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

May 10, 2023 by suresh84

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) [h^1/2c⁻^1/2G¹]

(2) [h⁻^1/2c^1/2G^1/2]

(3) [n^1/2c^1/2G⁻^1/2]

(4) [h¹c¹G⁻¹]

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.

(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 μ_s = 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 xβ₀.β₀ 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 NC⁻¹ m⁻¹. If a = 2 cm, the charge contained in the cubical volume is Q × 10⁻¹⁴ The value of Q is _______. (Take ϵ₀ = 9 × 10⁻¹² 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 × 10⁻³ m/s. The value of x will be _______.

Answer: (2)

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 Δ_egH, the incorrect option is :

1) Δ_eg H(Te) < Δ_egH(PO)

(2) 2. Δ_egH(Se) < Δ_egH(S)

(3) Δ_egH(Cl) < Δ_egH(F)

(4) Δ_egH(I) < Δ_egH(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 K⁻¹, R = 8.3JK⁻¹ mol⁻¹.

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

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 : K_SP(AgBr) = 4.9 × 10⁻¹³ 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 mol⁻¹

atomic mass of C = 12 g mol⁻¹

atomic mass of Cl = 35.5 g mol⁻¹

density of dibromine = 3.2 g cm⁻³

density of CCl₄ = 1.6 g cm⁻³]

Answer: (139)

Mathematics

SECTION-A

61. Let αx = exp(x^βy^γ) be the solution of the differential equation 2x²y dy – (1 – xy²) dx = 0, x > 0, y(2) = √log_e 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 ∩ B^C) ∪ (B ∩ A^C) = ∅ and AR₂B if A ∪ B^C = B ∪ A^C, ∀ 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) = x^x, x > 0, then y”(2) – 2yʹ(2) is equal to :

(1) 4log_e 2 + 2

(2) 8log_e 2 – 2

(3) 4(log_e 2)² + 2

(4) 4(log_e 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

May 10, 2023 by suresh84

JEE MAIN 31^st 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.

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 ms⁻¹

(2) 400 ms⁻¹

(3) 20 ms⁻¹

(4) 10 ms⁻¹

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 c_p/c_v 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 ms⁻¹

(2) 18.96 × 10³ ms⁻¹

(3) 3.65 × 10³ ms⁻¹

(4) 6.32 × 10³ ms⁻¹

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, X_L = 100Ω, and X_C= 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, |V_B – V_D| = ________ 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.

(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

(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

(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 min⁻¹. The time required for the initial concentration of the reactant to reduce to its 1/32 level is ______ 10⁻² (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 M₈₃O_1.00. The metal M can exist in two oxidation states + 2 and +3.In the sample of M_0.83O_1.00, 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 × 10⁻³⁴ Js, Speed of light (c) = 3.0 × 10⁸ m/s and Avogadro’s Constant (N_A) = 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) + O_2(g) → 2CO(g)

When 12 g carbon is burnt in 48 g of oxygen, the volume of carbon monoxide produced is × 10⁻¹ L at STP [nearest integer]

[Given : Assume co as ideal gas, Mass of c is 12 g mol⁻¹, Mass of O is 16 g mol⁻¹ and molar volume of an ideal gas STP is 22.7 L mol⁻¹]

Answer: (227)

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

Answer: (5)

Mathematics

SECTION-A

61. The equation e^4x + 8e^3x + 13e^2x − 8e^x + 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)− cos⁻¹(sin θ) > 0, θ ∈ (0, 2π), holds. If αx + βx + sin (x – 6x + 10) + cos⁻¹(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! – 4a_n(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 = [a_ij], a_ij ∈ 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 A⁻¹)) ∙ 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 x⁻^3l is 2^αβ, 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 x⁻⁶, 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⁺¹P_n₋₁ : ²ⁿ⁻¹P_n = 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

May 10, 2023May 10, 2023 by suresh84

JEE MAIN 30^th 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.

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) E_I = 0, E_II = 0, E_III = 0

(2) E_I = 0, E_II = 0, E_III ≠ 0

(3) E_I ≠ 0, E_II = 0, E_III ≠ 0

(4) E_I ≠ 0, E_II = 0, E_III = 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 (I_rms) 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 × 10⁻⁵ T

(2) 4√3 × 10⁻⁴ T

(3) 3√3 × 10⁻⁵ T

(4) √3 × 10⁻⁴ 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 = R₀A^1/3, 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 pK_a 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

(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 K_f = 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

(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. N_A = 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 loga_b, log_c a and log_b 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 lim_x_→₁g(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 P^T = 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

May 10, 2023May 10, 2023 by suresh84

JEE MAIN 29^th 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.

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.

(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.

(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

(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 × 10⁻³ J

(2) 2.5 × 10⁻³ J

(3) 5 × 10⁻³ J

(4) 1.0 × 10⁻⁴ 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 ms⁻²)

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, μ_r = 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 C_rO₅ 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

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 × 10⁻⁵ L mol⁻¹ s⁻¹. 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 × 10⁻¹³

Based on above equilibria, the equilibrium constant of the reaction, is _______ × 10⁻³³ (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 = {w_n : 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 a_n = a_{n – 1}+ (n – 1), b_n = b_{n – 1}+ a_{n – 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 ∑_Z_∈_A_∩_B(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 {a_k} and {b_k}, k ∈ ℕ, be two G. P.s with common ratios r₁ and r₂ respectively such that a₁ = b₁ = 4 and r₁ < r₂. Let c_k = a_k + b_k, 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

May 10, 2023May 10, 2023 by suresh84

JEE MAIN 25^th 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.

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 ms⁻¹

(2) 20 ms⁻¹

(3) 8 ms⁻¹

(4) 25 ms⁻¹

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

(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.

(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 (R_e), 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 u_i is the surface energy of the original drop and u_f 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 X_L= 70Ω, and a capacitor of capacitive reactance X_C = 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.

(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.

(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 : ∆H_C(CH₄) = −900 kJ mol⁻¹

∆H_c(C₂H₄) = −1400 kJ mol⁻¹

Answer: (847)

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

The value of a is

Given : E^θM³⁺/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)

(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.

(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.

