √3 cos ωt]cm respectively. The amplitude of the resultant wave is ________ cm.

Answer: (20)

Chemistry

SECTION-A

31. Which one of the following statements is incorrect ?

(1) van Arkel method is used to purify tungsten.

(2) The malleable iron is prepared from cast iron by oxidising impurities in a reverberatory furnace.

(3) Cast iron is obtained by melting pig iron with scrap iron and coke using hot air blast.

(4) Boron and Indium can be purified by zone refining method.

Answer: (1)

32. Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : The first ionization enthalpy of 3 d series elements is more than that of group 2 metals

Reason (R) : In 3d series of elements successive filling of d-orbitals takes place.

In the light of the above statements, choose the correct answer from the options given below :

(1) Both (A) and (R) are true but (R) is not the correct explanation of (A)

(2) Both (A) and (R) are true and (R) is the correct explanation of (A)

(3) (A) is true but (R) is false

(4) (A) is false but (R) is true

Answer: (2)

33. Given below are two statements :

Statement I : H2O2 is used in the synthesis of Cephalosporin

Statement II : H2O2 is used for the restoration of aerobic conditions to sewage wastes.

In the light of the above statements, choose the most appropriate answer from the options given below:

(1) Both Statement I and Statement II are incorrect

(2) Statement I is incorrect but Statement II is correct

(3) Statement I is correct but Statement II is incorrect

(4) Both Statement I and Statement II are correct

Answer: (4)

34. A hydrocarbon ‘X’ with formula C6H8 uses two moles H2 on catalytic hydrogenation of its one mole. On ozonolysis, ‘X’ yields two moles of methane dicarbaldehyde. The hydrocarbon ‘X’ is :

(1) cyclohexa-1, 4-diene

(2) cyclohexa – 1, 3 – diene

(3) 1-methylcyclopenta-1, 4-diene

(4) hexa-1, 3, 5-triene

Answer: (1)

35. Evaluate the following statements for their correctness.

(A) The elevation in boiling point temperature of water will be same for 0.1MNaCl and 0.1M urea.

(B) Azeotropic mixtures boil without change in their composition.

(C) Osmosis always takes place from hypertonic to hypotonic solution.

(D) The density of 32% H2SO4 solution having molarity 4.09M is approximately 1.26 g mL−1

(E) A negatively charged sol is obtained when KI solution is added to silver nitrate solution.

Choose the correct answer from the options given below :

(1)   A, B and D only

(2)   B and D only

(3)   B, D and E only

(4)   A and C only

Answer: (2)

36. The Lewis acid character of boron tri halides follows the order :

(1) BI3 > BBr3 > BCl3 > BF3

(2) BBr3 > BI3 > BCl3 > BF3

(3) BCl3 > BF3 > BBr3 > BI3

(4) BF3 > BCl3 > BBr3 > BI3

Answer: (1)

37. When a hydrocarbon A undergoes complete combustion it requires 11 equivalents of oxygen and produces 4 equivalents of water. What is the molecular formula of A ?

(1)   C5H8

(2)   C11H4

(3)   C9H8

(4)   C11H8

Answer: (3)

38. Arrange the following orbitals in decreasing order of energy.

(A) n = 3, l = 0, m = 0

(B) n = 4, l = 0, m = 0

(C) n = 3, l = 1, m = 0

(D) n = 3, l = 2, m = 1

The correct option for the order is :

(1) D > B > C > A

(2)D > B > A > C

(3)A > C > B > D

(4) B > D > C > A

Answer: (1)

39. The element playing significant role in neuromuscular function and interneuronal transmission is :

(1)   Li

(2)   Mg

(3)   Be

(4)   Ca

Answer: (4)

40. Given below are two statements :

Statement I : Upon heating a borax bead dipped in cupric sulphate in a luminous flame, the colour of the bead becomes green

Statement II : The green colour observed is due to the formation of copper(I) metaborate

In the light of the above statements, choose the most appropriate answer from the options given below:

(1) Both Statement I and Statement II are true

(2) Statement I is true but Statement II is false

(3) Statement 𝐈 is false but Statement II is true

(4) Both Statement I and Statement II are false

Answer: (4)

41. Which of the following compounds are not used as disinfectants ?

(A) Chloroxylenol          (B) Bithional

(C) Veronal                    (D) Prontosil

(E) Terpineol

Choose the correct answer from the options given below :

(1)   C, D

(2)   B, D, E

(3)   A, B

(4)   A, B E

Answer: (1)

42. Incorrect statement for the use of indicators in acid-base titration is :

(1) Methyl orange may be used for a weak acid vs weak base titration.

(2) Phenolphthalein is a suitable indicator for a weak acid vs strong base titration.

(3) Methyl orange is a suitable indicator for a strong acid vs weak base titration.

(4) Phenolphthalein may be used for a strong acid vs strong base titration.

Answer: (1)

43. An organic compound [A](C4H11N), shows optical activity and gives N2 gas on treatment with HNO2. The compound [A] reacts with PhSO2Cl producing a compound which is soluble in KOH.

Answer: (4)

44. The normal rain water is slightly acidic and its pH value is 5.6 because of which one of the following?

(1) CO2 + H2O → H2CO3

(2) 2SO2 + O2 + 2H2O → 2H2SO4

(3) 4NO2 + O2 + 2H2O → 4HNO3

(4) N2O5 + H2O → 2HNO3

Answer: (1)

45. Match List I with List II

Choose the correct answer from the options given below:

(1) A – II, B – I, C – IV, D – III

(2) A – IV, B – II, C – III, D – I

(3) A – II, B – III, C – I, D – IV

(4) A – III, B – IV, C – I, D – II

Answer: (1)

46. Cyclohexylamine when treated with nitrous acid yields (P).On treating (P) with PCC results in (Q). When (Q) is heated with dil. NaOH we get (R) The final product (R) is :

Answer: (2)

47. In the following halogenated organic compounds the one with maximum number of chlorine atoms in its structure is :

(1) Freon-12

(2) Gammaxene

(3) Chloropicrin

(4) Chloral

Answer: (2)

48. In Dumas method for the estimation of N2, the sample is heated with copper oxide and the gas evolved is passed over :

(1)   Copper oxide

(2)   Ni

(3)   Pd

(4)   Copper gauze

Answer: (2)

49. Which of the following elements have half-filled f-orbitals in their ground state ? (Given : atomic number Sm = 62; Eu = 63; Tb = 65; Gd = 64, Pm = 61 )

(A) Sm   (B) Eu (C) Tb  (D) Gd                        (E) Pm

Choose the correct answer from the options given below:

(1)   A and B only

(2)   A and E only

(3)   C and D only

(4)   B and D only

Answer: (4)

50. Compound A, C5H10O5, given a tetraacetate with AC2O and oxidation of A with Br2−H2O gives an acid, C5H10O6 .Reduction of A with HI gives isopentane. The possible structure of A is :

Answer: (3)

SECTION B

51. The rate constant for a first order reaction is 20 min1. The time required for the initial concentration of the reactant to reduce to its 1/32 level is ______ 102 (Nearest integer)

Given : ln 10 = 2.303, log 2 = 0.3010)

Answer: (17)

52. Enthalpies of formation of CCl4( g), H2O(g), CO2(g) and HCl(g) are −105, −242, −394 and − 92 kJ mol−1 The magnitude of enthalpy of the reaction given below is kJmol−1. (nearest integer)

CCl4(g) + 2H2O(g) → CO2(g) + 4HCl(g)

Answer: (173)

53. A sample of a metal oxide has formula M83O1.00. The metal M can exist in two oxidation states + 2 and +3.In the sample of M0.83O1.00, the percentage of metal ions existing in + 2 oxidation state is %. (nearest integer)

Answer: (59)

54. The resistivity of a 0.8M solution of an electrolyte is 5 × 10−3 Ω cm. Its molar conductivity is ______ × 104Ω−1 cm2 mol−1 (Nearest integer)

Answer: (25)

55. At 298 K, the solubility of silver chloride in water is 1.434 × 10−3 g L−1.The value of −log Ksp for silver chloride is ____ (Given mass of Ag is 107.9 g mol−1 and mass of Cl is 35.5 g mol−1)

Answer: (10)

56. If the CFSE of [Ti(H2O)6]3+ is −96.0 kJ/mol, this complex will absorb maximum at wavelength _____ nm. (nearest integer)

Assume Planck’s constant (h) = 6.4 × 1034 Js, Speed of light (c) = 3.0 × 108 m/s and Avogadro’s Constant (NA) = 6 × 1023/mol

Answer: (480)

57. The number of alkali metal(s), from Li, K, Cs, Rb having ionization enthalpy greater than 400 kJ mol−1 and forming stable super oxide is _____

Answer: (2)

58. The number of molecules which gives haloform test among the following molecules is

Answer: (3)

59. Assume carbon burns according to following equation :

2C(g)  + O2(g) → 2CO(g)

When 12 g carbon is burnt in 48 g of oxygen, the volume of carbon monoxide produced is × 101 L at STP [nearest integer]

[Given : Assume co as ideal gas, Mass of c is 12 g mol1, Mass of O is 16 g mol1 and molar volume of an ideal gas STP is 22.7 L mol1]

Answer: (227)

60. Amongst the following, the number of species having the linear shape is

Answer: (5)

Mathematics

SECTION-A

61. The equation e4x + 8e3x + 13e2x − 8ex + 1 = 0, x ∈ ℝ has :

(1) four solutions two of which are negative

(2) two solutions and only one of them is negative

(3) two solutions and both are negative

(4) no solution

Answer: (3)

62. Among the relations

and T = {(a, b): a, b ∈ ℝ, a2 – b2 ∈ ℤ},

(1)   neither S nor T is transitive

(2)   S is transitive but T is not

(3)   T is symmetric but S is not

(4)   both S and T are symmetric

Answer: (3)

63. Let α > 0. If  then α is equal to :

(1)   4

(2)   2√2

(3)   √2

(4)   2

Answer: (3)

64. The complex number  is equal to :

Answer: (3)

65. Let y = y(x) be the solution of the differential equation (3y – 5x)y dx + 2x(x − y)dy = 0 such that y(1) = 1. Then |(y(2)) – 12y(2)| is equal to :

(1)   16√2

(2)   32√2

(3)   32

(4)   64

Answer: (2)

66.

(1)   does not exist

(2)   is equal to 27

(3)   is equal to 27/2

(4)   is equal to 9

Answer: (2)

67. The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the points A,B,C is (2, a ,4), a ∈ If the volume of the tetrahedron OABC is 144 unit, then which of the following points is NOT on P ?

(1)   (0, 6, 3)

(2)   (0, 4, 4)

(3)   (2, 2, 4)

(4)   (3, 0, 4)

Answer: (4)

68. Let (a, b) ⊂ (0, 2π) be the largest interval for which sin (sin θ)− cos1(sin θ) > 0, θ ∈ (0, 2π), holds. If αx + βx + sin (x – 6x + 10) + cos1(x – 6x + 10) = 0 and α – β = b − a, then α is equal to :

(1)   π/16

(2)   π/48

(3)   π/12

(4)   π/8

Answer: (3)

69. Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and α ( > 0 ), and the mean and standard deviation of marks of class B of n students be respectively 55 and 30 −α. If the mean and variance of the marks of the combined class of 100 + n students are respectively 50 and 350, then the sum of variances of classes A and B is :

(1)   650

(2)   450

(3)   900

(4)   500

Answer: (4)

70. The absolute minimum value, of the function f(x) = |x2 – x + 1| + [x2 – x + 1], where [t] denotes the greatest integer function, in the interval [−1, 2], is

(1)   1/4

(2)   3/2

(3)   5/4

(4)   3/4

Answer: (4)

71. Let H be the hyperbola, whose foci are (1 ± √2, 0) and eccentricity is √2. Then the length of its latus rectum is

(1)   3/2

(2)   2

(3)   3

(4)   5/2

Answer: (2)

72. Let a1, a2, a3, … be an A.P. If a7 = 3, the product a1a4 is minimum and the sum of its first n terms is zero, then n! – 4an(n+2) is equal to :

(1)   9

(2)   33/4

(3)   381/4

(4)   24

Answer: (4)

73. If a point P(α, β, γ) satisfying

lies on the plane 2x + 4y + 3z = 5, then 6α + 9β + 7γ is equal to :

(1)   −1

(2)   11/5

(3)   5/4

(4)   11

Answer: (4)

74. Let :  and  be there vectors. If  is a vector such that,  then  is equal to

(1)   560

(2)   449

(3)   339

(4)   336

Answer: (3)

75. Let the plane P : 8x + α1y + α2y + α2z + 12 = 0 be parallel to the line  If the intercept of P on the y-axis is 1, then the distance between P and L is :

(1)

(2)

(3)   6/√14

(4)   √14

Answer: (4)

76. Let P be the plane, passing through the point (1, −1, −5) and perpendicular to the line joining the points (4, 1, −3) and (2, 4, 3). Then the distance of P from the point (3, −2, 2) is

(1)   5

(2)   4

(3)   7

(4)   6

Answer: (1)

77. The number of values of r ∈ {p, q, ~p, ~q} for which ((p ⋀ q) ⇒ (r ⋁ q)) ⋀ ((p ⋀ r) ⇒ q) is a tautology, is:

(1)   3

(2)   4

(3)   1

(4)   2

Answer: (4)

78. The set of all values of a2 for which the line x + y = 0 bisects two distinct chords drawn from a point  on the circle 2x2 + 2y2 – (1 + a)x – (1 – a)y = 0, is equal to:

(1)   (0, 4]

(2)   (4, ∞)

(3)   (2, 12]

(4)   (8, ∞)

Answer: (4)

79. If  x > 0, then ∅ʹ(π/4) is equal to:

Answer: (1)

80. Let f : ℝ − {2, 6} → ℝ be real valued function defined as  Then range of f is

Answer: (4)

SECTION-B

81. Let A = [aij], aij ∈ Z ∩ [0, 4], 1 ≤ i, j ≤ The number of matrices A such that the sum of all entries is a prime number p ∈ (2, 13) is

Answer: (204)

82. Let A be a n × n matrix such that |A| = 2. If the determinant of the matrix Adj(2 ∙ Adj(2 A1)) ∙ is 284, then n is equal to

Answer: (84)

83. If the constant term in the binomial expansion of  is −84 and the coefficient of x3l is 2αβ, where β < 0 is an odd number, then |αl – β| is equal to

Answer: (98)

84. Let S be the set of all a ∈ N such that the area of the triangle formed by the tangent at the point P(b, c), b, c ∈ ℕ, on the parabola y = 2ax and the lines x = b, y = 0 is 16 unit2, then  is equal to

Answer: (146)

85. Let the area of the region {(x, y) : |2x – 1| ≤ y ≤ |x2 – x|, 0 ≤ x ≤ 1} be A. Then (6A + 11)2 is equal to

Answer: (125)

86. The coefficient of x6, in the expansion of  is

Answer: (5040)

87. Let A be the event that the absolute difference between two randomly choosen real numbers in the sample space [0, 60] is less than or equal to a . If  then a is equal to

Answer: (10)

88. If 2n+1Pn1 : 2n1Pn = 11 : 21, then n2 + n + 15 is equal to :

Answer: (45)

89. Let  be three vectors such that  and  If the angle between  is equal to

Answer: (3)

90. The sum 12 – 2 ∙ 32 + 3 ∙ 52 – 4 ∙ 72 + 5 ∙ 92 − … + 15 ∙ 292 is

Answer: (6952)

## JEE Main Session 2 30th January 2023 Shift 2 Question Paper and Answer Key

JEE MAIN 30th January 2023 Shift 2

Physics

SECTION-A

IMPORTANT INSTRUCTIONS:

(1) The test is of 3 hours duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) Section-A: This section contains 20 multiple choice questions which have only one correct answer. each question carries 4 marks for correct answer and 1 mark for wrong answer.

(ii) Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of 10. The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and 1 mark for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. A current carrying rectangular loop PQRS is made of uniform wire. The length PR = QS =5 cm and = RS =100 cm. If ammeter current reading changes from I to 2I, the ratio of magnetic forces per unit length on the wire PQ due to wire RS in the two cases respectively  is :

(1)   1 : 2

(2)   1 : 3

(3)   1 : 4

(4)   1 : 5

Answer: (3)

2. The output Y for the inputs A and B of circuit is given by

Truth table of the shown circuit is:

Answer: (3)

3. Given below are two statements: one is labelled as Assertion 𝐀 and the other is labelled as Reason R

Assertion A: Efficiency of a reversible heat engine will be highest at −273∘C temperature of cold reservoir.