(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) ⁵¹C₄ – ⁴⁵C₄

(2) ⁵²C₃ – ⁴⁵C₃

(3) ⁵²C₄ – ⁴⁵C₄

(4) ⁵¹C₃ – ⁴⁵C₃

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 = A^TBA, then the inverse of the matrix AM²⁰²³ A^T 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 + 6log_eμ + μ⁶ – e⁶^λ 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) = log_√_m{√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 + 60^1/4x + 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

May 10, 2023May 9, 2023 by suresh84

JEE MAIN 24^th 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.

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 = r^aρ^bs^c. 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 × 10⁻⁷ TmA⁻¹)

(1) 88 × 10⁻⁴ T

(2) 352 × 10⁻⁴ T

(3) 176 × 10⁻⁴ T

(4) 1232 × 10⁻⁴ 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

E_x = E₀sin(kz − ωt)

B_y = B₀sin(kz − ωt)

Then the correct relation between E_o and B_o is given by

(1) E_oB_o = ωk

(2) E_o = kB_o

(3) kE_o = ωB_o

(4) ωE_o = kB_o

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) λ_α < λ_p < λ_e

(2) λ_α = λ_p = λ_e

(3) λ_α > λ_p > λ_e

(4) λ_α > λ_p < λ_e

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 × 10⁻^x 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.

(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.

(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

(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)

(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 r_i. 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 lim_x_→_a([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

May 10, 2023May 9, 2023 by suresh84

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.

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 E_I, E_II and E_III 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 × 10⁻⁵ J

(2) 17.5 × 10⁻⁵ J

(3) 5 × 10⁻⁵ J

(4) 2.26 × 10⁻⁵ 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 υ_e 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 × 10⁻¹⁵ eVs, c = 3 × 10⁸ ms⁻¹).

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 X_L= 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.

(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.

(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) [F_eF₆]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

(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.

(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 V_c ( cm³ mol⁻¹) and critical pressure P_c (atm) is highest for Krypton but the compressibility factor at critical point Z_c 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) K_p1 = 3

(ii) A(g) ⇌ 2B(g) K_p2 = 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, m_e = 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.

(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 : K_f = 2.0 K kg mol⁻¹)

Assume (1) 100% ionization and

(2) density of the aqueous solution as 1 g mL⁻¹

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° = XkJmol⁻¹

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 – 2sin⁻¹ (√3/4)

(2) π – sin⁻¹ (√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) log_e 2

(2) log_e (3/2)

(3) log_e (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 + tan⁻¹ x and Then

(1) min fʹ(x) = 1 + maxgʹ(x)

(2) max f(x) > max g(x)

(3) there exist 0 < x₁ < x₂ < 3 such that f(x) < g(x), ∀x ∈ (x₁, x₂)

(4) there exists

Answer: (2)

76. If the orthocentre of the triangle, whose vertices are (1, 2) (2, 3) and (3, 1) is (α, β), then the quadratic equation whose roots are α + 4β and 4α + β, is

(1) x² – 20x + 99 = 0

(2) x² – 19x + 90 = 0

(3) x² – 22x + 120 = 0

(4) x² – 18x + 80 = 0

Answer: (1)

77. Let S = {x: x ∈ ℝ and

Then n(S) is equal to

(1) 4

(2) 0

(3) 6

(4) 2

Answer: (1)

78. If the center and radius of the circle are respectively (α, β) and γ. Then 3(α + β + γ) is equal to

(1) 11

(2) 12

(3) 9

(4) 10

Answer: (2)

79. Let If α and β respectively are the maximum and the minimum values of f, then

(1) α² + β² = 9/2

(2) β² − 2√α = 19/4

(3) α² – β² = 4√3

(4) β² + 2√α = 19/4

Answer: (2)

80. The negation of the expression q ∨ ((∼q) ∧ p) is equivalent to

(1) (~p) ∨ (~q)

(2) p ∧ (~q)

(3) (~p) ∨ q

(4) (~p) ∧ (~q)

Answer: (4)

SECTION B

81. Let and be a vector such that If the minimum value of the scalar triple product and where m and n are coprime natural numbers, then m + n is equal to

Answer: (3501)

82. The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is

Answer: (50400)

83. The remainder, when 19²⁰⁰ + 23²⁰⁰ is divided by 49 is _____

Answer: (29)

84. The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is

Answer: (514)

85. Let f : ℝ → ℝ be a differentiable function such that If f(0) = e⁻², then 2f(0) – f(2) is equal to

Answer: (1)

86. If f(x) = x² + gʹ(1)x + gʺ(2) and g(x) = f(1)x² + xfʹ(x) + fʺ(x), then the value of f(4) – g(4) is equal to

Answer: (14)

87. Let A be the area bounded by the curve y = x|x − 3|, the x-axis and the ordinates x = −1 and x = 2. Then 12A is equal to

Answer: (62)

88. If where l, m, n ∈ ℕ, m and n are coprime then l + m + n is equal to

Answer: (63)

89. Let a₁ = 8, a₂, a₃, …, a_n be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is

Answer: (754)

90. A(2, 6, 2), B(−4, 0, λ), C(2, 3, −1) and D(4, 5, 0), |λ| ≤ 5 are the vertices of a quadrilateral ABCD. If its area is 18 square units, then 5 − 6λ is equal to

Answer: (11)

JEE Main Session 1 31st January 2023 Shift 1 Question Paper and Answer Key

May 10, 2023May 9, 2023 by suresh84

JEE MAIN 31^th January 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.

1. The maximum potential energy of a block executing simple harmonic motion is 25 J. A is amplitude of oscillation. At A/2, the kinetic energy of the block is :

(1) 18.75 J

(2) 9.75 J

(3) 37.5 J

(4) 12.5 J

Answer: (1)

2. The drift velocity of electrons for a conductor connected in an electrical circuit is V_d. The conductor in now replaced by another conductor with same material and same length but double the area of cross section. The applied voltage remains same. The new drift velocity of electrons will be

(1) V_d

(2) V_d/4

(3) 2V_d

(4) V_d/2

Answer: (1)

3. The initial speed of a projectile fired from ground is u. At the highest point during its motion, the speed of projectile is The time of flight of the projectile is :

(1) 2u/g

(2) u/2g

(3) √3u/g

(4) u/g

Answer: (4)

4. The correct relation between γ = c_p/c_v and temperature T is :

(1) γαT⁰

(2) γαT

(3)

(4)

Answer: (1)

5. The effect of increase in temperature on the number of electrons in conduction band (n_e) and resistance of a semiconductor will be as:

(1) Both n_e and resistance increase

(2) Both n_e and resistance decrease

(3) n_e decreases, resistance increases

(4) n_e increases, resistance decreases

Answer: (4)

6. The amplitude of 15sin⁡(1000πt) is modulated by 10sin⁡(4πt) signal. The amplitude modulated signal contains frequency (ies) of

(A) 500 Hz (B) 2 Hz (C) 250 Hz (D) 498 Hz (E) 502 Hz

Choose the correct answer from the options given below:

(1) A only

(2) B only

(3) A and B only

(4) A, D and E only

Answer: (4)

7. Two polaroide A and B are placed in such a way that the pass-axis of polaroids are perpendicular to each other. Now, another polaroid C is placed between A and B bisecting angle between them. If intensity of unpolarized light is I₀ then intensity of transmitted light after passing through polaroid B will be:

(1) I₀/4

(2) I₀/2

(3) Zero

(4) I₀/8

Answer: (4)

8. As shown in figure, a 70 kg garden roller is pushed with a force of at an angle of 30° with horizontal. The normal reaction on the roller is

(Given g = 10 ms⁻²)

(1) 800√2 N

(2) 200√3 N

(3) 600 N

(4) 800 N

Answer: (4)

9. If 1000 droplets of water of surface tension 0.07 N/m, having same radius 1 mm each, combine to from a single drop. In the process the released surface energy is- (Take π = 22/7)

(1) 8.8 × 10⁻⁵ J

(2) 7.92 × 10⁻⁴ J

(3) 7.92 × 10⁻⁶ J

(4) 9.68 × 10⁻⁴ J

Answer: (2)

10. The pressure of a gas changes linearly with volume from A to B as shown in figure. If no heat is supplied to or extracted from the gas then change in the internal energy of the gas will be

(1) −4.5 J

(2) zero

(3) 4.5 J

(4) 6 J

Answer: (C)

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

Assertion A: The beam of electrons show wave nature and exhibit interference and diffraction.

Reason R: Davisson Germer Experimentally verified the wave nature of electrons.

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) 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)

12. A free neutron decays into a proton but a free proton does not decay into neutron. This is because

(1) proton is a charged particle

(2) neutron is an uncharged particle

(3) neutron is a composite particle made of a proton and an electron

(4) neutron has larger rest mass than proton

Answer: (4)

13. Spherical insulating ball and a spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of the following Assume negligible air friction}

(1) Insulating ball will reach the earth’s surface earlier than the metal ball

(2) Metal ball will reach the earth’s surface earlier than the insulating ball

(3) Both will reach the earth’s surface simultaneously.

(4) Time taken by them to reach the earth’s surface will be independent of the properties of their materials

Answer: (1)

14. If R, X_L, and X_C represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :

(1) R/X_LX_C

(2)

(3)

(4) RX_LX_C

Answer: (2)

15. 100 balls each of mass m moving with speed ν simultaneously strike a wall normally and reflected back with same speed, in time t sec. The total force exerted by the balls on the wall is

(1) 100mν/t

(2) 200mνt

(3) mν/100t

(4) 200mν/t

Answer: (4)

16. If a source of electromagnetic radiation having power 15 kW produces 10¹⁶ photons per second, the radiation belongs to a part of spectrum is.

(Take Planck constant h=6 × 10⁻³⁴Js )

(1) Micro waves

(2) Ultraviolet rays

(3) Gamma rays

(4) Radio waves

Answer: (3)

17. Which of the following correctly represents the variation of electric potential (V) of a charged spherical conductor of radius (R) with radial distance (r) from the center?

Answer: (1)

18. A bar magnet with a magnetic moment 5.0 Am² is placed in parallel position relative to a magnetic field of 0.4 T. The amount of required work done in turning the magnet from parallel to antiparallel position relative to the field direction is _______.

(1) 1 J

(2) 4 J

(3) 2 J

(4) zero

Answer: (2)

19. At a certain depth “d ” below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height 3R above earth surface. Where R is Radius of earth (Take R = 6400 km ). The depth d is equal to

(1) 4800 km

(2) 2560 km

(3) 640 km

(4) 5260 km

Answer: (A)

20. A rod with circular cross-section area 2 cm² and length 40 cm is wound uniformly with 400 turns of an insulated wire. If a current of 0.4 A flows in the wire windings, the total magnetic flux produced inside windings is 4π × 10⁻⁶ The relative permeability of the rod is (Given : Permeability of vacuum μ₀ = 4π × 10⁻⁷NA⁻²)

(1) 5/16

(2) 12.5

(3) 125

(4) 32/5

Answer: (3)

SECTION-B

21. In a medium the speed of light wave decreases to 0.2 times to its speed in free space The ratio of relative permittivity to the refractive index of the medium is x : 1. The value of x is (Given speed of light in free space =3 × 10⁸ ms⁻¹ and for the given medium μ_r = 1)

Answer: (5)

22. A solid sphere of mass 1 kg rolls without slipping on a plane surface. Its kinetic energy is 7 × 10⁻³ The speed of the centre of mass of the sphere is _________ cms⁻¹

Answer: (10)

23. A lift of mass M = 500 kg is descending with speed of 2 ms⁻¹. Its supporting cable begins to slip thus allowing it to fall with a constant acceleration of 2 ms⁻². The kinetic energy of the lift at the end of fall through to a distance of 6 m will be _______ kJ.

Answer: (7)

24. In the figure given below, a block of mass M = 490 g placed on a frictionless table is connected with two springs having same spring constant (K = 2 N m⁻¹). If the block is horizontally displaced through ‘X’ m then the number of complete oscillations it will make in 14π seconds will be ________.

Answer: (20)

25. An inductor of 0.5mH, a capacitor of 20𝜇F and resistance of 20Ω are connected in series with a 220 V ac source. If the current is in phase with the emf, the amplitude of current of the circuit is √x The value of x is-

Answer: (242)

26. The speed of a swimmer is 4 kmh⁻¹ in still water. If the swimmer makes his strokes normal to the flow of river of width 1 km, he reaches a point 750 m down the stream on the opposite bank. The speed of the river water is _______ kmh⁻¹.

Answer: (3)

27. For hydrogen atom, 𝜆₁ and 𝜆₂ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of 𝜆₁ and 𝜆₂ is x/32. The value of x is

Answer: (27)

28. Two identical cells, when connected either in parallel or in series gives same current in an external resistance 5Ω. The internal resistance of each cell will be ______ Ω.