Reason R: The efficiency of Carnot’s engine depends not only on temperature of cold reservoir but it depends on the temperature of hot reservoir too and is given as

In the light of the above statements, choose the correct answer from the options given below

(1) Both 𝐀 and 𝐑 are true but 𝐑 is NOT the correct explanation of 𝐀

(2) Both 𝐀 and 𝐑 are true and 𝐑 is the correct explanation of 𝐀

(3) A is false but 𝐑 is true

(4)  A is true but 𝐑 is false

Answer: (2)

4. As shown in the figure, a point charge Q is placed at the centre of conducting spherical shell of inner radius a and outer radius b. The electric field due to charge Q in three different regions I, II and III is given by: (I: r < a, II: a < r < b, III: r > b)

(1)   EI = 0, EII = 0, E­III = 0

(2)   EI = 0, EII = 0, E­III ≠ 0

(3)   EI ≠ 0, EII = 0, E­III ≠ 0

(4)   EI ≠ 0, EII = 0, E­III = 0

Answer: (3)

5. The equivalent resistance between A and B is

Answer: (4)

6. A vehicle travels 4 km with speed of 3 km/h and another 4 km with sped of 5 km/h, then its average speed is

(1)   3.50 km/h

(2)   4.25 km/h

(3)   4.00 km/h

(4)   3.75 km/h

Answer: (4)

7. In the given circuit, rms value of current (Irms) through the resistor R is:

(1)   2√2A

(2)   2 A

(3)   20 A

(4)

Answer: (2)

8. A point source of 100 W emits light with 5% efficiency. At a distance of 5 m from the source, the intensity produced by the electric field component is:

Answer: (4)

9. A block of √3 kg is attached to a string whose other end is attached to the wall. An unknown force F is applied so that the string makes an angle of 30° with the wall. The tension T is: (Given g = 10 ms−2)

(1)   20 N

(2)   10 N

(3)   15 N

(4)   25 N

Answer: (1)

10. Match List I with List II

Choose the correct answer from the options given below:

(1)  A-IV, B-III, C-I, D-II

(2) A-I, B-II, C-III, D-IV

(3) A-IV, B-III, C-II, D-I

(4) A-II, B-III, C-IV, D-I

Answer: (3)

11. An electron accelerated through a potential difference V1 has a de-Broglie wavelength of 𝜆. When the potential is changed to V2, its de-Broglie wavelength increases by 50%. The value of (V1/V2) is equal to

(1)   3

(2)   3/2

(3)   4

(4)   9/4

Answer: (4)

12. A flask contains hydrogen and oxygen in the ratio of 2:1 by mass at temperature 27°C. The ratio of average kinetic energy per molecule of hydrogen and oxygen respectively is:

(1)   2 : 1

(2)   1 : 1

(3)   1 : 4

(4)   4 : 1

Answer: (2)

13. As shown in the figure, a current of 2 A flowing in an equilateral triangle of side 4√3 cm. The magnetic field at the centroid O of the triangle is

(Neglect the effect of earth’s magnetic field)

(1)   1.4√3 × 105 T

(2)   4√3 × 104 T

(3)   3√3 × 105 T

(4)   √3 × 104 T

Answer: (3)

14. An object is allowed to fall from a height R above the earth, where R is the radius of earth. Its velocity when it strikes the earth’s surface, ignoring air resistance, will be

Answer: (4)

15. Match List I with List II:

Choose the correct answer from the options given below:

(1) A−IV,B−I,C−III,D−II

(2) A−IV,B−III,C−I,D−II

(3) A−IV,B−I,C−II,D−III

(4) A−I,B−IV,C−III,D−II

Answer: (1)

16. Given below are two statements: one is labelled as Assertion 𝐀 and the other is labelled as Reason 𝐑

Assertion A: The nuclear density of nuclides  can be arranged as

Reason R: The radius R of nucleus is related to its mass number A as R = R0A1/3, where R0 is a constant.

In the light of the above statements, choose the correct answer from the options given below

(1) A is false but 𝐑 is true

(2) A is true but 𝐑 is false

(3) Both 𝐀 and 𝐑 are true but 𝐑 is NOT the correct explanation of 𝐀

(4) Both 𝐀 and 𝐑 are true and 𝐑 is the correct explanation of 𝐀

Answer: (1)

17. A force is applied to a steel wire ‘A’, rigidly clamped at one end. As a result elongation in the wire is 0.2 mm. If same force is applied to another steel wire ‘B’ of double the length and a diameter 2.4 times that of the wire ‘A’, the elongation in the wire ‘ B ‘ will be (wires having uniform circular cross sections)

(1) 6.06 × 10−2 mm

(2) 2.77 × 10−2 mm

(3) 3.0 × 10−2 mm

(4) 6.9 × 10−2 mm

Answer: (4)

18. A thin prism, P1 with an angle 6th and made of glass of refractive index 1.54 is combined with another prism P2 made from glass of refractive index 1.72 to produce dispersion without average deviation. The angle of prism P2 is

(1)   1.3°

(2)   6°

(3)   4.5°

(4)   7.8°

Answer: (3)

19. A machine gun of mass 10 kg fires 20 g bullets at the rate of 180 bullets per minute with a speed of 100 ms−1 The recoil velocity of the gun is

(1)   1.5 m/s

(2)   0.6 m/s

(3)   2.5 m/s

(4)   0.02 m/s

Answer: (2)

20. For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is 1 kg, the angular frequency is ω1. When the mass block is 2 kg the angular frequency is ω2. The ratio ω2/ ω1 is

(1)   1/√2

(2)   √2

(3)   2

(4)   1/2

Answer: (1)

SECTION-B

21. A uniform disc of mass 0.5 kg and radius r is projected with velocity 18 m/s at t = 0 s on a rough horizontal surface. It starts off with a purely sliding motion at t = 0 s. After 2 s it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after 2 s will be _______ 𝐉 (given, coefficient of friction is 0.3 and g = 10 m/s2).

Answer: (54)

22. If the potential difference between B and D is zero, the value of x is  The value of n is _______.

Answer: (2)

23. A stone tied to 180 cm long string at its end is making 28 revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is  The value of x ________. (Take π = 22/7)

Answer: (125)

24. A radioactive nucleus decays by two different process. The half life of the first process is 5 minutes and that of the second process is 30 s. The effective half-life of the nucleus is calculated to be  The value of α is ________.

Answer: (300)

25. A faulty thermometer reads 5°C in melting ice and 95°C in stream. The correct temperature on absolute scale will be _______ K when the faulty thermometer reads 41°

Answer: (313)

26. In an ac generator, a rectangular coil of 100 turns each having area 14 × 10−2 m2 is rotated at 360rev/min about an axis perpendicular to a uniform magnetic field of magnitude 3.0 T. The maximum value of the emf produced will be _______ V. (Take π = 22/7)

Answer: (1584)

27. A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 s is  The value of α will be _______.

Answer: (4)

28. As shown in figure, a cuboid lies in a region with electric field  The magnitude of charge within the cuboid is n ∈0

The value of n is  _______ (if dimension of cuboid is 1 × 2 × 3 m3).

Answer: (12)

29. In a Young’s double slit experiment, the intensities at two points, for the path differences λ/4 and λ/3 (λ being the wavelength of light used) are I1 and I2 If I0 denotes the intensity produced by each one of the individual slits, then

Answer: (3)

30. The velocity of a particle executing SHM varies with displacement (x) as 4v2 = 50 − 𝑥2. The time period of oscillations is  The value of x is _______. (Take π = 22/7)

Answer: (88)

Chemistry

SECTION-A

31. The Cl−Co−Cl bond angle values in a fac- [Co(NH3)3Cl3] complex is/are:

(1)   90°

(2)   90° & 120°

(3)   180°

(4)   90° & 180°

Answer: (1)

32. The correct order of pKa values for the following compounds is:

(1)   c > a > d > b

(2)   b > a > d > c

(3)   b > d > a > c

(4)   a > b > c > d

Answer: (3)

33. Given below are two statements:

Statement I : During Electrolytic refining, the pure metal is made to act as anode and its impure metallic form is used as cathode.

Statement II : During the Hall-Heroult electrolysis process, purified Al2O3 is mixed with Na3AlF6 to lower the melting point of the mixture.  In the light of the above statements, choose the most appropriate answer from the options given below:

1) Statement I  is correct but Statement II is incorrect

(2) Both Statement I and Statement II are incorrect

(3) Both Statement I and Statement II are correct

(4) Statement I is incorrect but Statement II is correct

Answer: (4)

34. Match List I with List II:

(1) A-IV, B-I, C-III, D-II

(2) A-III, B-IV, C-I, D-II

(3) A-III, B-I, C-IV, D-II

(4) A-II, B-I, C-III, D-IV

Answer: (2)

35. 1 L, 0.02M solution of [Co(NH3)5SO4]Br is mixed with 1 L, 0.02M solution of [Co(NH3)5Br]SO4. The resulting solution is divided into two equal parts (X) and treated with excess of AgNO3 solution and BaCl2 solution respectively as shown below:

1 L solution (X) + AgNO3 solution (excess) → Y

1 L Solution (X) + BaCl2 solution (excess) → Z

The number of moles of Y and Z respectively are

(1)   0.02, 0.01

(2)   0.01, 0.01

(3)   0.01, 0.02

(4)   0.02, 0.02

Answer: (2)

36. Decreasing order towards SN 1 reaction for the following compounds is:

(1)   a > c > d > b

(2)   b > d > c > a

(3)   a > b > c > d

(4)   d > b > c > a

Answer: (2)

37. Which of the following reaction is correct?

Answer: (4)

38. Boric acid is solid, whereas BF3 is gas at room temperature because of

(1) Strong van der Waal’s interaction in Boric acid

(2) Strong covalent bond in BF3

(3) Strong ionic bond in Boric acid

(4) Strong hydrogen bond in Boric acid

Answer: (4)

39. Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason 𝐑.

Assertion A:  Antihistamines do not affect the secretion of acid in stomach.

Reason  : Antiallergic and antacid drugs work on different receptors.

In the light of the above statements, choose the correct answer from the options given below:

(1) A is false but R is true

(2) Both A and R are true but R is not the correct explanation of A

(3) Both A and R  are true and R is the correct explanation of A

(4) A is true but R is false

Answer: (3)

40. Formulae for Nessler’s reagent is:

(1)   HgI2

(2)   K2HgI4

(3)   KHgI3

(4)   KHg2I2

Answer: (2)

41. Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A:  can be easily reduced using Zn-Hg/HCl to

Reason R: Zn−Hg/HCl is used to reduce carbonyl group to −CH2− group.

In the light of the above statements, choose the correct answer from the options given below:

(1) A is true but R is false

(2) Both A and R are true and R is the correct explanation of A

(3) A is false but R is true

(4) Both A and R are true but R is not the correct explanation of A

Answer: (2)

42. Maximum number of electrons that can be accommodated in shell with n = 4

(1)   16

(2)   32

(3)   72

(4)   50

Answer: (2)

43. The wave function (Ψ) of 2 s is given by

At r = r0, radial node is formed. Thus, r0 in terms of a0

(1)   r0 = 4a0

(2)   r0 = a0/2

(3)   r0 = a0

(4)   r0 = 2a0

Answer: (4)

44.

In the above conversion of compound (X) to product (Y), the sequence of reagents to be used will be:

(1) (i) Br2(aq) (ii) LiAIH4 (iii) H3O+

(2) (i) Br2, Fe (ii) Fe, H+ (iii) LiAIH4

(3) (i) Fe, H+ (ii) Br2 (aq)  (iii) HNO2 (iv) H3PO2

(4) (i) Fe, H+  (ii) Br2 (aq)  (iii) HNO2 (iv) CuBr

Answer: (3)

45. Match List I with List II:

(1) A-I, B-II, C-IV, D-III

(2) A-II, B-I, C-III, D-IV

(3) A-II, B-I, C-IV, D-III

(4) A-I, B-II, C-III, D-IV

Answer: (1)

46. The most stable carbocation for the following is:

(1)   a

(2)   b

(3)   c

(4)   c

Answer: (3)

47. Chlorides of which metal are soluble in organic solvents:

(1)   K

(2)   Be

(3)   Mg

(4)   Ca

Answer: (2)

48. KMnO4 oxidises I in acidic and neutral/faintly alkaline solution, respectively, to

(1)   IO3 & IO3

(2)   I2 & IO3

(3)   I2 & I2

(4)   IO3 & I2

Answer: (2)

49. Bond dissociation energy of “E-H” bond of the “H2E ” hydrides of group 16 elements (given below), follows order.

(A) O

(B) S

(C) Se

(D) Te

Choose the correct from the options given below:

(1) B > A > C > D

(2) A > B > D > C

(3) A > B > C > D

(4) D > C > B > A

Answer: (3)

50. The water quality of a pond was analysed and its BOD was found to be 4. The pond has

(1)   Highly polluted water

(2)   Slightly polluted water

(3)   Water has high amount of fluoride compounds

(4)   Very clean water

Answer: (4)

SECTION B

51. Number of compounds from the following which will not dissolve in cold NaHCO3 and NaOH solutions but will dissolve in hot NaOH solution is

Answer: (3)

52. 1 mole of ideal gas is allowed to expand reversibly and adiabatically from a temperature of 27° The work done is 3 kJ mol−1. The final temperature of the gas is _______ K (Nearest integer). Given CV = 20 J mol–1 K–1

Answer: (150)

53. A short peptide on complete hydrolysis produces 3 moles of glycine (G), two moles of leucine (L) and two moles of valine (V) per mole of peptide. The number of peptide linkages in it are

Answer: (6)

54. Lead storage battery contains 38% by weight solution of H2SO4. The van’t Hoff factor is 2.67 at this concentration. The temperature in Kelvin at which the solution in the battery will freeze is __ (Nearest integer). Given Kf = 1.8 K kg mol−1

Answer: (243)

55. The strength of 50 volume solution of hydrogen peroxide is ___________ g/L  (Nearest integer).

Given:

Molar mass of H2O2 is 34 g mol−1  Molar volume of gas at STP = 22.7 L.

Answer: (150)

56. The electrode potential of the following half cell at 298 K

X|X2+(0.001M||Y2+(0.01M)|Y is____________ × 10−2 V (Nearest integer).

Answer: (275)

57. An organic compound undergoes first order decomposition. If the time taken for the 60% decomposition is 540 s, then the time required for 90% decomposition will be is______ s. (Nearest integer).

Given: ln 10 = 2.3; log 2 = 0.3

Answer: (1350)

58. Consider the following equation:

2SO2(g) + O2(g) ⇌ 2SO3(g), Δ𝐻=−190 kJ

The number of factors which will increase the yield of SO3 at equilibrium from the following is

(A) Increasing temperature

(B) Increasing pressure

(C) Adding more SO2

(D) Adding more O2

(E) Addition of catalyst

Answer: (3)

59. Iron oxide FeO, crystallises in a cubic lattice with a unit cell edge length of 5.0Å. If density of the FeO in the crystal is 4.0 g cm−3, then the number of FeO units present per unit cell is______ (Nearest integer)

Given: Molar mass of Fe and O is 56 and 16 g mol−1 respectively. NA = 6.0 × 1023 mol−1

Answer: (4)

60. The graph of  for an adsorption process is a straight line inclined at an angle of 45° with intercept equal to 0.6020. The mass of gas adsorbed per unit mass of adsorbent at the pressure of 0.4 atm is_______ ×10−1 (Nearest integer)

Answer: (16)

Mathematics

SECTION-A

61. A vector  in the first octant is inclined to the x-axis at 60∘, to the y-axis at 45 and to the z-axis at an acute angle. If a plane passing through the points (√2, −1, 1) and (a, b, c), is normal to , then

(1) √2a + b + c = 1

(2) a + √2b + c = 1

(3) a + b + √2c = 1

(4) √2a – b + c = 1

Answer: (2)

62. Let a, b, c > 1, a3, b3 and c3 be in A.P., and logab, logc a and logb c be in G.P. If the sum of first 20 terms of an A.P., whose first term is  and the common difference is  then abc is equal to :

(1)   125/8

(2)   216

(3)   343

(4)   343/8

Answer: (2)

63. Let a1 = 1, a2, a3, a4, ….. be consecutive natural numbers. Then  is equal to

Answer: (3)

64. Let λ ∈ ℝ,

If  then  is equal to

(1)   132

(2)   136

(3)   140

(4)   144

Answer: (3)

65. Let q be the maximum integral value of p in [0, 10] for which the roots of the equation  are rational. Then the area of the region {(x, y): 0 ≤ y ≤ (x – q)2, 0 ≤ x ≤ q} is

(1)   243

(2)   164

(3)   125/3

(4)   25

Answer: (1)

66. Let f, g and h be the real valued functions defined on ℝ as

and h(x) = 2[x] − f(x), where [x] is the greatest integer ≤ x.