Answer: (5)

29. Expression for an electric field is given by The electric flux through the cube of side 20 cm when placed in electric field (as shown in the figure) is _____ Vcm.

Answer: (640)

30. A thin rod having a length of 1 m and area of cross-section 3 × 10⁻⁶ m² is suspended vertically from one end. The rod is cooled from 210°C to 160° After cooling, a mass M is attached at the lower end of the rod such that the length of rod again becomes 1 m. Young’s modulus and coefficient of linear expansion of the rod are 2 × 10¹¹ Nm⁻² and 2 × 10⁻⁵ K⁻¹, respectively. The value of M is _____ kg. (Take g = 10 ms⁻²)

Answer: (60)

Chemistry

SECTION-A

31. Match items of column I and II

Correct match is

(1) A-(ii), B-(iii), C-(iv), D-(i)

(2) A-(i), B-(iii), C-(ii), D-(iv)

(3) A-(ii), B-(iv), C-(i), D-(iii)

(4) A-(iii), B-(iv), C-(ii), D-(i)

Answer: (1)

32.

Consider the above reaction and identify the product B. Options

Answer: (1)

33. An organic compound ‘A’ with emperical formula C₆H₆O gives sooty flame on burning. Its reaction with bromine solution in low polarity solvent results in high yield of B.B is

Answer: (3)

34. When Cu²⁺ ion is treated with KI, a white precipitate, X appears in solution. The solution is titrated with sodium thiosulphate, the compound Y is formed. X and Y respectively are

(1) X=CuI₂ Y=Na₂ S₄O₆

(2) X=CuI₂ Y=Na₂ S₂O₃

(3) X=Cu₂I₂ Y=Na₂ S₄O₅

(4) X=Cu₂I₂ Y=Na₂ S₄O₆

Answer: (4)

35. Choose the correct set of reagents for the following conversion. trans⁡(Ph – CH = CH − CH₃) → cis⁡(Ph – CH = CH − CH₃)

(1) Br₂, aq ⋅ KOH, NaNH₂, Na(LiqNH₃)

(2) Br₂, alc ⋅ KOH, NaNH₂, H₂ Lindlar Catalyst

(3) Br₂, aq⋅KOH, NaNH₂, H₂ Lindlar Catalyst

(4) Br₂, alc ⋅ KOH, NaNH₂, Na(LiqNH₃)

Answer: (2)

36. Consider the following reaction

The correct statement for product B is. It is

(1) optically active alcohol and is neutral

(2) racemic mixture and gives a gas with saturated NaHCO₃ solution

(3) optically active and adds one mole of bromine

(4) racemic mixture and is neutral

Answer: (2)

37. The methods NOT involved in concentration of ore are

(A) Liquation

(B) Leaching

(D) Hydraulic washing

(E) Froth floatation

Choose the correct answer from the options given below :

(1) C, D and E only

(2) B, D and C only

(3) B, D and C only

(4) B, D and C only

Answer: (3)

38. A protein ‘X’ with molecular weight of 70,000 u, on hydrolysis gives amino acids. One of these amino acid is

Answer: (4)

39. Nd²⁺ =

(1) 4f³

(2) 4f⁴6 s²

(3) 4f⁴

(4) 4f²6 s²

Answer: (3)

40. Match List I with List II

Choose the correct answer from the options given below:

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

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

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

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

Answer: (4)

41. Identify X,Y and Z in the following reaction. (Equation not balanced)

(1) X = ClONO₂, Y = HOCl, Z = HNO₃

(2) X = ClONO₂, Y = HOCl, Z = NO₂

(3) X = ClNO₂, Y = HCl, Z = HNO₃

(4) X = ClNO₃, Y = Cl₂, Z = NO₂

Answer: (1)

42. The correct increasing order of the ionic radii is

(1) S²⁻ < Cl⁻ < Ca²⁺ < K⁺

(2) K⁺ < S²⁻ < Ca²⁺ < Cl⁻

(3) Ca²⁺ < K⁺ < Cl⁻ < S²⁻

(4) Cl⁻ < Ca²⁺ < K⁺ < S²⁻

Answer: (3)

43. Cobalt chloride when dissolved in water forms pink colored complex X which has octahedral geometry. This solution on treating with conc HCl forms deep blue complex, Y which has a Z X, Y and Z, respectively, are

(1) X = [Co(H₂O)₆]²⁺, Y = [CoCl₄]²⁻, Z = Tetrahedral

(2) X = [Co(H₂O)₆]²⁺, Y = [CoCl₆]³⁻, Z = Octahedral

(3) X = [Co(H₂O)₄Cl₂]⁺, Y = [CoCl₄]²⁻, Z = Tetrahedral

(4) X = [Co(H₂O)₆]³⁺, Y = [CoCl₆]³⁻, Z= Octahedral

Answer: (1)

44. H₂O₂ acts as a reducing agent in

(1) 2NaOCl + H₂O₂ → 2NaCl + H₂O + O₂

(2) Na₂S + 4H₂O₂ → Na₂SO₄ + 4H₂O

(3) 2Fe²⁺ + 2H⁺ + H₂O₂ → 2Fe³⁺ + 2H₂O

(4) Mn²⁺ + 2H₂O₂ → MnO₂ + 2H₂O

Answer: (1)

45. Adding surfactants in non polar solvent, the micelles structure will look like

(1) a

(2) d

(3) b

(4) c

Answer: (1)

46. The correct order of melting points of dichlorobenzenes is

Answer: (2)

47. The correct order of basicity of oxides of vanadium is

(1) V₂O₅ > V₂O₄ > V₂O₃

(2) V₂O₄ > V₂O₃ > V₂O₅

(3) V₂O₃ > V₂O₅ > V₂O₄

(4) V₂O₃ > V₂O₄ > V₂O₅

Answer: (4)

48. Which of the following artificial sweeteners has the highest sweetness value in comparison to cane sugar ?

(1) Sucralose

(2) Aspartame

(3) Alitame

(4) Saccharin

Answer: (3)

49. Which one of the following statements is correct for electrolysis of brine solution?

(1) Cl₂ is formed at cathode

(2) O₂ is formed at cathode

(3) H₂ is formed at anode

(4) OH⁻ is formed at cathode

Answer: (4)