Then the value of limx1g(h(x – 1)) is

(1)   −1

(2)   0

(3)   sin(1)

(4)   1

Answer: (4)

67. Let S be the set of all values of a1 for which the mean deviation about the mean of 100 consecutive positive integers a1, a2, a3, …. a100 is 25 . Then S is

(1)   N

(2)   ϕ

(3)   {99}

(4)   {9}

Answer: (1)

68. For α, β ∈ ℝ, suppose the system of linear equations

x – y + z = 5

2x + 2y + αz = 8

3x – y + 4z = β

has infinitely many solutions. Then α and β are the roots of

(1)   x2 + 14x + 24 = 0

(2)   x2 + 18x + 56 = 0

(3)   x2 – 18x + 56 = 0

(4)   x2 – 10x + 16 = 0

Answer: (3)

69. Let  be two vectors, let  If  then the value of  is

(1)   −24

(2)   −84

(3)   −48

(4)   −60

Answer: (3)

70. If the functions  and  have a common extreme point, then a + 2b + 7 is equal to :

(1)   3/2

(2)   3

(3)   4

(4)   6

Answer: (4)

71. If P is a 3×3 real matrix such that PT = aP + (a − 1)I, where a > 1, then

(1)   |Adj P| = 1/2

(2)   |Adj P| = 1

(3)   P is a singular matrix

(4)   |Adj P| > 1

Answer: (2)

72. The number of ways of selecting two numbers a and b, a ∈ {2, 4, 6, …., 100} and b ∈ {1, 3, 5, …., 99} such that 2 is the remainder when a + b is divided by 23 is

(1)   268

(2)   108

(3)   54

(4)   186

Answer: (2)

73.  is equal to

(1)   12

(2)   19/3

(3)   0

(4)   19

Answer: (4)

74. Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2 + y2 = 8 and y2 = 16x. If one of these tangents touches the two curves at Q and R, then (QR)2 is equal to

(1)   81

(2)   72

(3)   76

(4)   64

Answer: (2)

75. If a plane passes through the points (−1, k, 0), (2, k, −1), (1, , 2) and is parallel to the line  then the value of  is

(1)   17/5

(2)   13/6

(3)   6/13

(4)   5/17

Answer: (2)

76. The range of the function  is:

(1)   [2√2, √11]

(2)   [√5, √13]

(3)   [√2, √7]

(4)   [√5, √10]

Answer: (4)

77. The solution of the differential equation  is

Answer: (2)

78. The parabolas : ax2 + 2bx + cy = 0 and dx2 + 2ex + fy = 0 intersect on the line y=1. If a, b, c, d, e, f are positive real numbers and a, b, c are in G.P., then

(1)   d, e, f are in G.P.

(2)   d/a, e/b, f/c are in A.P.

(3)   d, e, f are in A.P.

(4)   d/a, e/b, f/c are in G.P.

Answer: (2)

79. Consider the following statements:

P : I have fever

Q: I will not take medicine

R : I will take rest.

The statement “If I have fever, then I will take medicine and I will take rest” is equivalent to:

(1)   ((∼P) ∨ ∼Q) ∧ ((∼P) ∨ R)

(2)   (P ∨ Q) ∧ ((∼P) ∨ R)

(3)   ((∼P) ∨ ∼Q) ∧ ((∼P) ∨ ∼R)

(4)   (P ∨ ∼Q) ∧ (P ∨ ∼R)

Answer: (1)

80. x = (8√3 + 13)13 and y = (7√2 + 9)9. If [t] denotes the greatest integer ≤ t, then

(1) [x] is odd but [y] is even

(2) [x] + [y] is even

(3) [x] and [y] are both odd

(4) [x] is even but [y] is odd

Answer: (2)

SECTION-B

81. Let a line L pass through the point P(2, 3, 1) and be parallel to the line x + 3y − 2z – 2 = 0 = x – y + 2z. If the distance of L from the point (5, 3, 8) is α, then 3α2 is equal to ______.

Answer: (158)

82. A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q. If p : q = m : n, where m and n are coprime, then m + n is equal to ______.

Answer: (14)

83. Let P(a1, b1) and Q(a2, b2) be two distinct points on a circle with center C(√2,√3). Let O be the origin and OC be perpendicular to both CP and CQ.

If the area of the triangle OCP is √35/2, then a12 + a22 + b12 + b22 is equal to _______.

Answer: (24)

84. Let A be the area of the region {(x, y) : y ≥ x2, y ≥ (1 − x)2, y ≤ 2x(1−x)}. Then 540 A is equal to ______.

Answer: (25)

85. The 8th common term of the series

S1 = 3 + 7 + 11 + 15 + 19 + …

S2 = 1 + 6 + 11 + 16 + 21 + …

is _______.

Answer: (151)

86. Let A = {1, 2, 3, 5, 8,9}. Then the number of possible functions f:A→A such that f(m ⋅ n) = f(m) ⋅ f(n) for every m, n ∈ A with m ⋅ n ∈ A is equal to ______.

Answer: (1)

87. If  constant, then β – α is equal to ______.

Answer: (1)

88. If the value of real number a>0 for which x2 − 5ax + 1 = 0 and x2 – ax – 5 = 0⁡have a common real root is 3/√2β then β is equal to _______.

Answer: (13)

89. 50th root of a number x is 12 and 50th root of another number y is 18 . Then the remainder obtained on dividing (x + y) by 25 is ______.

Answer: (23)

90. The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is ______.

Answer: (240)

## JEE Main Session 2 29th January 2023 Shift 2 Question Paper and Answer Key

JEE MAIN 29th January 2023 Shift 2

Physics

SECTION-A

IMPORTANT INSTRUCTIONS:

(1) The test is of 3 hours duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) Section-A: This section contains 20 multiple choice questions which have only one correct answer. each question carries 4 marks for correct answer and 1 mark for wrong answer.

(ii) Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of 10. The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and 1 mark for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. Substance A has atomic mass number 16 and half-life of 1 day. Another substance 𝐵 has atomic mass number 32 and half life of 1/2 day. If both 𝐴 and 𝐵 simultaneously start undergo radio activity at the same time with initial mass 320 g each, how many total atoms of A and B combined would be left after 2 days

(1)   3.38 × 1024

(2)   1.69 × 1024

(3)   6.76 × 1024

(4)   6.76 × 1023

Answer: (1)

2. For the given logic gates combination, the correct truth table will be

Answer: (3)

3. The time taken by an object to slide down 45° rough inclined plane is n times as it takes to slide down a perfectly smooth 45∘ incline plane. The coefficient of kinetic friction between the object and the incline plane is:

Answer: (3)

4. Heat energy of 184 kJ is given to ice of mass 600 g at −12∘ Specific heat of ice is 2222.3 J kg−1C−1 and latent heat of ice in 336 kJkg−1

(A) Final temperature of system will be 0∘C.

(B) Final temperature of the system will be greater than 0∘C.

(C) The final system will have a mixture of ice and water in the ratio of 5:1.

(D) The final system will have a mixture of ice and water in the ratio of 1:5.  E. The final system will have water only.

Choose the correct answer from the options given below:

(1)   A and D only

(2)   A and E only

(3)   A and C only

(4)   B and D only

Answer: (1)

5. Identify the correct statements from the following:

(A) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.

(B) Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative.

(C) Work done by friction on a body sliding down an inclined plane is positive.

(D) Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity in zero.

(E) Work done by the air resistance on an oscillating pendulum in negative.

Choose the correct answer from the options given below:

(1)   B, D and E only

(2)   A and C only

(3)   B and D only

(4)   B and E only

Answer: (4)

6. A scientist is observing a bacteria through a compound microscope. For better analysis and to improve its resolving power he should. (Select the best option)

(1) Increase the refractive index of the medium between the object and objective lens

(2) Decrease the diameter of the objective lens

(3) Increase the wave length of the light

(4) Decrease the focal length of the eye piece.

Answer: (1)

7. With the help of potentiometer, we can determine the value of emf of a given cell. The sensitivity of the potentiometer is

(A) directly proportional to the length of the potentiometer wire

(B) directly proportional to the potential gradient of the wire

(C) inversely proportional to the potential gradient of the wire

(D) inversely proportional to the length of the potentiometer wire

Choose the correct option for the above statements:

(1)   A only

(2)   C only

(3)   A and C only

(4)   B and D only

Answer: (3)

8. A force acts for 20 s on a body of mass 20 kg, starting from rest, after which the force ceases and then body describes 50 m in the next 10 s. The value of force will be:

(1)   40 N

(2)   5 N

(3)   20 N

(4)   10 N

Answer: (2)

9. The modulation index for an A.M. wave having maximum and minimum peak-to-peak voltages of 14 mV and 6 mV respectively is:

(1)   0.4

(2)   0.6

(3)   0.2

(4)   1.4

Answer: (1)

10. Given below are two statements:

Statement I: Electromagnetic waves are not deflected by electric and magnetic field.

Statement II: The amplitude of electric field and the magnetic field in electromagnetic waves are related to each other as

In the light of the above statements, choose the correct answer from the options given below :

(1) Statement I is true but statement II is false

(2) Both Statement I and Statement II are false

(3) Statement I is false but statement II is true

(4) Both Statement I and Statement II are true

Answer: (1)

11. A square loop of area 25 cm2 has a resistance of 10 Ω. The loop is placed in uniform magnetic field of magnitude 40.0 T. The plane of loop is perpendicular to the magnetic field. The work done in pulling the loop out of the magnetic field slowly and uniformly in 1.0sec, will be

(1)   1.0 × 103 J

(2)   2.5 × 103 J

(3)   5 × 103 J

(4)   1.0 × 104 J

Answer: (1)

12. For the given figures, choose the correct options:

(1) At resonance, current in (b) is less than that in (a)

(2) The rms current in circuit (b) can never be larger than that in (a)

(3) The rms current in figure(a) is always equal to that in figure (b)

(4) The rms current in circuit (b) can be larger than that in (a)

Answer: (2)

13. A fully loaded boeing aircraft has a mass of 5.4 × 105 Its total wing area is 500 m2. It is in level flight with a speed of 1080 km/h. If the density of air ρ is 1.2 kg m−3, the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface in percentage will be. (g = 10 m/s2)

(1)   16

(2)   10

(3)   8

(4)   6

Answer: (2)

14. The ratio of de-Broglie wavelength of an α particle and a proton accelerated from rest by the same potential is 1/√m, the value of m is-

(1)   16

(2)   4

(3)   2

(4)   8

Answer: (4)

15. The time period of a satellite of earth is 24 hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value, then its new time period will become.

(1)   4 hours

(2)   6 hours

(3)   3 hours

(4)   12 hours

Answer: (3)

16. The electric current in a circular coil of four turns produces a magnetic induction 32 T at its centre. The coil is unwound and is rewound into a circular coil of single turn, the magnetic induction at the centre of the coil by the same current will be :

(1)   16 T

(2)   2 T

(3)   8 T

(4)   4 T

Answer: (2)

17. A point charge 2 × 10−2 C is moved from P to S in a uniform electric field of 30NC−1 directed along positive x-axis. If coordinates of P and S are (1, 2, 0)m and (0, 0, 0)m respectively, the work done by electric field will be

(1)   1200 mJ

(2)   −1200 mJ

(3)   −600 mJ

(4)   600 mJ

Answer: (3)

18. An object moves at a constant speed along a circular path in a horizontal plane with center at the origin. When the object is at =+2 m, its velocity is

The object’s velocity (v) and acceleration ( a ) at x = −2 m will be

Answer: (3)

19. At 300 K the rms speed of oxygen molecules is  times to that of its average speed in the gas. Then, the value of α will be (used = 22/7)

(1)   28

(2)   24

(3)   32

(4)   27

Answer: (1)

20. The equation of a circle is given by x2 + y2 = a2, where 𝑎 is the radius. If the equation is modified to change the origin other than (0, 0), then find out the correct dimensions of A and B in a new equation : The dimensions of t is given as [T−1].

(1) A=[LT], B=[L−1 T−1]

(2) A=[L−1 T−1], B=[LT]

(3) A=[L−1 T], B=[LT−1]

(4) A=[L−1 T−1], B=[LT−1]

Answer: (1)

SECTION-B

21. A particle of mass 100 g is projected at time t = 0 with a speed 20 ms−1 at an angle 45∘ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time t = 2 s is found to be √K kgm2/s. The value of K is ________. (Take g = 10 ms2)

Answer: (800)

22. Unpolarised light is incident on the boundary between two dielectric media, whose dielectric constants are 2.8 (medium −1) and 6.8 (medium −2), respectively. To satisfy the condition, so that the reflected and refracted rays are perpendicular to each other, the angle of incidence should be  the value of θ is ________.

(Given for dielectric media, μr = 1)

Answer: (7)

23. A particle of mass 250 g executes a simple harmonic motion under a periodic force F = (−25x)N. The particle attains a maximum speed of 4 m/s during its oscillation. The amplitude of the motion is ______ cm.

Answer: (40)

24. A car is moving on a circular path of radius 600 m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is moving with an initial speed of 54 km/hr is t(1 – e−π/2)S. The value of t is ________.

Answer: (40)

25. When two resistances R1 and R2 connected in series and introduced into the left gap of a meter bridge and a resistance of 10Ω is introduced into the right gap, a null point is found at 60 cm from left side. When R1 and R2 are connected in parallel and introduced into the left gap, a resistance of 3Ω is introduced into the right-gap to get null point at 40 cm from left end. The product of R1R2 is _______ Ω2

Answer: (30)

26. In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab, a student measures real thickness of the glass slab as 5.25 mm and apparent thickness of the glass slab as 5.00 mm. Travelling microscope has 20 divisions in one cm on main scale and 50 divisions on vernier scale is equal to 49 divisions on main scale. The estimated uncertainty in the measurement of refractive index of the slab is  where x is _______.

Answer: (41)

27. An inductor of inductance 2μH is connected in series with a resistance, a variable capacitor and an AC source of frequency 7kHz. The value of capacitance for which maximum current is drawn into the circuit is  where the value of x is _______. (Take π = 22/7)

Answer: (3872)

28. A null point is found at 200 cm in potentiometer when cell in secondary circuit is shunted by 5Ω. When a resistance of 15Ω is used for shunting, null point moves to 300 cm. The internal resistance of the cell is _______ Ω.

Answer: (5)

29. For a charged spherical ball, electrostatic potential inside the ball varies with r as V = 2ar2 + b. Here, 𝑎 and 𝑏 are constant and r is the distance from the center. The volume charge density inside the ball is −λaε. The value of 𝜆 is ________.

ε = permittivity of the medium

Answer: (12)

30. A metal block of base area 0.20 m2 is placed on a table, as shown in figure. A liquid film of thickness 0.25 mm is inserted between the block and the table. The block is pushed by a horizontal force of 0.1 N and moves with a constant speed. If the viscosity of the liquid is 5.0 × 10−3 Pl, the speed of block is ________ × 10−3 m/s.

Answer: (25)

Chemistry

SECTION-A

31. According to MO theory the bond orders for O22−, CO and NO+ respectively, are

(1)   1, 2 and 3

(2)   1, 3 and 2

(3)   2, 3 and 3

(4)   1, 3 and 3

Answer: (4)

32. A doctor prescribed the drug Equanil to a patient. The patient was likely to have symptoms of which disease?

(1) Hyperacidity

(2) Anxiety and stress

(3) Depression and hypertension

(4) Stomach ulcers

Answer: (3)

33. Reaction of propanamide with Br2/KOH(aq) produces :

(1)   Propylamine

(2)   Ethylnitrile

(3)   Propanenitrile

(4)   Ethylamine

Answer: (4)

34. The one giving maximum number of isomeric alkenes on dehydrohalogenation reaction is (excluding rearrangement)

(1) 2-Bromopropane

(2) 2-Bromo-3,3-dimethylpentane

(3) 1-Bromo-2-methylbutane

(4) 2-Bromopentane

Answer: (4)

35. An indicator ‘ X ‘ is used for studying the effect of variation in concentration of iodide : on the rate of reaction of iodide ion with H2O2 at room temp. The indicator ‘ X ‘ forms blue colored complex with compound ‘ A ‘ present in the solution. The indicator ‘ X ‘ and compound ‘A’ respectively are

(1) Methyl orange and H2O2

(2) Starch and iodine

(3) Starch and H2O2

(4) Methyl orange and iodine

Answer: (2)

36. The major component of which of the following ore is sulphide based mineral?

(1)   Siderite

(2)   Sphalerite

(3)   Malachite

(4)   Calamine

Answer: (2)

37. A solution of CrO5 in amyl alcohol has a _______ colour.

(1)   Green

(2)   Orange-Red

(3)   Yellow

(4)   Blue

Answer: (4)

38. The set of correct statements is :

(i) Manganese exhibits +7 oxidation state in its oxide.

(ii) Ruthenium and Osmium exhibit +8 oxidation in their oxides.

(iii) Sc shows +4 oxidation state which is oxidizing in nature.

(iv) Cr shows oxidising nature in +6 oxidation state.

(1)  (ii) and (iii)

(2) (i), (ii) and (iv)

(3) (ii), (iii) and (iv)

(4) (i) and (iii)

Answer: (2)

39. Following tetrapeptide can be represented as

(F, L, D, Y, I, Q, P are one letter codes for amino acids)

(1)  PLDY

(2) FIQY

(3) YQLF

(4) FLDY

Answer: (4)

40. Find out the major product for the following reaction.

Answer: (4)

41. Match List I with List II

Choose the correct answer from the options given below :

(1)  A-I, B-III, C-II, D-IV

(2) A-III, B-I, C-IV, D-II

(3) A-III, B-I, C-II, D-IV

(4) A-III, B-II, C-I, D-IV

Answer: (3)

42. Correct order of spin only magnetic moment of the following complex ions is: (Given At.no. Fe: 26, Co : 27)

(1) [FeF6]3− > [Co(C2O4)3]3− > [CoF6]3−

(2) [FeF6]3− > [CoF6]3− > [Co(C2O4)3]3−

(3) [Co(C2O4)3]3− > [CoF6]3− > [FeF6]3−

(4) [CoF6]3− > [FeF6]3− > [Co(C2O4)3]3−

Answer: (2)

43. Match List I with List II

Choose the correct answer from the options given below :

(1) A-II, B-III, C-I, D-IV

(2) A-IV, B-III, C-I, D-II

(3) A-IV, B-I, C-III, D-II

(4) A-II, B-I, C-IV, D-III

Answer: (2)

44. The concentration of dissolved Oxygen in water for growth of fish should be more than X ppm and Biochemical Oxygen Demand in clean water should be less than Y X and Y in ppm are, respectively.

Answer: (2)

45. Find out the major products from the following reaction sequence.

Answer: (4)

46. When a hydrocarbon A undergoes combustion in the presence of air, it requirs 9.5 equivalents of oxygen and produces 3 equivalents of water. What is the molecular formula of A ?

(1)   C9H9

(2)   C8H6

(3)   C9H6

(4)   C6H6

Answer: (2)

47. Given below are two statements:

Statement I : Nickel is being used as the catalyst for producing syn gas and edible fats.

Statement II : Silicon forms both electron rich and electron deficient hydrides.

In the light of the above statements, choose the most appropriate answer from the options given below:

(1) Statement I is correct but statement II is incorrect

(2) Both the statements I and II are incorrect

(3) Statement I is incorrect but statement II is correct

(4) Both the statements I and II are correct

Answer: (1)

48. Which of the following relations are correct?

(A) ΔU = q + pΔV         (B) ΔG = ΔH −TΔS

(C) ΔS = qrev/T                (D) ΔH=ΔU−ΔnRT

Choose the most appropriate answer from the options given below:

(1) B and D Only

(2) A and B Only

(3) B and C Only

(4) C and D Only

Answer: (3)

49. Given below are two statements :

Statement I : The decrease in first ionization enthalpy from B to Al is much larger than that from Al to Ga.

Statement II : The d orbitals in Ga are completely filled.

In the light of the above statements, choose the most appropriate answer from the options given below

(1)  Statement I is incorrect but statement II is correct

(2) Both the statements I and II are correct

(3) Both the statements I and II are incorrect

(4) Statement I is correct but statement II is incorrect

Answer: (1)

50. Match List I and List II

Choose the correct answer from the options given below :

(1)  A-I, B-III, C-IV, D-II

(2) A-III, B-I, C-IV, D-II

(3) A-III, B-I, C-II, D-IV

(4) A-I, B-III, C-II, D-IV

Answer: (2)

SECTION-B

51. Assume that the radius of the first Bohr orbit of hydrogen atom is 0.6Å. The radius of the third Bohr orbit of He+ is _______ picometer. (Nearest Integer)

Answer: (270)

52. Total number of acidic oxides among

N2O3, NO2, N2O, Cl2O7, SO2, CO, CaO, Na2O and NO is _______

Answer: (4)

53. The denticity of the ligand present in the Fehling’s reagent is _______

Answer: (4)

54. The equilibrium constant for the reaction Zn(s) + Sn2+(aq) ⇌ Zn2+(aq) + Sn(s) is 1 × 1020 at 298 K. The magnitude of standard electrode potential of Sn/Sn2+ if  is _______ × 10−2 V (Nearest integer).

Answer: (17)

55. The volume of HCl, containing 73 g L−1, required to completely neutralise NaOH obtained by reacting 0.69 g of metallic sodium with water, is _______ mL.( Nearest Integer) (Given : molar Masses of Na, Cl, O, H, are 23, 35.5, 16 and 1 g mol−1 respectively)

Answer: (15)

56. For conversion of compound A→B, the rate constant of the reaction was found to be 6 × 105 L mol1 s1. The order of the reaction is _________.

Answer: (2)

57. On heating, LiNO3 gives how many compounds among the following? _______ LiO2, N2, O2, LiNO2, NO2

Answer: (3)

58. When 0.01 mol of an organic compound containing 60% carbon was burnt completely, 4.4 g of CO2 was produced. The molar mass of compound is _______ gmol−1 (Nearest integer).

Answer: (200)

59. At 298 K

N2(g) + 3H2(g) ⇌ 2NH3 ( g), K1 = 4 × 105

N2( g) + O2( g) ⇌ 2NO(g), K2 = 1.6 × 1012

K3 = 1.0 × 1013

Based on above equilibria, the equilibrium constant of the reaction,  is _______ × 1033 (Nearest integer).

Answer: (4)

60. A metal M forms hexagonal close-packed structure. The total number of voids in 0.02 mol of it is _______ × 1021 (Nearest integer). (Given NA = 6.02 × 1023 )

Answer: (36)

Mathematics

SECTION-A

61. The statement B ⇒ ((∼A) ∨ B) is equivalent to :

(1) A ⇒ (A ⇔ B)

(2) A ⇒ ((∼A) ⇒ B)

(3) B ⇒(A ⇒ B)

(4) B ⇒ ((∼A) ⇒ B)

Answer: (1, 3 or 4)

62. The value of the integral  is

Answer: (4)

63. The set of all values of λ for which the equation cos2⁡2x − 2sin4⁡x − 2cos2⁡x = λ has a real solution x, is

(1)   [−2, −1]

(2)   [−1, −1/2]

(3)   [−3/2, −1]

(4)   [−2, −3/2]

Answer: (3)

64. Let R be a relation defined on ℕ as a R b if 2a + 3b is a multiple of 5, a, b ∈ ℕ. Then R is

(1)   an equivalence relation

(2)   transitive but not symmetric

(3)   not reflexive

(4)   symmetric but not transitive

Answer: (1)

65. Consider a function f : ℕ → ℝ, satisfying f(1) + 2f(2) + 3f(3) + … + xf(x) = x(x + 1) f(x); x ≥ 2 with f(1) = 1. Then  is equal to

(1)   8100

(2)   8400

(3)   8000

(4)   8200

Answer: (1)

66. If  and  is equal to

(1)   32

(2)   30

(3)   36

(4)   34

Answer: (4)

67. The shortest distance between the lines  and

(1)   5√3

(2)   2√3

(3)   3√3

(4)   4√3

Answer: (4)

68. The plane 2x – y + z = 4 intersects the line segment joining the points A(a, −2, 4) and B(2, b, −3) at the point C in the ratio 2:1 and the distance of the point C from the origin is √5. If ab < 0 and P is the point (a − b, b, 2b − a) then CP2 is equal to

(1)   97/3

(2)   17/3

(3)   16/3

(4)   73/3

Answer: (2)

69. The value of the integral  is equal to

Answer: (1)

70. The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is

(1)   84

(2)   79

(3)   89

(4)   86

Answer: (3)

71. The set of all values of t ∈ ℝ, for which the matrix  is invertible, is

(1)   ℝ

(2)

(3)   {kπ, k ∈ ℤ}

(4)

Answer: (1)

72. The area of the region A = {(x, y): |cos x – sin x| ≤ y ≤ sin x, 0 ≤ x ≤ π/2}  is

(1)   √5 + 2√2 – 4.5

(2)

(3)

(4)   √5 – 2√2 + 1

Answer: (4)

73. The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

(1)   507

(2)   432

(3)   472

(4)   400

Answer: (2)

74. If the lines  and  intersect at the point P, then the distance of the point P from the plane z = a is :

(1)   28

(2)   16

(3)   10

(4)   22

Answer: (1)

75. Let y = y(x) be the solution of the differential equation  If y(2) = 2, then y(e) is equal to

Answer: (2)

76. Let f and g be twice differentiable functions on ℝ such that

fʹʹ(x) = gʹʹ(x) + 6x

fʹ(1) = 4gʹ(1) – 3 = 9

f(2) = 3g(2) = 12.

Then which of the following is NOT true?

(1)   There exists x0 ∈ (1, 3/2) such that f(x0) = g(x0)

(2)   |fʹ(x) – gʹ(x)| < 6 ⇒ −1 < x < 1

(3)   If −1 < x < 2, then |f(x) − g(x)| < 8

(4)   g(−2) − f(−2) = 20

Answer: (3)

77. If the tangent at a point P on the parabola y2 = 3x is parallel to the line x + 2y = 1 and the tangents at the points Q and R on the ellipse  are perpendicular to the line x – y = 2, then the area of the triangle PQR is :

(1)

(2)   3√5

(3)   9/√5

(4)   5√3

Answer: (2)

78. Let  If  is a vector such that  and projection of  then the projection of  equals

(1)   1/5

(2)   5/√2

(3)   3/√2

(4)   1/√2

Answer: (2)

79. Let S = {w1, w2, …….} be the sample space associated to a random experiment. Let  Let A = {2k + 3l; k. l ∈ ℕ} and B = {wn : n ∈ A}. Then P(B) is equal to

(1)   3/64

(2)   1/16

(3)   1/32

(4)   3/32

Answer: (1)

80. Let K be the sum of the coefficients of the odd powers of x in the expansion of (1 + x)99. Let a be the middle term in the expansion of  where m and n are odd numbers, then the ordered pair (l, n) is equal to

(1)   (50, 51)

(2)   (50, 101)

(3)   (51, 99)

(4)   (51, 101)

Answer: (2)

SECTION-B

81. The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is

Answer: (3000)

82. Let a1 = b1 = 1 and an = an – 1 + (n – 1), bn = bn – 1 + an – 1, ∀n ≥ If  then 27 (2S – T) is equal to

Answer: (461)

83. A triangle is formed by the tangents at the point (2, 2) on the curves y2 = 2x and x2 + y2 = 4x, and the line x + y + 2 = 0. If r is the radius of its circumcircle, then r2 is equal to

Answer: (10)

84. Let α1, α2, …., α7 be the roots of the equation x7 + 3x5 – 13x3 – 15x = 0 and |α1| ≥ | α2| ≥ ⋯ ≥ | α7|. Then α1 α2 − α3 α4 + α5α6 is equal to

Answer: (3)

85. Let X = {11, 12, 13, …, 40, 41} and Y = {61, 62, 63, …, 90, 91} be the two sets of observations. If are their respective means and σ2 is the variance of all the observations in X ∪ Y, then  is equal to

Answer: (603)

86. If the equation of the normal to the curve  at the point (1, −3) is x – 4y = 13, then the value of a + b is equal to

Answer: (6)

87. Let A be a symmetric matrix such that |A| = 2 and  If the sum of the diagonal elements of A is s, then βs/α2 is equal to

Answer: (5)

88. Let α = 8 – 14i,  and B = {z ∈ ℂ: |z + 3i| = 4}. Then ∑ZAB(Re z = Im z) is equal to

Answer: (14)

89. A circle with centre (2, 3) and radius 4 intersects the line x + y = 3 at the points P and Q. If the tangents at P and Q intersect at the point S(α, β), then 4α − 7β is equal to

Answer: (11)

90. Let {ak} and {bk}, k ∈ ℕ, be two G. P.s with common ratios r1 and r2 respectively such that a1 = b1 = 4 and r1 < r2. Let ck = ak + bk, k ∈ ℕ. If c2 = 5 and c3 = 13/4 then  is equal to

Answer: (9)

## JEE Main Session 2 25th January 2023 Shift 2 Question Paper and Answer Key

JEE MAIN 25th January 2023 Shift 2

Physics

SECTION-A

IMPORTANT INSTRUCTIONS:

(1) The test is of 3 hours duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) Section-A: This section contains 20 multiple choice questions which have only one correct answer. each question carries 4 marks for correct answer and 1 mark for wrong answer.

(ii) Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of 10. The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and 1 mark for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. According to law of equipartition of energy the molar specific heat of a diatomic gas at constant volume where the molecule has one additional vibrational mode is:-

Answer: (3)

2. A wire of length 1 m moving with velocity 8 m/s at right angles to a magnetic field of 2 T. The magnitude of induced emf, between the ends of wire will be

(1)   20 V

(2)   8 V

(3)   12 V

(4)   16 V

Answer: (4)

3. The energy levels of an atom is shown in figure.

Which one of these transitions will result in the emission of a photon of wavelength 124.1 nm ?  Given (h = 6.62 × 10−34Js)

(1)   D

(2)   B

(3)   C

(4)   A

Answer: (1)

4. Given below are two statements :

Statement I: Stopping potential in photoelectric effect does not depend on the power of the light source.

Statement II: For a given metal, the maximum kinetic energy of the photoelectron depends on the wavelength of the incident light.

In the light of above statements, choose the most appropriate answer from the options given below

(1) Statement I is incorrect but statement II is correct

(2) Statement 𝐈 is correct but statement 𝐈𝐈 is incorrect

(3) Both Statement 𝐈 and statement II are correct

(4) Both Statement I and Statement II are incorrect

Answer: (3)

5. The distance travelled by a particle is related to time t as x = 4t2. The velocity of the particle at t = 5 s is:-

(1)   40 ms1

(2)   20 ms1

(3)   8 ms1

(4)   25 ms1

Answer: (1)

6. Match List I with List II

Choose the correct answer from the options given below: options

(1) A-I, B-II, C-III, D-IV

(2) A-II, B-III, C-IV, D-I

(3) A-I, B-III, C-IV, D-II

(4) A-III, B-I, C-II, D-IV

Answer: (4)

7. Match List I with List II

Choose the correct answer from the options given below:

(1) A-III, B-IV, C-II, D-I

(2) A-III, B-II, C-I, D-IV

(3) A-I, B-IV, C-III, D-II

(4) A-I, B-II, C-IV, D-III

Answer: (1)

8. The light rays from an object have been reflected towards an observer from a standard flat mirror, the image observed by the observer are:-

(A) Real

(B) Erect

(C) Smaller in size then object

(D) Laterally inverted

Choose the most appropriate answer from the options given below:

(1)   A, C, and D only

(2)   B and D only

(3)   A and D only

(4)   B and C only

Answer: (2)

9. The graph between two temperature scales P and Q is shown in the figure. Between upper fixed point and lower fixed point there are 150 equal divisions of scale P and 100 divisions on scale Q. The relationship for conversion between the two scales is given by:-

Answer: (4)

10. Consider a block kept on an inclined plane (inclined at 45°) as shown in the figure. If the force required to just push it up the incline is 2 times the force required to just prevent it from sliding down, the coefficient of friction between the block and inclined plane (μ) is equal to :

(1)   0.25

(2)   0.50

(3)   0.60

(4)   0.33

Answer: (4)

11. Every planet revolves around the sun in an elliptical orbit:-

(A) The force acting on a planet is inversely proportional to square of distance from sun.