50. Which transition in the hydrogen spectrum would have the same wavelength as the Balmer type transition from n = 4 to n = 2 of He⁺ spectrum

(1) n = 2 to n = 1

(2) n = 1 to n = 2

(3) n = 3 to n = 4

(4) n = 1 to n = 3

Answer: (1)

SECTION B

51. The oxidation state of phosphorus in hypophosphoric acid is +

Answer: (4)

52. The enthalpy change for the conversion of is (−) kJmol⁻¹ (Nearest integer)

Given :

Answer: (610)

53. The logarithm of equilibrium constant for the reaction Pd²⁺ + 4Cl⁻ ⇌ PdCl₄²⁻ is (Nearest integer)

Answer: (6)

54. On complete combustion, 0.492 g of an organic compound gave 0.792 g of CO₂. The % of carbon in the organic compound is _____ (Nearest integer)

Answer: (44)

55. Zinc reacts with hydrochloric acid to give hydrogen and zinc chloride. The volume of hydrogen gas produced at STP from the reaction of 11.5 g of zinc with excess HCl is L (Nearest integer) (Given : Molar mass of Zn is 65.4 g mol⁻¹ and Molar volume of H₂ at STP = 22.7 L )

Answer: (4)

56. A → B

The rate constants of the above reaction at 200 K and 300 K are 0.03 min⁻¹ and 0.05 min⁻¹ respectively. The activation energy for the reaction is J⁡(Nearest integer) (Given : ln⁡10 = 2.3 R = 8.3 J K⁻¹ mol⁻¹

log 5 = 0.70

log 3 = 0.48

log 2 = 0.30)

Answer: (2520)

57. For reaction :

K_p = 2 × 10¹² at 27°C and 1 atm pressure. The K_c for the same reaction is × 10¹³. (Nearest integer) (Given R=0.082 L atm K⁻¹ mol⁻¹)

Answer: (1)

58. The total pressure of a mixture of non-reacting gases X(0.6 g) and Y(0.45 g) in a vessel is 740 mm of Hg. The partial pressure of the gas X is _____ mm of Hg. (Nearest Integer)

(Given : molar mass X = 20 and Y = 45 g mol⁻¹ )

Answer: (555)

59. How many of the transformations given below would result in aromatic amines ?

Answer: (3)

60. At 27∘C, a solution containing 2.5 g of solute in 250.0 mL of solution exerts an osmotic pressure of 400 Pa. The molar mass of the solute is _____ gmol⁻¹ (Nearest integer)

(Given : R=0.083 L_bar K⁻¹ mol⁻¹)

Answer: (62250)

Mathematics

SECTION-A

61. If the maximum distance of normal to the ellipse from the origin is 1, then the eccentricity of the ellipse is :

(1) 1/2

(2) √3/4

(3) √3/2

(4) 1/√2

Answer: (3)

62. Let a differentiable function f satisfy Then 12f(8) is equal to :

(1) 34

(2) 1

(3) 17

(4) 19

Answer: (3)

63. For all z ∈ C on the curve C₁ : |z| = 4, let the locus of the point be the curve C₂. Then :

(1) the curve C₁ lies inside C₂

(2) the curve C₂ lies inside C₁

(3) the curves C₁ and C₂ intersect 4 points

(4) the curves C₁ and C₂ intersect at 2 points

Answer: (3)

64. Then, at x = 1,

(1) √2yʹ − 3π²y = 0

(2) yʹ + 3π²y = 0

(3) 2yʹ + 3π²y = 0

(4) 2yʹ + √3π²y = 0

Answer: (3)

65. A wire of length 20 m is to be cut into two pieces. A piece of length 𝑙₁ is bent to make a square of area 𝐴1 and the other piece of length 𝑙₂ is made into a circle of area A₂. If 2A₁+ 3A₂ is minimum then (π𝑙₁) : 𝑙₂ is equal to :

(1) 1 : 6

(2) 6 : 1

(3) 3 : 1

(4) 4 : 1

Answer: (2)

66. Let a circle C₁ be obtained on rolling the circle x² + y² – 4x – 6y + 11 = 0 upwards 4 units on the tangent 𝑇 to it at the point (3, 2). Let C₂ be the image of C₁ in T. Let A and B be the centers of circles C₁ and C₂ respectively, and 𝑀 and N be respectively the feet of perpendiculars drawn from A and B on the x-axis. Then the area of the trapezium AMNB is :

(1) 4(1 + √2)

(2) 3 + 2√2

(3) 2(1 + √2)

(4) 2(2 + √2)

Answer: (1)

67. A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is

(1) 3/7

(2) 5/7

(3) 5/6

(4) 2/7

Answer: (2)

68. Let y = f(x) represent a parabola with focus (−1/2, 0) and directrix y = −1/2. Then

(1) contains exactly two elements

(2) contains exactly one element

(3) is an empty set

(4) is an infinite set

Answer: (1)

69. Let be two nonzero vectors such that and Consider the following two statements:

Then

(1) both (A) and (B) are correct

(2) only (A) is correct

(3) neither (A) nor (B) is correct

(4) only (B) is correct

Answer: (2)

70. The value of is equal to

Answer: (4)

71. Let the shortest distance between the lines and L₁ : x + 1 = y – 1 = 4 – z be 2√ If (α, β, γ) lies on L, then which of the following is NOT possible ?

(1) α − 2γ = 19

(2) 2α + γ = 7

(3) 2α – γ = 9

(4) α + 2γ = 24

Answer: (4)

72. For the system of linear equations

x + y + z = 6 αx + βy + 7z = 3 x + 2y + 3z = 14 which of the following is NOT true ?

(1) If α = β and α ≠ 7, then the system has a unique solution

(2) If α = β = 7, then the system has no solution

(3) For every point (α, β) ≠ (7, 7) on the line x – 2y + 7 = 0, the system has infinitely many solutions

(4) There is a unique point (α, β) on the line x + 2y + 18 = 0 for which the system has infinitely many solutions

Answer: (3)

73. If the domain of the function where [x] is greatest integer ≤ x, is [2, 6), then its range is

Answer: (3)

74. Let R be a relation on N × N defined by (a, b) R (c, d) if and only if ad(b − c) = bc(a − d). Then R is

(1) transitive but neither reflexive nor symmetric

(2) symmetric but neither reflexive nor transitive

(3) symmetric and transitive but not reflexive

(4) reflexive and symmetric but not transitive

Answer: (2)

75. (S1) (p ⇒ q) ∨ (p ∧ (∼q)) is a tautology (S2) ((∼p) ⇒ (∼q)) ∧ ((∼p) ∨ q) is a contradiction.