(B) Force acting on planet is inversely proportional to product of the masses of the planet and the sun.

(C) The Centripetal force acting on the planet is directed away from the sun.

(D) The square of time period of revolution of planet around sun is directly proportional to cube of semi-major axis of elliptical orbit.

Choose the correct answer from the options given below:

(1)   B and C only

(2)   A and C only

(3)   A and D only

(4)   C and D only

Answer: (3)

12. For a moving coil galvanometer, the deflection in the coil is 0.05 rad when a current of 10 mA is passed through it. If the torsional constant of suspension wire is 4.0 × 10−5 N m rad−1, the magnetic field is 0.01 T and the number of turns in the coil is 200 , the area of each turn (in cm2) is :

(1)   1.0

(2)   2.0

(3)   1.5

(4)   0.5

Answer: (1)

13. Match List I with List II

Choose the correct answer from the options given below:

(1) A-IV, B-I, C-II, D-III

(2) A-II, B-III, C-IV, D-I

(3) A-III, B-IV, C-I, D-II

(4) A-I, B-II, C-III, D-IV

Answer: (1)

14. Two objects are projected with same velocity ‘u’ however at different angles α andβwith the horizontal. If α + β = 90°, the ratio of horizontal range of the first object to the 2nd object will be:

(1)   2 : 1

(2)   1 : 2

(3)   1 : 1

(4)   4 : 1

Answer: (3)

15. A particle executes simple harmonic motion between x = −A and x = +A. If time taken by particle to go from x = 0 to A/2 is 2 s; then time taken by particle in going from x = A/2 to A is

(1)   4 S

(2)   1.5 S

(3)   2 S

(4)   3 S

Answer: (1)

16. Match List I with List II

Choose the correct answer from the options given below:

(1) A-I, B-II, C-III, D-IV

(2) A-II, B-I, C-IV, D-III

(3) A-II, B-I, C-III, D-IV

(4) A-I, B-II, C-IV, D-III

Answer: (2)

17. Statement I: When a Si sample is doped with Boron, it becomes P type and when doped by Arsenic it becomes N-type semi conductor such that P-type has excess holes and N-type has excess electrons.

Statement II: When such P-type and N-type semi-conductors, are fused to make a junction, a current will automatically flow which can be detected with an externally connected ammeter.

In the light of above statements, choose the most appropriate answer from the options given below

(1) Both Statement I and statement II are correct

(2) Statement 𝐈 is incorrect but statement II is correct

(3) Both Statement I and Statement II are incorrect

(4) Statement I is correct but statement II is incorrect

Answer: (4)

18. A point charge of 10μC is placed at the origin. At what location on the X-axis should a point charge of 40μC be placed so that the net electric field is zero at x = 2 cm on the X-axis?

(1)   x = −4 cm

(2)   x = 6 cm

(3)   x = 4 cm

(4)   x = 8 cm

Answer: (2)

19. The resistance of a wire is 5Ω. It’s new resistance in ohm if stretched to 5 times of it’s original length will be :

(1)   25

(2)   125

(3)   5

(4)   625

Answer: (2)

20. A body of mass is taken from earth surface to the height h equal to twice the radius of earth (Re), the increase in potential energy will be: (g = acceleration due to gravity on the surface of Earth)

Answer: (3)

SECTION-B

21. Two long parallel wires carrying currents 8 A and 15 A in opposite directions are placed at a distance of 7 cm from each other. A point P is at equidistant from both the wires such that the lines joining the point P to the wires are perpendicular to each other. The magnitude of magnetic field at P is _____× 10−6 T

(Given : √2=1⋅4)

Answer: (60)

22. A spherical drop of liquid splits into 1000 identical spherical drops. If ui is the surface energy of the original drop and uf is the total surface energy of the resulting drops, the (ignoring evaporation),  Then value of x is _______.

Answer: (1)

23. A nucleus disintegrates into two smaller parts, which have their velocities in the ratio 3:2. The ratio of their nuclear sizes will be (x/3)1/3. The value of ‘x’ is:-

Answer: (2)

24. A train blowing a whistle of frequency 320 Hz approaches an observer standing on the platform at a speed of 66 m/s. The frequency observed by the observer will be (given speed of sound =330 ms−1) _______ Hz.

Answer: (400)

25. A body of mass 1 kg collides head on elastically with a stationary body of mass 3 kg. After collision, the smaller body reverses its direction of motion and moves with a speed of 2 m/s. The initial speed of the smaller body before collision is ________ ms−1.

Answer: (4)

26. A series LCR circuit is connected to an AC source of 220 V,50 Hz. The circuit contains a resistance R= 80Ω, an inductor of inductive reactance XL = 70Ω, and a capacitor of capacitive reactance XC = 130Ω. The power factor of circuit is x/10. The value of x is :

Answer: (8)

27. If a solid sphere of mass 5 kg and a disc of mass 4 kg have the same radius. Then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be x/7. The value of x is _____.

Answer: (5)

28. An object is placed on the principal axis of convex lens of focal length 10 cm as shown. A plane mirror is placed on the other side of lens at a distance of 20 cm. The image produced by the plane mirror is 5 cm inside the mirror. The distance of the object from the lens is cm

Answer: (30)

29. A capacitor has capacitance 5𝜇F when it’s parallel plates are separated by air medium of thickness d. A slab of material of dielectric constant 1.5 having area equal to that of plates but thickness d/2 is inserted between the plates. Capacitance of the capacitor in the presence of slab will be μ

Answer: (6)

30. Two cells are connected between points A and B as shown. Cell 1 has emf of 12 V and internal resistance of 3Ω. Cell 2 has emf of 6 V and internal resistance of 6Ω. An external resistor R of 4Ω is connected across A and B. The current flowing through R will be __________ A.

Answer: (1)

Chemistry

SECTION-A

31. When the hydrogen ion concentration [H+]changes by a factor of 1000 , the value of pH of the solution

(1)   increases by 2 units

(2)   increases by 1000 units

(3)   decreases by 2 units

(4)   decreases by 3 units

Answer: (4)

32. Find out the major product from the following reaction.

Answer: (4)

33. Given below are two statements, one is labelled as Assertion 𝐀 and the other is labelled as Reason 𝐑

Assertion A: Carbon forms two important oxides – CO and CO2.CO is neutral whereas CO2 is acidic in nature

Reason 𝐑: CO2 can combine with water in a limited way to form carbonic acid, while CO is sparingly soluble in water.

In the light of the above statements, choose the most appropriate answer from the options given below

(1) Both A and R are correct but R is NOT the correct explanation of A

(2) A is correct but R is not correct

(3) Both A and R are correct and R is the correct explanation of A

(4) A is not correct but R is correct

Answer: (3)

34. Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason 𝐑

Assertion A: The alkali metals and their salts impart characteristic colour to reducing flame.

Reason R: Alkali metals can be detected using flame tests.

In the light of the above statements, choose the most appropriate answer from the options given below

(1) A is not correct but R is correct

(2) Both A and R are correct but R is NOT the correct explanation of A

(3) A is correct but R is not correct

(4) Both A and R are correct and R is the correct explanation of A

Answer: (1)

35. Potassium dichromate acts as a strong oxidizing agent in acidic solution. During this process, the oxidation state changes from

(1)   +2 to +1

(2)   +3 to +1

(3)   +6 to +2

(4)   +6 to +3

Answer: (4)

36. Match List I with List II

Choose the correct answer from the options given below:

(1)  A-III, B-IV, C-I, D-II

(2) A-III, B-II, C-IV, D-I

(3) A-III, B-I, C-IV, D-II

(4) A-III, B-IV, C-II, D-I

Answer: (4)

37. Which of the following represents the correct order of metallic character of the given elements ?

(1) Si < Be < Mg < K

(2) Be < Si < K < Mg

(3) Be < Si < Mg < K

(4) K < Mg < Be < Si

Answer: (1)

38. Match List I with List II

Choose the correct answer from the options given below:

(1) A-IV, B-I, C-II, D-III

(2) A-IV, B-III, C-II, D-I

(3) A-II, B-III, C-IV, D-I

(4) A-IV, B-III, C-I, D-II

Answer: (2)

39. Match List I with List II

Choose the correct answer from the options given below:

(1) A-III, B-IV, C-II, D-I

(2) A-III, B-II, C-I, D-IV

(3) A-I, B-IV, C-II, D-III

(4) A-III, B-II, C-IV, D-I

Answer: (1)

40. Match List I with List II

Choose the correct answer from the options given below:

(1) A-II, B-III, C-I, D-IV

(2) A-III, B-I, C-IV, D-II

(3) A-III, B-IV, C-I, D-II

(4) A-IV, B-III, C-I, D-II

Answer: (2)

41. What is the mass ratio of ethylene glycol (C2H6O2, molar mass =62 g/mol) required for making 500 g of 0.25 molal aqueous solution and 250 mL of 0.25 molal aqueous solution?

(1)   1 : 1

(2)   2 : 1

(3)   1 : 2

(4)   3 : 1

Answer: (2)

42. Match list I with List II

Choose the correct answer from the options given below:

(1) A-III, B-I, C-II, D-IV

(2) A-IV, B-I, C-III, D-II

(3) A-III, B-II, C-I, D-IV

(4) A-II, B-III, C-IV, D-I

Answer: (3)

43. Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason 𝐑

Assertion A : Butylated hydroxy anisole when added to butter increases its shelf life.

Reason R : Butylated hydroxy anisole is more reactive towards oxygen than food.

In the light of the above statements, choose the most appropriate answer from the options given below

(1) A is correct but R is not correct

(2) A is not correct but R is correct

(3) Both A and R are correct and R is the correct explanation of A

(4) Both A and R are correct but R is NOT the correct explanation of A

Answer: (3)

44. The isomeric deuterated bromide with molecular formula C4H8DBr having two chiral carbon atoms is

(1) 2 – Bromo – 2 – deuterobutane

(2) 2 – Bromo-1-deuterobutane

(3) 2 – Bromo – 1 – deutero – 2 – methylpropane

(4) 2 – Bromo −3 – deuterobutane

Answer: (4)

45. A chloride salt solution acidified with dil. HNO3 gives a curdy white precipitate, [A], on addition of AgNO3⋅[A] on treatment with NH4OH gives a clear solution, B. A and B are respectively

(1) AgCl & (NH4)[Ag(OH)2]

(2) AgCl & [Ag(NH3)2]Cl

(3) H[AgCl3] & (NH4)[Ag(OH)2]

(4) H[AgCl3] & [Ag(NH3)2]Cl

Answer: (2)

46. Statement I : Dipole moment is a vector quantity and by convention it is depicted by a small arrow with tail on the negative centre and head pointing towards the positive centre.

Statement II : The crossed arrow of the dipole moment symbolizes the direction of the shift of charges in the molecules.

In the light of the above statements, choose the most appropriate answer from the options given below:

(1) Statement I is incorrect but Statement II is correct

(2) Statement I is correct but Statement II is incorrect

(3) Both Statement I and Statement II are incorrect

(4) Both Statement I and Statement II are correct

Answer: (2)

47. ꞌA’ in the given reaction is

Answer: (4)

48. (A) Ammonium salts produce haze in atmosphere.

(B) Ozone gets produced when atmospheric oxygen reacts with chlorine radicals.

(C) Polychlorinated biphenyls act as cleansing solvents.

(D) ‘Blue baby’ syndrome occurs due to the presence of excess of sulphate ions in water.

Choose the correct answer from the options given below:

(1)   A and D only

(2)   A, B and C only

(3)   A and C only

(4)   B and C only

Answer: (3)

49. Given below are two statements:

Statement I: In froth floatation method a rotating paddle agitates the mixture to drive air out of it.

Iron pyrites are generally avoided Statement II: for extraction of iron due to environmental reasons.

In the light of the above statements, choose the correct answer from the options given below:

(1) Statement I is false but Statement II is true

(2) Both Statement I and Statement II are false

(3) Statement I is true but Statement II is false

(4) Both Statement I and Statement II are true

Answer: (1)

50. Which one among the following metals is the weakest reducing agent?

(1)   Li

(2)   K

(3)   Rb

(4)   Na

Answer: (4)

Section B

51. Total number of moles of AgCl precipitated on addition of excess of AgNO3 to one mole each of the following complexes [Co(NH3)4Cl2]Cl,[Ni(H2O)6]Cl2,[Pt(NH3)2Cl2] and [Pd(NH3)4]Cl2 is ____

Answer: (5)

52. The number of incorrect statement/s from the following is/are

(A) Water vapours are adsorbed by anhydrous calcium chloride.

(B) There is a decrease in surface energy during adsorption.

(C) As the adsorption proceeds, ΔH becomes more and more negative.

(D) Adsorption is accompanied by decrease in entropy of the system.

Answer: (2)

53. Number of hydrogen atoms per molecule of a hydrocarbon A having 85.8% carbon is ____ (Given: Molar mass of A = 84 g mol−1)

Answer: (12)

54. The number of given orbitals which have electron density along the axis is ________

Answer: (5)

55. 28.0 L of CO2 is produced on complete combustion of 16.8 L gaseous mixture of ethene and methane at 25°C and 1 atm. Heat evolved during the combustion process is________ kJ.

Given : ∆HC(CH4) = −900 kJ mol1

∆Hc(C2H4) = −1400 kJ mol1

Answer: (847)

56. Pt(s) |H2(g) (1bar)| |H + (aq) (1M)| |M3+(aq), M+(aq)|Pt(s)

The E cell for the given cell is 0.1115 V at 298 K when

The value of a is

Given : EθM3+/M+ = 0.2 V

Answer: (3)

57. The number of pairs of the solutions having the same value of the osmotic pressure from the following is (Assume 100% ionization)

(A) 0.500 M C2H5OH (aq) and 0.25 M KBr (aq)

(B) 0.100 M K4[Fe(CN)6] (aq) and 0.100 M FeSO4(NH4)2SO4 (aq)

(C) 0.05 M K4[Fe(CN)6] (aq) and 0.25 M NaCl (aq)

(D) 0.15 M NaCl(aq) and 0.1 M BaCl2(aq)

(E) 0.02 M KCl⋅MgCl2⋅6H2O(aq) and 0.05 M KCl(aq)

Answer: (4)

58. A first order reaction has the rate constant, = 4.6 × 10−3 s−1. The number of correct statement/s from the following is/are

Given: log 3 = 0.48

(A) Reaction completes in 1000 s.

(B) The reaction has a half-life of 500 s.

(C) The time required for 10% completion is 25 times the time required for 90% completion.

(D) The degree of dissociation is equal to (1 – e−kt)

(E) The rate and the rate constant have the same unit.

Answer: (1)

59. Based on the given figure, the number of correct statement/s is/are ___________

(A) Surface tension is the outcome of equal attractive and repulsive forces acting on the liquid molecule in bulk.

(B) Surface tension is due to uneven forces acting on the molecules present on the surface.

(C) The molecule in the bulk can never come to the liquid surface.

(D) The molecules on the surface are responsible for vapours pressure if system is a closed system.

Answer: (2)

60. Number of compounds giving (i) red colouration with ceric ammonium nitrate and also (ii) positive iodoform test from the following is

Answer: (3)

Mathematics

SECTION-A

61. Let Δ, ∇ ∈ {∧, ∨} be such that (p → q) Δ (p ∇ q) is a tautology. Then

(1)  Δ = V, ∇ = V

(2) Δ = V,∇ = Λ

(3) Δ = Λ, ∇ = V

(4) Δ = Λ, ∇ = Λ

Answer: (1)

62. If the four points, whose position vectors are  and  are coplanar, then α is equal to

(1)   73/17

(2)   107/17

(3)   −73/17

(4)   −107/17

Answer: (1)

63. The foot of perpendicular of the point (2, 0, 5) on the line  is (α, β, γ). Then, which of the following is NOT correct?

Answer: (1)

64. The equations of two sides of a variable triangle are x = 0 and y = 3, and its third side is a tangent to parabola y2 = 6x. The locus of its circumcentre is:

(1)   4y2 – 18y – 3x – 18 = 0

(2)   4y2 – 18y – 3x + 18 = 0

(3)   4y2 – 18y + 3x + 18 = 0

(4)   4y2 + 18y + 3x + 18 = 0

Answer: (3)

65. Let f(x) = 2Xn + λ, λ ∈ ℝ, n ∈ ℕ, and f(4) = 133, f(5) 255. Then the sum of all the positive integer divisors of (f(3) – f(2)) is

(1)   60

(2)   59

(3)   61

(4)   58

Answer: (1)

66. is equal to

(1)   51C445C4

(2)   52C345C3

(3)   52C445C4

(4)   51C345C3

Answer: (3)

67. Let the function f(x) = 2x3 + (2p − 7) x2 + 3(2p − 9) x − 6 have a maxima for some value of x < 0 and a minima for some value of x > 0.Then,the set of all values of p is

(1)   (0, 9/2)

(2)   (−∞, 9/2)

(3)   (−9/2, 9/2)

(4)   (9/2, ∞)

Answer: (2)

68. Let  and  where i = √−1.

If M = ATBA, then the inverse of the matrix AM2023 AT is

Answer: (4)

69. Let  and  Then  is equal to

Answer: (3)

70. The integral  is equal to

Answer: (2)

71. Let T and C respectively be the transverse and conjugate axes of the hyperbola 16x2 − y2 + 64x + 4y + 44 = 0.Then the area of the region above the parabola x2 = y + 4,   below the transverse axis T and on the right of the conjugate axis C is:

Answer: ()

72. Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that N−2,√3N,   N+2 are in geometric progression be k/48. Then the value of k is

(1)   8

(2)   16

(3)   2

(4)   4

Answer: (4)

73. If the function  is continuous at x = π/2, then 9λ + 6logeμ + μ6 – e6λ is equal to

(1)   10

(2)   2e4 + 8

(3)   11

(4)   8

Answer: (*)

74. The number of functions f:{1, 2, 3, 4} → {a∈ : ℤ|a| ≤ 8} satisfying  ∀ n ∈ {1, 2, 3) is

(1)   1

(2)   4

(3)   2

(4)   3

Answer: (2)

75. Let y = y(t) be a solution of the differential equation  where, α > 0, β > 0 and γ > 0. Then

(1)   is −1

(2)   is 1

(3)   does not exist

(4)   is 0

Answer: (4)

76. Let z be a complex number such that  z ≠ − Then z lies on the circle of radius 2 and centre

(1)   (2, 0)

(2)   (0, 2)

(3)   (0, −2)

(4)   (0, 0)

Answer: (3)

77. Let A, B, C be 3 × 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) A13 B26 − B26 A13 is symmetric

(S2)A26C13 − C13 A26 is symmetric

Then,

(1) Only S2 is true

(2) Both S1 and S2 are false

(3) Only S1 is true

(4) Both S1 and S2 are true

Answer: (1)

78. The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1,3,5,7,9 without repetition, is

(1)   12

(2)   120

(3)   72

(4)   6

Answer: (3)

79. Let f : ℝ → ℝ be a function defined by

f(x) = logm{√2(sin x – cos x) + m – 2}, for some m, such that the range of f is [0, 2]. Then the value of m is

(1)   5

(2)   4

(3)   3

(4)   2

Answer: (1)

80. The shortest distance between the lines x + 1 = 2y = −12z and x = y + 2 = 6z − 6 is

(1)   3/2

(2)   2

(3)   5/2

(4)   3

Answer: (2)

SECTION-B

81. 25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is k/10. Then the value of k is.

Answer: (9)

82. The remainder when (2023)2023 is divided by 35 is

Answer: (7)

83. Let a ∈ ℝ and let α, β be the roots of the equation x2 + 601/4x + a = 0. If α4 + β4 = −30, then the product of all possible values of a is

Answer: (45)

84. For the two positive numbers a, b is a, b and 1/18 are in a geometric progression, while 1/a, 10 and 1/b are in an arithmetic progression, then 16a + b is equal to

Answer: (3)

85. If m and n respectively are the numbers of positive and negative values of q in the interval [–p, p] that satisfy the equation  then mn is equal to

Answer: (25)

86. If the shortest distance between the line joining the points (1,2,3) and (2,3,4),and the line then  28a2 is equal to

Answer: (18)

87. Points P(–3,2),Q(9,10) and (a,4) lie on a circle C with PR as its diameter, The tangents to C at the points Q and R intersect at the point S. If S lies on the line 2x – ky = 1, then k is equal to

Answer: (3)

88. Suppose Anil’s mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil’s mother can offer 5 fruits to Anil is

Answer: (6860)

89. If  where m and n are coprime natural numbers, then m2 + n2 − 5 is equal to

Answer: (20)

90. A triangle is formed by X- axis, Y-axis and the line 3x + 4y = 4y = 60. Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is

Answer: (31)

## JEE Main Session 2 24th January 2023 Shift 2 Question Paper and Answer Key

JEE MAIN 24th January 2023 Shift 2

Physics

SECTION-A

IMPORTANT INSTRUCTIONS:

(1) The test is of 3 hours duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) Section-A: This section contains 20 multiple choice questions which have only one correct answer. each question carries 4 marks for correct answer and 1 mark for wrong answer.

(ii)  Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of 10. The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and 1 mark for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: A pendulum clock when taken to Mount Everest becomes fast.

Reason : The value of g (acceleration due to gravity) is less at Mount Everest than its value on the surface of earth.

In the light of the above statements, choose the most appropriate answer from the options given below

(1) Both 𝐀 and 𝐑 are correct but 𝐑 is NOT the correct explanation of 𝐀

(2) A is correct but 𝐑 is not correct

(3) Both 𝐀 and 𝐑 are correct and 𝐑 is the correct explanation of 𝐀

(4) 𝐀 is not correct but 𝐑 is correct

Answer: (4)

2. The frequency (v) of an oscillating liquid drop may depend upon radius (r) of the drop, density (ρ) of liquid and the surface tension (s) of the liquid as : v = raρbsc. The values of a, b and c respectively are

(1)   (−3/2, 1/2, 1/2)

(2)   (3/2, −1/2, 1/2)

(3)   (−3/2, −1/2, 1/2)

(4)   (3/2, 1/2, −1/2)

Answer: (3)

3. Given below are two statements:

Statement I : Acceleration due to earth’s gravity decreases as you go ‘up’ or ‘down’ from earth’s surface.

Statement II : Acceleration due to earth’s gravity is same at a height ‘h’ and depth ‘d’ from earth’s surface, if h = d.

In the light of above statements, choose the most appropriate answer form the options given below

(1) Both Statement I and Statement II are incorrect

(2) Statement I is incorrect but statement II is correct

(3) Both Statement I and II are correct

(4) Statement I is correct but statement II is incorrect

Answer: (4)

4. A long solenoid is formed by winding 70 turns cm–1. If 2.0 A current flows, then the magnetic field produced inside the solenoid is _______ (μ0 = 4π × 107 TmA1)

(1)   88 × 104 T

(2)   352 × 104 T

(3)   176 × 104 T

(4)   1232 × 104 T

Answer: (3)

5. The electric potential at the centre of two concentric half rings of radii R1 and R2, having same linear charge density 𝜆 is :

(1)   λ/2ε0

(2)   λ/4ε0

(3)   2λ/ε0

(4)   λ/ε0

Answer: (1)

6. If the distance of the earth from Sun is 1.5 × 106 Then the distance of an imaginary planet from Sun, if its period of revolution is 2.83 years is :

(1)   6 × 106 km

(2)   3 × 106 km

(3)   3 × 107 km

(4)   6 × 107 km

Answer: (2)

7. A photon is emitted in transition from n = 4 to n = 1 level in hydrogen atom. The corresponding wavelength for this transition is (given, h = 4 × 10−15 eVs ) :

(1)   99.3 nm

(2)   941 nm

(3)   974 nm

(4)   94.1 nm

Answer: (4)

8. A cell of emf 90 V is connected across series combination of two resistors each of 100Ω resistance. A voltmeter of resistance 400Ω is used to measure the potential difference across each resistor. The reading of the voltmeter will be:

(1)   90 V

(2)   45 V

(3)   80 V

(4)   40 V

Answer: (2)

9. If two vectors  and  are perpendicular to each other. Then, the value of m will be:

(1)   −1

(2)   3

(3)   2

(4)   1

Answer: (3)

10. The electric field and magnetic field components of an electromagnetic wave going through vacuum is described by

Ex = E0sin(kz − ωt)

By = B0sin(kz − ωt)

Then the correct relation between Eo and Bo is given by

(1)   Eo Bo = ωk

(2)   Eo = kBo

(3)   kEo = ωBo

(4)   ωEo = kBo

Answer: (3)

11. The logic gate equivalent to the given circuit diagram is :

(1)   NAND

(2)   OR

(3)   AND

(4)   NOR

Answer: (1)

12. Let γ1 be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and γ2 be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio, γ1/ γ2 is :

(1)   25/21

(2)   35/27

(3)   21/25

(4)   27/35

Answer: (1)

13. When a beam of white light is allowed to pass through convex lens parallel to principal axis, the different colours of light converge at different point on the principle axis after refraction. This is called:

(1) Spherical aberration

(2) Polarisation

(3) Chromatic aberration

(4) Scattering

Answer: (*)

14. A metallic rod of length ‘L’ is rotated with an angular speed of ‘ω’ normal to a uniform magnetic field ‘B’ about an axis passing through one end of rod as shown in figure. The induced emf will be:

Answer: (4)

15. An a-particle, a proton and an electron have the same kinetic energy. Which one of the following is correct in case of their de-Broglie wavelength:

(1)   λα < λp < λe

(2)   λα = λp = λe

(3)   λα > λp > λe

(4)   λα > λp < λe

Answer: (1)

16. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason

Assertion A : Steel is used in the construction of buildings and bridges.

Reason R : Steel is more elastic and its elastic limit is high.

In the light of above statements, choose the most appropriate answer from the options given below

(1) Both 𝐀 and 𝐑 are correct and 𝐑 is the correct explanation of 𝐀

(2) Both 𝐀 and 𝐑 are correct but 𝐑 is NOT the correct explanation of 𝐀

(3) A is correct but 𝐑 is not correct

(4) A is not correct but 𝐑 is correct

Answer: (1)

17. In an Isothermal change, the change in pressure and volume of a gas can be represented for three different temperature; T3 > T2 > T1 as:

Answer: (3)

18. Match List I with List II

Choose the correct answer from the options given below:

(1) A-II, B-I, C-IV, D-III

(2) A-I, B-III, C-II, D-IV

(3) A-IV, B-III, C-I, D-II

(4) A-II, B-III, C-I, D-IV

Answer: (1)

19. A body of mass 200 g is tied to a spring of spring constant 12.5 N/m, while the other end of spring is fixed at point O. If the body moves about O in a circular path on a smooth horizontal surface with constant angular speed 5rad/s. Then the ratio of extension in the spring to its natural length will be :

(1)   2 : 5

(2)   1 : 1

(3)   2 : 3

(4)   1 : 2

Answer: (3)

20. The velocity time graph of a body moving in a straight line is shown in figure.

The ratio of displacement to distance travelled by the body in time 0 to 10 s is :

(1)   1 : 1

(2)   1 : 2

(3)   1 : 3

(4)   1 : 4

Answer: (3)

SECTION-B

21. A body of mass 1 kg begins to move under the action of a time dependent force

Answer: (100)

22. A convex lens of refractive index 1.5 and focal length 18 cm in air is immersed in water. The change in focal length of the lens will be ________ cm

(Given refractive index of water = 4/3)

Answer: (54)

23. The energy released per fission of nucleus of ⁡240X is 200MeV. The energy released if all the atoms in 120 g of pure ⁡240X undergo fission is ______ × 1025MeV (Given NA = 6 × 1023)

Answer: (6)

24. A uniform solid cylinder with radius R and length L has moment of inertia I1, about the axis of the cylinder. A concentric solid cylinder of radius Rꞌ = R/2 and Length Lꞌ = L/2 is carved out of the original cylinder. If I2 is the moment of inertia of the carved out portion of the cylinder then I1/I2 = ________ (Both I1 and I2 are about the axis of the cylinder)

Answer: (32)

25. A parallel plate capacitor with air between the plate has a capacitance of 15pF. The separation between the plate becomes twice and the space between them is filled with a medium of dielectric constant 3.5. Then the capacitance becomes x/4pF. The value of x is _______

Answer: (105)

26. A single turn current loop in the shape of a right angle triangle with sides 5 cm,12 cm,13 cm is carrying a current of 2 A. The loop is in a uniform magnetic field of magnitude 0.75 T whose direction is parallel to the current in the 13 cm side of the loop. The magnitude of the magnetic force on the 5 cm side will be x/130 N. The value of x is ______

Answer: (9)

27. A mass m attached to free end of a spring executes SHM with a period of 1 s. If the mass is increased by 3 kg the period of oscillation increases by one second, the value of mass m is _____ kg.

Answer: (1)

28. If a copper wire is stretched to increase its length by 20%. The percentage increase in resistance of the wire is _________ %

Answer: (44)

29. Three identical resistors with resistance R = 12 Ω and two identical inductors with self inductance L = 5mH are connected to an ideal battery with emf of 12 V as shown in figure. The current through the battery long after the switch has been closed will be _______ A.

Answer: (3)

30. A Spherical ball of radius 1 mm and density 10.5 g/cc is dropped in glycerine of coefficient of viscosity 9.8 poise and density 1.5 g/cc. Viscous force on the ball when it attains constant velocity is 3696 × 10x The value of x is (Given, g = 9.8 m/s2 and π = 22/7)

Answer: (7)

Chemistry

SECTION-A

31. Identify the correct statements about alkali metals.

(A) The order of standard reduction potential (M + ∣M) for alkali metal ions is Na>Rb>Li.

(B) CsI is highly soluble in water.

(C) Lithium carbonate is highly stable to heat.

(D) Potassium dissolved in concentrated liquid ammonia is blue in colour and paramagnetic.

(E) All the alkali metal hydrides are ionic solids.

Choose the correct answer from the options given below:

(1)   C and E only

(2)   A, B and E only

(3)   A, B, D only

(4)   A and E only

Answer: (4)

32. Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason 𝐑

Assertion A: Beryllium has less negative value of reduction potential compared to the other alkaline earth metals.