Then

(1) both (S1) and (S2) are correct

(2) only ( S1) is correct

(3) only (S2) is correct

(4) both (S1) and (S2) are wrong

Answer: (2)

76. If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is

(1) 7

(2) 3

(3) 9/2

(4) 14

Answer: (1)

77. Let Then the sum of the diagonal elements of the matrix (A + I)¹¹ is equal to

(1) 6144

(2) 2050

(3) 4097

(4) 4094

Answer: (3)

78. The number of real roots of the equation is :

(1) 3

(2) 1

(3) 2

(4) 0

Answer: (2)

79. If 0 < α < 13, then sin⁻¹(sin α) + cos⁻¹(cos α) is equal to

(1) 16

(2) 0

(3) π

(4) 16 – 5π

Answer: (3)

80. Let α ∈ (0, 1) and β = log_e(1 – α). Let x ∈ (0, 1). Then the integral is equal to

(1) β + P₅₀(α)

(2) P₅₀(α) – β

(3) β – P₅₀(α)

(4) −(β + P₅₀(α))

Answer: (4)

SECTION B

81. Let α > 0, be the smallest number such that the expansion of has a term βx^−α, β ∈ ℕ. Then α is equal to

Answer: (2)

82. Let for x ∈ ℝ

Then area bounded by the curve y = (f ° g) (x) and the lines y = 0, 2y – x = 15 is equal to

Answer: (72)

83. Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

Answer: (710)

84. If the variance of the frequency distribution

is 3, then α is equal to

Answer: (5)

85. Let θ be the angle between the planes and Let L be the line that meets P₂ at the point (4, −2, 5) and makes an angle θ with the normal of P₂. If α is the angle between L and P₂, then (tan² θ) (cot² α) is equal to

Answer: (9)

86. Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is

Answer: (2997)

87. Let be two vectors such that and Then is equal to

Answer: (36)

88. Let the line intersect the plane 2x + y + 3z = 16 at the point P. Let the point Q be the foot of perpendicular from the point R(1, −1, −3) on the line L. If α is the area of triangle PQR, then α² is equal to

Answer: (180)

89. Let a₁, a₂, …, a_n be in A.P. If a₅ = 2a₇ and a₁₁= 18, then is equal to

Answer: (8)

90. The remainder on dividing 5⁹⁹ by 11 is :

Answer: (9)

JEE Main Session 1 30th January 2023 Shift 1 Question Paper and Answer Key

May 10, 2023May 9, 2023 by suresh84

JEE MAIN 30^th January 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.

1. The magnetic moments associated with two closely wound circular coils A and B of radius r_A = 10 cm and r_B= 20 cm respectively are equal if : (Where N_A, I_A and N_B, I_B are number of turn and current of A and B respectively)

(1) 4N_AI_A = N_BI_B

(2) N_A = 2N_B

(3) N_AI_A = 4N_BI_B

(4) 2N_AI_A = N_BI_B

Answer: (3)

2. The figure represents the momentum time (p-t) curve for a particle moving along an axis under the influence of the force. Identify the regions on the graph where the magnitude of the force is maximum and minimum respectively ?

If (t₃ – t₂) < t₁

(1) c and b

(2) b and c

(3) a and b

(4) c and a

Answer: (1)

3. Two isolated metallic solid spheres of radii R and 2R are charged such that both have same charge density σ. The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is σ′. The ratio σʹ/σ is :

(1) 4/3

(2) 5/3

(3) 5/6

(4) 9/4

Answer: (3)

4. A person has been using spectacles of power −1.0 dioptre for distant vision and a separate reading glass of power 2.0 dioptres. What is the least distance of distinct vision for this person :

(1) 40 cm

(2) 30 cm

(3) 10 cm

(4) 50 cm

Answer: (4)

5. A small object at rest, absorbs a light pulse of power 20 mW and duration 300 ns. Assuming speed of light as 3 × 10⁸ m/s, the momentum of the object becomes equal to :

(1) 3 × 10⁻¹⁷ kg m/s

(2) 2 × 10⁻¹⁷ kg m/s

(3) 1 × 10⁻¹⁷ kg m/s

(4) 0.5 × 10⁻¹⁷ kg m/s

Answer: (2)

6. Match Column-I with Column-II :

Choose the correct answer from the options given below:

(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- II, B-IV, C-III, D-I

Answer: (4)

7. The pressure (P) and temperature (T) relationship of an ideal gas obeys the equation PT² = constant. The volume expansion coefficient of the gas will be :

(1) 3/T³

(2) 3/T²

(3) 3 T²

(4) 3/T

Answer: (4)

8. Heat is given to an ideal gas in an isothermal process.

(A) Internal energy of the gas will decrease.

(B) Internal energy of the gas will increase.

(D) The gas will do positive work. (E) The gas will do negative work.

Choose the correct answer from the options given below :

(1) C and D only

(2) C and E only

(3) A and E only

(4) B and D only

Answer: (1)

9. If the gravitational field in the space is given as (−K/r²). Taking the reference point to be at r = 2 cm with gravitational potential V = 10 J/kg. Find the gravitational potential at r = 3 cm in SI unit (Given, that K = 6Jcm/kg )

(1) 9

(2) 10

(3) 11

(4) 12

Answer: (3)

10. In a series LR circuit with X_L = R, power factor is P₁. If a capacitor of capacitance C with X_C = X_L is added to the circuit the power factor becomes P₂. The ratio of P₁ to P₂ will be :

(1) 1 : 3

(2) 1 : 2

(3) 1 : √2

(4) 1 : 1

Answer: (3)

11. As per the given figure, a small ball P slides down the quadrant of a circle and hits the other ball Q of equal mass which is initially at rest. Neglecting the effect of friction and assume the collision to be elastic, the velocity of ball Q after collision will be :

(1) 0

(2) 4 m/s

(3) 2 m/s

(4) 0.25 m/s

Answer: (3)

12. A ball of mass 200 g rests on a vertical post of height 20 m. A bullet of mass 10 g, travelling in horizontal direction, hits the centre of the ball. After collision both travels independently. The ball hits the ground at a distance 30 m and the bullet at a distance of 120 m from the foot of the post. The value of initial velocity of the bullet will be (if g = 10 m/s²) :

(1) 360 m/s

(2) 400 m/s

(3) 60 m/s

(4) 120 m/s

Answer: (1)

13. The output waveform of the given logical circuit for the following inputs A and B as shown below, is

Answer: (3)

14. The charge flowing in a conductor changes with time as Q(t) = αt – βt² + γt³. Where α, β and γ are constants. Minimum value of current is :

Answer: (3)

15. Choose the correct relationship between Poisson ratio (σ), bulk modulus (K) and modulus of rigidity (η) of a given solid object :

Answer: (2)

16. Speed of an electron in Bohr’s 7th orbit for Hydrogen atom is 3.6 × 10⁶ m/s. The corresponding speed of the electron in 3rd orbit, in m/s is:

(1) (1.8 × 10⁶)

(2) (3.6 × 10⁶)

(3) (7.5 × 10⁶)

(4) (8.4 × 10⁶)

Answer: (4)

17. A massless square loop, of wire of resistance 10 Ω, supporting a mass of 1 g, hangs vertically with one of its sides in a uniform magnetic field of 103G, directed outwards in the shaded region. A dc voltage V is applied to the loop. For what value of V, the magnetic force will exactly balance the weight of the supporting mass of 1 g ?

(If sides of the loop =10 cm, g = 10 ms⁻²)

(1) 1/10 V

(2) 100 V

(3) 10 V

(4) 1 V

Answer: (3)

18. Electric field in a certain region is given by The SI unit of A and B are :

(1) Nm³C^–1; Nm²C^–1

(2) Nm²C^–1; Nm³C^–1

(3) Nm³C; Nm²C

(4) Nm²C; Nm³C

Answer: (2)

19. The height of liquid column raised in a capillary tube of certain radius when dipped in liquid A vertically is, 5 cm. If the tube is dipped in a similar manner in another liquid B of surface tension and density double the values of liquid A, the height of liquid column raised in liquid B would be m

(1) 0.05

(2) 0.10

(3) 0.20

(4) 0.5

Answer: (1)

20. A sinusoidal carrier voltage is amplitude modulated. The resultant amplitude modulated wave has maximum and minimum amplitude of 120 V and 80 V respectively. The amplitude of each sideband is :

(1) 20 V

(2) 15 V

(3) 10 V

(4) 5 V

Answer: (3)

SECTION-B

21. The general displacement of a simple harmonic oscillator is x = A sin ω Let T be its time period. The slope of its potential energy (U)-time (t) curve will be maximum when t = T/β. The value of β is

Answer: (8)

22. A thin uniform rod of length 2 m, cross sectional area ‘A’ and density ‘d’ is rotated about an axis passing through the centre and perpendicular to its length with angular velocity ω. If value of ω in terms of its rotational kinetic energy E is then value of α is

Answer: (3)

23. A horse rider covers half the distance with 5 m/s speed. The remaining part of the distance was travelled with speed 10 m/s for half the time and with speed 15 m/s for other half of the time. The mean speed of the rider averaged over the whole time of motion is x/7 m/s. The value of x is

Answer: (50)

24.

As per the given figure, if then the value of V_AB at this instant will be V.

Answer: (30)

25. A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of 24 W. The radius of curvature of hemisphere is 10 cm and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is _____ 10⁻⁸ N

Answer: (4)

26. In the following circuit, the magnitude of current I₁, is ______ A.

Answer: (1.5)

27. In a screw gauge, there are 100 divisions on the circular scale and the main scale moves by 0.5 mm on a complete rotation of the circular scale. The zero of circular scale lies 6 divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, 4 linear scale divisions are clearly visible while 46th division the circular scale coincide with the reference line. The diameter of the wire is _______ × 10⁻² mm

Answer: (220)

28. In Young’s double slit experiment, two slits S₁ and S₂ are ‘ d ‘ distance apart and the separation from slits to screen is D (as shown in figure). Now if two transparent slabs of equal thickness 0.1 mm but refractive index 1.51 and 1.55 are introduced in the path of beam (λ = 4000 Å) from S₁ and S₂ respectively. The central bright fringe spot will shift by number of fringes.

Answer: (10)

29. A capacitor of capacitance 900μF is charged by a 100 V battery. The capacitor is disconnected from the battery and connected to another uncharged identical capacitor such that one plate of uncharged capacitor connected to positive plate and another plate of uncharged capacitor connected to negative plate of the charged capacitor. The loss of energy in this process is measured as x × 10⁻² The value of x is

Answer: (225)

30. In an experiment for estimating the value of focal length of converging mirror, image of an object placed at 40 cm from the pole of the mirror is formed at distance 120 cm from the pole of the mirror. These distances are measured with a modified scale in which there are 20 small divisions in 1 cm. The value of error in measurement of focal length of the mirror is 1/K cm. The value of K is

Answer: (32)

Chemistry

SECTION-A

31. Lithium aluminium hydride can be prepared from the reaction of

(1) LiH and Al(OH)₃

(2) LiH and Al₂Cl₆

(3) LiCl and Al₂H₆

(4) LiCl,Al and H₂

Answer: (2)

32. Amongst the following compounds, which one is an antacid?

(1) Terfenadine

(2) Meprobamate

(3) Brompheniramine

(4) Ranitidine

Answer: (4)

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

Assertion (A): In expensive scientific instruments, silica gel is kept in watch-glasses or in semipermeable membrane bags.

Reason (R): Silica gel adsorbs moisture from air via adsorption, thus protects the instrument from water corrosion (rusting) and / or prevents malfunctioning.

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

(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)

34. Match List I with List II

Choose the correct answer from the options given below:

(1) A – IV, B – III, C – II, D – I

(2) A – II, B – IV, C – I, D – III

(3) A – IV, B – II, C – I, D – III

(4) A – I, B – III, C – IV, D – II

Answer: (3)

35. What is the correct order of acidity of the protons marked A-D in the given compounds ?

(1) H_C > H_A > H_D > H_B

(2) H_D > H_C > H_B > H_A

(3) H_C > H_D > H_B > H_A

(4) H_C > H_D > H_A > H_B

Answer: (4)

36. Which of the following compounds would give the following set of qualitative analysis?

(i) Fehling’s Test : Positive

(ii) Na fusion extract upon treatment with sodium nitroprusside gives a blood red colour but not prussian blue.