Reason  : Beryllium has large hydration energy due to small size of Be2+ but relatively large value of atomization enthalpy

In the light of the above statements, choose the most appropriate answer from the options given below

(1) A is not correct but R is correct

(2) A is correct but 𝑅 is not correct

(3) Both A and R are correct and R is the correct explanation of A

(4) Both A and R are correct but R is NOT the correct explanation of A

Answer: (3)

33. A student has studied the decomposition of a gas AB3 at 25∘ He obtained the following data.

The order of the reaction is

(1)   0(zero)

(2)   0.5

(3)   1

(4)   2

Answer: (2)

34. K2Cr2O7 paper acidified with dilute H2SO4 turns green when exposed to

(1)   Carbon dioxide

(2)   Sulphur trioxide

(3)   Sulphur dioxide

(4)   Hydrogen sulphide

Answer: (3)

35. Which will undergo deprotonation most readily in basic medium?

(1)   c only

(2)   a only

(3)   Both a and c

(4)   b only

Answer: (2)

36. The hybridization and magnetic behaviour of cobalt ion in [Co(NH3)6]3+ complex, respectively is

(1)   d2sp3 and paramagnetic

(2)   sp3d2 and diamagnetic

(3)   d2sp3 and diamagnetic

(4)   sp3d2 and paramagnetic

Answer: (3)

37. Given below are two statements:

In the light of the above statements, choose the correct answer from the options given below :

(1) Statement I is false but Statement II is true

(2) Statement I is true but Statement II is false

(3) Both Statement I and Statement II are true

(4) Both Statement I and Statement II are false

Answer: (2)

38. Which of the following cannot be explained by crystal field theory?

(1)   The order of spectrochemical series

(2)   Stability of metal complexes

(3)   Magnetic properties of transition metal complexes

(4)   Colour of metal complexes

Answer: (1)

39. The number of s-electrons present in an ion with 55 protons in its unipositive state is

(1)   8

(2)   10

(3)   9

(4)   12

Answer: (2)

40. Which one amongst the following are good oxidizing agents?

(A) Sm2+  (B) Ce2+ (C) Ce4+ (D) Tb4+

Choose the most appropriate answer from the options given below:

(1)   D only

(2)   C only

(3)   C and D only

(4)   A and B only

Answer: (3)

41. Which one amongst the following are good oxidizing agents?

Answer: (1)

42. Match List I with List II

Choose the correct answer from the options given below:

(1) A-I, B-III, C-II, D-IV

(2) A-IV, B-III, C-II, D-I

(3) A-I, B-II, C-III, D-IV

(4)A-II, B-I, C-III, D-IV

Answer: (3)

43. Find out the major products from the following reaction

Answer: (2)

44. Given below are two statements, one is labelled as Assertion 𝐀 and the other is labelled as Reason 𝐑

Assertion : Benzene is more stable than hypothetical cyclohexatriene

Reason : The delocalized π electron cloud is attracted more strongly by nuclei of carbon atoms.

In the light of the above statements, choose the correct answer from the options given below:

(1) Both A and R are correct and R is the correct explanation of A

(2) Both A and R are correct but R is NOT the correct explanation of A

(3) A is false but R is true

(4) A is true but 𝑅 is false

Answer: (1)

45. In which of the following reactions the hydrogen peroxide acts as a reducing agent?

(1)   PbS + 4H2O2 → PbSO4 + 4H2O

(2)   Mn2+ + H2O2 → Mn4+ + 2OH

(3)   HOCl + H2O2 → H3O+ + Cl + O2

(4)   2Fe2+ + H2O2 → 2Fe3+ + 2OH

Answer: (3)

46. Given below are two statements:

Statement I : Pure Aniline and other arylamines are usually colourless.

Statement II : Arylamines get coloured on storage due to atmospheric reduction

In the light of the above statements, choose the most appropriate answer from the options given below:

(1) Both Statement I and Statement II are incorrect

(2) Statement I is incorrect but Statement II is correct

(3) Statement I is correct but Statement II is incorrect

(4) Both Statement I and Statement II are correct

Answer: (3)

47. Correct statement is:

(1) An average human being consumes nearly 15 times more air than food

(2) An average human being consumes 100 times more air than food

(3) An average human being consumes equal amount of food and air

(4) An average human being consumes more food than air

Answer: (1)

48. What is the number of unpaired electron(s) in the highest occupied molecular orbital of the following species : N2; N2+ ; O2; O2+?

(1)   2, 1, 0, 1

(2)   0, 1, 0, 1

(3)   0, 1, 0, 1

(4)   2, 1, 2, 1

Answer: (2)

49. The metal which is extracted by oxidation and subsequent reduction from its ore is:

(1)   Ag

(2)   Fe

(3)   Cu

(4)   Al

Answer: (1)

50. Choose the correct colour of the product for the following reaction.

(1)   White

(2)   Red

(3)   Blue

(4)   Yellow

Answer: (2)

SECTION-B

51. Following figure shows spectrum of an ideal black body at four different temperatures. The number of correct statement/s from the following is ____________.

(A) T4 > T3 > T2 > T1

(B) The black body consists of particles performing simple harmonic motion.

(C) The peak of the spectrum shifts to shorter wavelength as temperature increases.

(E) The given spectrum could be explained using quantisation of energy.

Answer: (2)

52. The number of units, which are used to express concentration of solutions from the following is______ Mass percent, Mole, Mole fraction, Molarity, ppm, Molality

Answer: (5)

53. The number of statement/s which are the characteristics of physisorption is______________

(A) It is highly specific in nature

(B) Enthalpy of adsorption is high

(C) It decreases with increase in temperature

(D) It results into unimolecular layer

(E) No activation energy is needed

Answer: (2)

54. Sum of π – bonds present in peroxodisulphuric acid and pyrosulphuric acid is:

Answer: (8)

55. If the pKa of lactic acid is 5, then the pH of 0.005M calcium lactate solution at 25°C is _________ × 10–1 (Nearest integer)

Answer: (85)

56. The total pressure observed by mixing two liquids A and B is 350 mmHg when their mole fractions are 0.7 and 0.3 respectively. The total pressure become 410 mmHg if the mole fractions are changed to 0.2 and 0.8 respectively for A and B. The vapour pressure of pure A is________ mm Hg. (Nearest integer) Consider the liquids and solutions behave ideally.

Answer: (314)

57. The number of statement/s, which are correct with respect to the compression of carbon dioxide from point (a) in the Andrews isotherm from the following is _________

(A) Carbon dioxide remains as a gas upto point (b)

(B) Liquid carbon dioxide appears at point (c)

(C) Liquid and gaseous carbon dioxide coexist between points (b) and (c)

(D) As the volume decreases from (b) to (c), the amount of liquid decreases

Answer: (4)

58. Maximum number of isomeric monochloro derivatives which can be obtained from 2, 2, 5, 5 tetramethylhexane by chlorination is ______

Answer: (3)

59. Total number of tripeptides possible by mixing of valine and proline is ________

Answer: (8)

60. One mole of an ideal monoatomic gas is subjected to changes as shown in the graph. The magnitude of the work done (by the system or on the system) is _______ J (nearest integer)

Answer: (6)

Mathematics

SECTION-A

61. If, f(x) = x3 – x2f ꞌ (1) + xf ꞌꞌ(2) – f ꞌꞌ(3), x ∈ ℝ then

(1) f(1) + f(2) + f(3) = f(0)

(2) 2f(0) − f(1) + f(3) = f(2)

(3) 3f(1) + f(2) = f(3)

(4)  f(3) − f(2)= f(1)

Answer: (2)

62. If the system of equations

X + 2y + 3z = 3

4x + 3y – 4z = 4

8x + 4y – λz = 9 + μ

has infinitely many solutions, then the ordered pair (λ, μ) is equal to :

(1)   (−72/5, 21/5)

(2)   (−72/5, −21/5)

(3)   (72/5, −21/5)

(4)   (72/5, 21/5)

Answer: (3)

63. If, then  then

(1)   1011

(2)   2010

(3)   1010

(4)   2011

Answer: (1)

64. Let  Let  be parallel to  be perpendicular to  then the value of  is

(1)   7

(2)   9

(3)   6

(4)   11

Answer: (1)

65. Let y = y(x) be the solution of the differential equation (x2 − 3y2)dx + 3xydy = 0, y(1) = 1.  Then 6y2(e) is equal to

(1)   2e2

(2)   3e2

(3)   e2

(4)

Answer: (1)

66. The locus of the mid points of the chords of the circle C1 : (x − 4)2 + (y − 5)2 = 4 which subtend an angle θ1 at the centre of the circle C1, is a circle of radius ri. If  and  then θ2 is equal to

(1)   π/4

(2)   π/2

(3)   π/6

(4)   3π/4

Answer: (2)

67. The number of real solutions of the equation  is

(1)   0

(2)   3

(3)   4

(4)   2

Answer: (1)

68. Let A be a 3×3 matrix such that |adj⁡(adj⁡(adj⁡A))|=124 Then |A−1adj⁡A| is equal to

(1)   √6

(2)   2√3

(3)   12

(4)   1

Answer: (2)

69. is equal to

(1)   2π

(2)   π/6

(3)   π/3

(4)   π/2

Answer: (1)

70. The number of square matrices of order 5 with entries form the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is

(1)   125

(2)   225

(3)   150

(4)   120

Answer: (4)

71. If (30C1)2 + 2(30C2)2 + 3(30C3)2 + … + 30(30C30)2 then α is equal to :

(1)   30

(2)   10

(3)   60

(4)   15

Answer: (4)

72. Let the plane containing the line of intersection of the planes P1: x + (λ + 4)y + z = 1 and P2: 2x + y+ z = 2 pass through the points (0, 1, 0) and (1, 0, 1). Then the distance of the point (2λ, λ ,−λ) from the plane P2 is

(1)   4√6

(2)   3√6

(3)   5√6

(4)   2√6

Answer: (2)

73. Let f(x) be a function such that and f(x + y) = f(x) ∙ f(y) for all x, y ∈ If f(1) = 3 and  then the value of n is

(1)   9

(2)   6

(3)   8

(4)   7

Answer: (4)

74. Let the six numbers a1, a2, a3, a4, a5, a6, be in A.P. and a1 + a3 = 10. If the mean of these six numbers is 19/2 and their variance is σ2, then 8σ2 is equal to :

(1)   210

(2)   220

(3)   200

(4)   105

Answer: (1)

75. The equations of the sides AB and AC of a triangle ABC are (λ + 1) x + λy = 4 and λx + (1 − λ) y + λ = 0 respectively. Its vertex A is on the y – axis and its orthocentre is (1,2). The length of the tangent from the point C to the part of the parabola y2 = 6x in the first quadrant is :

(1)   4

(2)   2

(3)   √6

(4)   2√2

Answer: (4)

76. Let p and q be two statements. Then ∼(p ∧ (p ⇒ ∼q)) is equivalent to

(1)   p ∨ (p ∧ q)

(2)   p ∨ (p ∧ (∼q))

(3)   (∼p) ∨ q

(4)   p ∨ ((∼p) ∧ q)

Answer: (3)

77. The set of all values of a for which limxa([x – 5] – [2x + 2]) = 0, where [∝] denotes the greatest integer less than or equal to α is equal to

(1) [−7.5, −6.5)

(2) [−7.5, −6.5]

(3) (−7.5, −6.5]

(4) (−7.5, −6.5)

Answer: (4)

78. If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes x + 2y + z = 0 and 3y – z = 3 is (α, β, γ), then α + β + γ is equal to

(1)   3

(2)   1

(3)   −1

(4)   5

Answer: (4)

79. The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is

(1)   168

(2)   220

(3)   120

(4)   48

Answer: (1)

80. The value of  is

Answer: (1)

SECTION-B

81. If the shortest distance between the lines  and  is 6, then the square of sum of all possible values of λ is

Answer: (384)

82. Three urns A, B and C contain 4 red, 6 black; 5 red, 5 black; and λ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4 then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola y2 = λx with one vertex at the vertex of the parabola, is

Answer: (432)

83. Let S ={θ ∈ [0, 2π):tan⁡(π cos⁡θ) + tan⁡(π sin θ) = 0}.

Then  is equal to

Answer: (2)

84. If  then value of n is

Answer: (5)

85. Let the sum of the coefficients of the first three terms in the expansion of  be 376. Then the coefficient of x4 is

Answer: (405)

86. The equations of the sides AB, BC and CA of a triangle ABC are : 2x + y = 0, x + py = 21a, (a ≠ 0) and x – y = 3 respectively. Let P(2, a) be the centroid of △ Then (BC)2 is equal to

Answer: (122)

87. Let  is equal to

Answer: (8)

88. The minimum number of elements that must be added to the relation R={(a, b),(b, c),(b, d)} on the set {a, b, c, d} so that it is an equivalence relation, is

Answer: (13)

89. If the area of the region bounded by the curves y2 − 2y = −x, x + y = 0 is A, then 8 A is equal to

Answer: (36)

90. Let f be a differentiable function defined on [0, π/2] such that f(x) > 0 and  is equal to

Answer: (27)

## JEE Main Session 1 1st February 2023 Shift 1 Question Paper and Answer Key

JEE MAIN 1st February 2023 Shift 1

Physics

SECTION-A

IMPORTANT INSTRUCTIONS:

(1) The test is of 3 hours duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of Physics, Chemistry and Mathematics having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) Section-A: This section contains 20 multiple choice questions which have only one correct answer. each question carries 4 marks for correct answer and 1 mark for wrong answer.

(ii) Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of 10. The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and 1 mark for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. A child stands on the edge of the cliff 10 m above the ground and throws a stone horizontally with an initial speed of 5 ms−1. Neglecting the air resistance, the speed with which the stone hits the ground will be __ ms−1 (given, g = 10 ms−2).

(1)   15

(2)   20

(3)   30

(4)   25

Answer: (1)

2. Let σ be the uniform surface charge density of two infinite thin plane sheets shown in figure. Then the electric fields in three different region EI, EII and EIII are:

Answer: (3)

3. A mercury drop of radius 10−3 m is broken into 125 equal size droplets. Surface tension of mercury is 0.45 Nm−1. The gain in surface energy is:

(1)   28 × 105 J

(2)   17.5 × 105 J

(3)   5 × 105 J

(4)   2.26 × 105 J

Answer: (4)

4. If earth has a mass nine times and radius twice to that of a planet P. Then  will be the minimum velocity required by a rocket to pull out of gravitational force of P, where υe is escape velocity on earth. The value of x is

(1)   1

(2)   3

(3)   18

(4)   2

Answer: (4)

5. A sample of gas at temperature T is adiabatically expanded to double its volume. The work done by the gas in the process is (given, γ = 3/2) :

(1)

(2)   W = RT[2 − √2]

(3)   W = TR[√2 – 2]

(4)

Answer: (2)

6. represents the equation of state of some gases. Where P is the pressure, 𝑉 is the volume, T is the temperature and a, b, R are the constants. The physical quantity, which has dimensional formula as that of b2/a, will be:

(1)   Compressibility

(2)   Energy density

(3)   Modulus of rigidity

(4)   Bulk modulus

Answer: (1)

7. The equivalent resistance between A and B of the network shown in figure:

(1)

(2)   21R

(3)   14R

(4)

Answer: (1)

8. Match List I with List II:

Choose the correct answer from the options given below:

(1) A-IV, B-III, C-I, D-II

(2) A-IV, B-II, C-I, D-III

(3) A-II, B-IV, C-I, D-III

(4) A-II, B-I, C-III, D-IV

Answer: (3)

9. An object moves with speed 𝑣1, 𝑣2 and 𝑣3 along a line segment AB, BC and CD respectively as shown in figure. Where AB = BC and AD = 3AB, then average speed of the object will be:

Answer: (3)

10. ʹn’ polarizing sheets are arranged such that each makes an angle 45° with the preceding sheet. An unpolarized light of intensity I is incident into this arrangement. The output intensity is found to be I/64. The value of n will be:

(1)   4

(2)   3

(3)   5

(4)   6

Answer: (4)

11. Match List I with List II:

Choose the correct answer from the options given below:

(1) A-I, B-III, C-IV, D-II

(2) A-IV, B-I, C-II, D-III

(3) A-IV, B-III, C-II, D-I

(4) A-I, B-II, C-III, D-IV

Answer: (2)

12. A proton moving with one tenth of velocity of light has a certain de Broglie wavelength of 𝜆. An alpha particle having certain kinetic energy has the same de-Brogle wavelength 𝜆. The ratio of kinetic energy of proton and that of alpha particle is:

(1)   2 : 1

(2)   1 : 2

(3)   1 : 4

(4)   4 : 1

Answer: (3)

13. A block of mass 5 kg is placed at rest on a table of rough surface. Now, if a force of 30 N is applied in the direction parallel to surface of the table, the block slides through a distance of 50 m in an interval of time 10 s. Coefficient of kinetic friction is (given, g = 10 ms−2):

(1)   0.60

(2)   0.25

(3)   0.75

(4)   0.50

Answer: (4)

14. Given below are two statements:

Statement I: Acceleration due to gravity is different at different places on the surface of earth.

Statement II: Acceleration due to gravity increases as we go down below the earth’s surface.

In the light of the above statements, choose the correct answer from the options given below

(1) Statement I is false but Statement II is true

(2)  Statement I is true but Statement II is false

(3) Both Statement I and Statement II are false

(4) Both Statement I and Statement II are true

Answer: (2)

15. Which of the following frequencies does not belong to FM broadcast.

(1)   64MHz

(2)   89MHz

(3)   99MHz

(4)   106MHz

Answer: (1)

16. The mass of proton, neutron and helium nucleus are respectively 1.0073u, 1.0087u and 4.0015u. The binding energy of helium nucleus is:

(1)   28.4MeV

(2)   56.8 MeV

(3)   14.2 MeV

(4)   7.1 MeV

Answer: (1)

17. A steel wire with mass per unit length 7.0 × 10−3 kg m−1 is under tension of 70 N. The speed of transverse waves in the wire will be:

(1)   100 m/s

(2)   10 m/s

(3)   50 m/s

(4)   200 πm/s

Answer: (1)

18. Match List I with List II:

Choose the correct answer from the options given below:

(1) A-II, B-III, C-I, D-IV

(2) A-I, B-II, C-III, D-IV

(3) A-II, B-I, C-III, D-IV

(4) A-III, B-I, C-II, D-IV

Answer: (1)

19. Find the magnetic field at the point P in figure. The curved portion is a semicircle connected to two long straight wires.

Answer: (2)

20. The average kinetic energy of a molecule of the gas is

(1) proportional to absolute temperature

(2) proportional to pressure

(3) proportional to volume

(4) dependent on the nature of the gas

Answer: (1)

SECTION-B

21. A small particle moves to position  from its initial position  under the action of force  The value of work done will be ______ J.

Answer: (40)

22. A certain pressure ‘P’ is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to 0.01% whereas the liquid gets compressed to 0.03%. The ratio of Bulk modulus of water to that of the liquid is 3/x.

The value of x is _______.

Answer: (1)

23. A light of energy 12.75eV is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is  The value of x is _______ (use h = 4.14 × 1015 eVs, c = 3 × 108 ms1).

Answer: (828)

24. A charge particle of 2μC accelerated by a potential difference of 100 V enters a region of uniform magnetic field of magnitude 4mT at right angle to the direction of field. The charge particle completes semicircle of radius 3 cm inside magnetic field. The mass of the charge particle is ______ × 10−18

Answer: (144)

25. The amplitude of a particle executing SHM is 3 cm. The displacement at which its kinetic energy will be 25% more than the potential energy is: ________ cm.

Answer: (2)

26. In an experiment to find emf of a cell using potentiometer, the length of null point for a cell of emf 1.5 V is found to be 60 cm. If this cell is replaced by another cell of emf E, the length-of null point increases by 40 cm. The value of E is  The value of x is ________.

Answer: (25)

27. A thin cylindrical rod of length 10 cm is placed horizontally on the principle axis of a concave mirror of focal length 20 cm. The rod is placed in a such a way that mid point of the rod is at 40 cm from the pole of mirror. The length of the image formed by the mirror will be x/3 cm. The value of x is ______.

Answer: (32)

28. A solid cylinder is released from rest from the top of an inclined plane of inclination 30° and length 60 cm. If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is ________ ms−1.

(Given g=10 ms−2 )

Answer: (2)

29. A series LCR circuit is connected to an ac source of 220 V,50 Hz. The circuit contain a resistance R = 100Ω and an inductor of inductive reactance XL = 79.6 Ω. The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be ________ μ

Answer: (40)

30. Two equal positive point charges are separated by a distance 2a. The distance of a point from the centre of the line joining two charges on the equatorial line (perpendicular bisector) at which force experienced by a test charge q0 becomes maximum is a/√ The value of x is ________.

Answer: (2)

Chemistry

SECTION-A

31. A solution of FeCl3 when treated with K4[Fe(CN)6] gives a prussian blue precipitate due to the formation of

(1) K[Fe2(CN)6]

(2) Fe4[Fe(CN)6]3

(3) Fe[Fe(CN)6]

(4) Fe3[Fe(CN)6]2

Answer: (2)

32. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason 𝐑

Assertion A: Hydrogen is an environment friendly fuel.

Reason R: Atomic number of hydrogen is 1 and it is a very light element.

In the light of the above statements, choose the correct answer from the options given below

(1)  A is true but 𝐑 is false

(2) 𝐀 is false but 𝐑 is true

(3) Both 𝐀 and 𝐑 are true and 𝐑 is the correct explanation of 𝐀

(4) Both 𝐀 and 𝐑 are true but 𝐑 is NOT the correct explanation of 𝐀

Answer: (4)

33. Resonance in carbonate ion (CO32) is

Which of the following is true?

(1) All these structures are in dynamic equilibrium with each other.

(2) It is possible to identify each structure individually by some physical or chemical method.

(3) Each structure exists for equal amount of time.

(4) CO32− has a single structure i.e., resonance hybrid of the above three structures.

Answer: (4)

34. Match List I with List II

Choose the correct answer from the options given below:

(1) (A) −IV,(B)−II,(C)−I,(D)−III

(2) (A) −II,(B)−I,(C)−III,(D) –IV

(3) (A) – III, (B) – I, (C) – II, (D) – IV

(4) (A) – II, (B) −IV,(C)−I,(D) −III

Answer: (3)

35. Identify the incorrect option from the following:

Answer: (4)

36. But-2-yne is reacted separately with one mole of Hydrogen as shown below:

(A) A is more soluble than

(B)  B. The boiling point & melting point of A are higher and lower than B respectively.

(C) A is more polar than B because dipole moment of A is zero.

(D) Br2 adds easily to B than A.

Identify the incorrect statements from the options given below:

Answer: (2)

37. In the following reaction, ‘ A ‘ is

Answer: (3)

38. Highest oxidation state of Mn is exhibited in Mn2O7. The correct statements about Mn2O7 are

(A) Mn is tetrahedrally surrounded by oxygen atoms.

(B) Mn is octahedrally surrounded by oxygen atoms.

(C) Contains Mn-O-Mn bridge.

(D) Contains Mn-Mn bond.

Choose the correct answer from the options given below:

(1)   A and C only

(2)   A and D only

(3)   B and C only

(4)   B and D only

Answer: (1)

39. Match List I with List II

Choose the correct answer from the options given below:

(1) (A) – III, (B) – IV, (C) – II, (D) – I

(2) (A) – III, (B) – II, (C) – IV, (D) – I

(3) (A) – I, (B) – IV, (C) – II, (D) – III

(4) (A) −II,(B) −IV, (C) – I, (D) – III

Answer: (4)

40. The correct representation in six membered pyranose form for the following sugar [X] is

Answer: (2)

41. Which of the following complex will show largest splitting of d-orbitals ?

(1)   [FeF6]3

(2)   [Fe(C2O4)3]3

(3)   [Fe(CN)6]3

(4)   [Fe(NH3)6]3+

Answer: (3)

42. Which of the following are the example of double salt?

(A) FeSO4 ⋅ (NH4)2SO4 ⋅ 6H2O

(B) CuSO4, 4NH3H2O

(C) K2SO4 ⋅ Al2(SO4)3 ⋅ 24H2O

(D) Fe(CN)2 . 4KCN

Choose the correct answer

(1)   B and D only

(2)   A and C only

(3)   A and B only

(4)   A, B and D only

Answer: (1)

43. Decreasing order of dehydration of the following alcohols is

(1)   b > a > d > c

(2)   a > d > b > c

(3)   d > b > c > a

(4)   b > d > c >a

Answer: (4)

44. Given below are two statements:

Statement I: Chlorine can easily combine with oxygen to form oxides; and the product has a tendency to explode.

Statement II: Chemical reactivity of an element can be determined by its reaction with oxygen and halogens.

In the light of the above statements, choose the correct answer from the options given below

(1) Both the Statements I and II are true

(2) Both the Statements I and II are false

(3) Statement I is false but Statement II is true

(4) Statement I is true but Statement II is false

Answer: (1)

45. Choose the correct statement(s):

(A) Beryllium oxide is purely acidic in nature.

(B) Beryllium carbonate is kept in the atmosphere of CO2.

(C) Beryllium sulphate is readily soluble in water.

(D) Beryllium shows anomalous behavior.  Choose the correct answer from the options given below:

(1)   B, C and D only

(2)   A only

(3)   A, B and C only

(4)   A and B only

Answer: (1)

46. Which of the following represents the lattice structure of A95O containing A2+, A3+ and O2− ions? ⊙ A2+ ⊙ A3+ ⊙ O2−

(1)   A only

(2)   B and C only

(3)   A and B only

(4)   B only

Answer: (1)

47. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason 𝐑

Assertion A: In an Ellingham diagram, the oxidation of carbon to carbon monoxide shows a negative slope with respect to temperature.

Reason R:   CO tends to get decomposed at higher temperature.

In the light of the above statements, choose the correct answer from the options given below

(1) Both 𝐀 and 𝐑 are correct but 𝐑 is NOT the correct explanation of 𝐀

(2) Both 𝐀 and 𝐑 are correct and 𝐑 is the correct explanation of 𝐀

(3) A is correct but 𝐑 is not correct

(4) A is not correct but 𝐑 is correct

Answer: (3)

48. Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason 𝐑

Assertion A: Amongst He, Ne, Ar and Kr; 1 g of activated charcoal adsorbs more of Kr.

Reason R: The critical volume Vc ( cm3 mol−1) and critical pressure Pc (atm) is highest for Krypton but the compressibility factor at critical point Zc is lowest for Krypton.

In the light of the above statements, choose the correct answer from the options given below

(1) 𝐀 is true but 𝐑 is false

(2) Both 𝐀 and 𝐑 are true and 𝐑 is the correct explanation of 𝐀

(3) A is false but 𝐑 is true

(4) Both 𝐀 and 𝐑 are true but 𝐑 is NOT the correct explanation of 𝐀

Answer: (1)

49. Match List I with List II

Choose the correct answer from the options given below:

(1) (A) − III,(B) − IV,(C) − I,(D) − II

(2) (A) −II,(B) −I,(C) −III, (D) – IV

(3) (A) −III,(B)−IV,(C)−II,(D)−I

(4) (A) −I,(B) −II,(C) −III,(D) −IV

Answer: (2)

50. How can photochemical smog be controlled?

(1) By using catalytic convertors in the automobiles/industry.

(2) By complete combustion of fuel.

(3) By using tall chimneys.

(4) By using catalyst.

Answer: (1)

SECTION-B

51. (i) X(g) ⇌ Y(g) + Z(g) Kp1 = 3

(ii) A(g) ⇌ 2B(g) Kp2 = 1

If the degree of dissociation and initial concentration of both the reactants X(g) and A(g) are equal, then the ratio of the total pressure at equilibrium (p1/p2) is equal to x : 1. The value of x is ______ (Nearest integer)

Answer: (12)

52. Electrons in a cathode ray tube have been emitted with a velocity of 1000 ms−1. The number of following statements which is/are true about the emitted radiation is

Given : h = 6 × 10−34 Js, me = 9 × 10−31 kg.

(A) The deBroglie wavelength of the electron emitted is 666.67 nm.

(B) The characteristic of electrons emitted depend upon the material of the electrodes of the cathode ray tube.

(C) The cathode rays start from cathode and move towards anode.

(D) The nature of the emitted electrons depends on the nature of the gas present in cathode ray tube.

Answer: (2)

53. A and B are two substances undergoing radioactive decay in a container. The half life of A is 15 min and that of B is 5 min. If the initial concentration of B is 4 times that of A and they both start decaying at the same time, how much time will it take for the concentration of both of them to be same? _____ min.

Answer: (15)

54. Sum of oxidation states of bromine in bromic acid and perbromic acid is

Answer: (12)

55. 25 mL of an aqueous solution of KCl was found to require 20 mL of 1M AgNO3 solution when titrated using K2CrO4 as an indicator. What is the depression in freezing point of KCl solutions of the given concentration? ______ (Nearest integer).

(Given : Kf = 2.0 K kg mol1)

Assume (1) 100% ionization and

(2) density of the aqueous solution as 1 g mL1

Answer: (3)

56. At 25∘C, the enthalpy of the following processes are given:

What would be the value of X for the following reaction?  (Nearest integer)

H2O(g) → H(g) + OH(g)∆H° = XkJmol1

Answer: (499)

57. At what pH, given half cell MnO4(0.1M) ∣ Mn2+(0.001M) will have electrode potential of 1.282 V ? (Nearest Integer)

Answer: (3)

58. The density of 3M solution of NaCl is 1.0 g mL−1. Molality of the solution is ____ × 10−2 (Nearest integer).

Given: Molar mass of Na and Cl is 23 and 35.5 g mol−1 respectively.

Answer: (364)

59. Number of isomeric compounds with molecular formula C9H10O which (i) do not dissolve in NaOH (ii)do not dissolve in HCl.(iii) do not give orange   precipitate with 2,4DNP (iv) on hydrogenation give identical compound with molecular   formula C9H12O is

Answer: (2)

60. The total number of chiral compound/s from the following is

Answer: (2)

Mathematics

SECTION-A

61. f y = y(x) is the solution curve of the differential equation  y(0) = 1, then y(π/6) is equal to

Answer: (2)

62. Let R be a relation on ℝ, given by R = {(a, b) : 3a – 3b + √7 is an irrational number}.

Then R is

(1) an equivalence relation

(2) reflexive and symmetric but not transitive

(3) reflexive but neither symmetric nor transitive

(4) reflexive and transitive but not symmetric

Answer: (3)

63. For a triangle ABC, the value of cos⁡2A + cos⁡2B + cos 2C is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

(1)   perimeter of ∆ ABC is 18√3

(2)   sin 2A + sin 2B + sin 2C = sin A + sin B + sin C

(3)

(4)   area of ∆ ABC is 27√3/2

Answer: (4)

64. Let S be the set of all solutions of the equation Then  is equal to

(1)   π – 2sin1 (√3/4)

(2)   π – sin1 (√3/4)

(3)   −2π/3

(4)   0

Answer: (*)

65. Let S denote the set of all real values of 𝜆 such that the system of equations

λx + y + z = 1

x + λy + z = 1

x + y + λz = 1

is inconsistent, then  is equal to

(1)   4

(2)   12

(3)   6

(4)   2

Answer: (3)

66. In a binomial distribution B(n, p), the sum and the product of the mean and the variance are 5 and 6 respectively, then 6(n + p – q) is equal to

(1)   52

(2)   50

(3)   51

(4)   53

Answer: (1)

67. The combined equation of the two lines ax + by + c = 0 and aʹx + bʹy + cʹ = 0 can be written as (ax + by + c) (aʹx + bʹy + cʹ) = 0.

The equation of the angle bisectors of the lines represented by the equation 2x2 + xy – 3y2 = 0 is

(1)   x2 – y2 – 10xy = 0

(2)   x2 – y2 + 10xy = 0

(3)   3x2 + 5xy + 2y2 = 0

(4)   3x2 + xy – 2y2 = 0

Answer: (1)

68. The area enclosed by the closed curve C given by the differential equation  y(1) = 0 is 4π.

Let P and Q be the points of intersection of the curve C and the 𝑦-axis. If normals at 𝑃 and Q on the curve C intersect 𝑥-axis at points R and S respectively, then the length of the line segment RS is

(1)   2

(2)   4√3/3

(3)   2√3

(4)   2√3/3

Answer: (2)

69. The value of  is :

(1)   250/51!

(2)   251/50!

(3)   250/50!

(4)   251/51!

Answer: (1)

70. The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5 then the sum of cubes of the remaining two observations is

(1)   1216

(2)   1072

(3)   1456

(4)   1792

Answer: (2)

71. The sum to 10 terms of the series  is

(1)   55/111

(2)   56/111

(3)   58/111

(4)   59/111

Answer: (1)

72. The shortest distance between the lines  and  is

(1)   5√3

(2)   7√3

(3)   6√3

(4)   4√3

Answer: (3)

73. is equal to

(1)   loge 2

(2)   loge (3/2)

(3)   loge (2/3)

(4)   0

Answer: (1)

74. Let the image of the point P(2, −1, 3) in the plane x + 2y – z = 0 be Q. Then the distance of the plane 3x + 2y + z + 29 = 0 from the point Q is

(1)   24√2/7

(2)   2√14

(3)   3√14

(4)   22√2/7

Answer: (3)

75. Let f(x) = 2x + tan1 x and  Th