Answer: (4)

37. The major products ‘ A’ and ‘ B ‘, respectively, are

Answer: (4)

38. During the qualitative analysis of SO₃²⁻ using dilute H₂SO₄, SO₂ gas is evolved which turns K₂Cr₂O₇ solution (acidified with dilute H₂SO₄ ):

(1) green

(2) blue

(3) red

(4) black

Answer: (1)

39. In the wet tests for identification of various cations by precipitation, which transition element cation doesn’t belong to group IV in qualitative inorganic analysis ?

(1) Ni²⁺

(2) Zn²⁺

(3) Co²⁺

(4) Fe³⁺

Answer: (4)

40. For OF₂ molecule consider the following :

(A) Number of lone pairs on oxygen is 2.

(B) FOF angle is less than 104.5∘.

(D) Molecule is bent ‘ V ‘ shaped.

(E) Molecular geometry is linear.

correct options are:

(1) A, C, D only

(2) C, D, E only

(3) A, B, D, only

(4) B, E, A only

Answer: (3)

41. Caprolactam when heated at high temperature in presence of water, gives

(1) Nylon 6, 6

(2) Nylon 6

(3) Teflon

(4) Dacron

Answer: (2)

42. Benzyl isocyanide can be obtained by :

Choose the correct answer from the options given below:

(1) A and D

(2) Only B

(3) B and C

(4) A and B

Answer: (4)

43. Formation of photochemical smog involves the following reaction in which A,B and C are respectively.

Choose the correct answer from the options given below:

(1) O, N₂O & NO

(2) O, NO & NO₃⁻

(3) NO, O & O₃

(4) N, O₂ & O₃

Answer: (3)

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

Assertion (A): Ketoses give Seliwanoff’s test faster than Aldoses.

Reason (R) : Ketoses undergo β-elimination followed by formation of furfural.

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: (2)

45. Match List I with List II

Choose the correct answer from the options given below:

(1) A – II, B – III, C – IV, D – I

(2) A – II, B – I, C – IV, D – III

(3) A – IV, B – I, C – II, D – III

(4) A – IV, B – III, C – II, D – I

Answer: (4)

46. To inhibit the growth of tumours, identify the compounds used from the following :

(A) EDTA

(B) Coordination Compounds of Pt C. D – Penicillamine

Choose the correct answer from the option given below:

(1) B and D only

(2) C and D only

(3) A and C only

(4) A and B only

Answer: (1)

47. The alkaline earth metal sulphate(s) which are readily soluble in water is/are :

(A) BeSO₄ (2) MgSO₄

(E) BaSO₄

Choose the correct answer from the options given below:

(1) B only

(2) A and B

(3) B and C

(4) A only

Answer: (2)

48. Which of the following is correct order of ligand field strength ?

(1) CO < en < NH₃ < C₂O₄²⁻ < S²⁻

(2) NH₃ < en < CO < S²⁻ < C₂O₄²⁻

(3) S²⁻ < C₂O₄²⁻ < NH₃ < en < CO

(4) S²⁻ < NH₃ < en < CO < C₂O₄²

Answer: (3)

49. 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 – III, B – II, C – IV, D – I

(4) A – II, B – I, C – III, D – IV

Answer: (1)

50. In the extraction of copper, its sulphide ore is heated in a reverberatory furnace after mixing with silica to:

(1) remove FeO as FeSiO₃

(2) decrease the temperature needed for roasting of Cu₂S

(3) separate CuO as CuSiO₃

(4) remove calcium as CaSiO₃

Answer: (1)

SECTION-B

51. 600 mL of 0.01MHCl is mixed with 400 mL of 0.01MH₂SO₄. The pH of the mixture is __________ ×10⁻². (Nearest integer)

[Given log 2 2 = 0.30

log 3 = 0.48

log 5 = 0.69

log 7 = 0.84

log 11 = 1.04]

Answer: (186)

52. The energy of one mole of photons of radiation of frequency 2 × 10¹² Hz in J mol^–1 is _____. (Nearest integer)

[Given : h = 6.626 × 10⁻³⁴ Js

N_A = 6.022 × 10²³ mol⁻¹]

Answer: (789)

53. Consider the cell Pt_(s) |H₂(g, 1 atm)| H⁺(aq, 1M)||Fe³⁺ (aq), Fe²⁺(aq)| Pt(s)

When the potential of the cell is 0.712 V at 298 K, the ratio [Fe²⁺] / [Fe³⁺] is______ (Nearest integer)

Given : Fe³⁺ + e⁻ = Fe²⁺, E^θFe³⁺, Fe²⁺ | Pt = 0.771

Answer: (10)

54. The number of electrons involved in the reduction of permanganate to manganese dioxide in acidic medium is

Answer: (3)

55. A 300 mL bottle of soft drink has 0.2MCO₂ dissolved in it. Assuming CO2 behaves as an ideal gas, the volume of the dissolved CO₂ at STP is _____ mL. (Nearest integer)

Given : At STP, molar volume of an ideal gas is 22.7 L mol⁻¹

Answer: (1362)

56. A trisubstituted compound ‘A’, C₁₀H₁₂O₂ gives neutral FeCl₃ test positive. Treatment of compound ‘A’ with NaOH and CH₃Br gives C₁₁H₁₄O₂, with hydroiodic acid gives methyl iodide and with hot conc. NaOH gives a compound B, C₁₀H₁₂O₂. Compound ‘A’ also decolorises alkaline KMnO₄. The number of π bond/s present in the compound ‘A’ is

Answer: (4)

57. If compound A reacts with B following first order kinetics with rate constant 2.011 × 10⁻³ s⁻¹. The time taken by A (in seconds) to reduce from 7 g to 2 g will be (Nearest Integer) [log 5 = 0.698, log 7 = 0.845, log 2 = 0.301]

Answer: (623)

58. A solution containing 2 g of a non-volatile solute in 20 g of water boils at 373.52 K. The molecular mass of the solute is ______ g mol^–1. (Nearest integer) Given, water boils at 373 K, K_b for water = 0.52 K kg mol⁻¹

Answer: (100)

59. When 2 litre of ideal gas expands isothermally into vacuum to a total volume of 6 litre, the change in internal energy is ______ J. (Nearest integer)

Answer: (0)

60. Some amount of dichloromethane (CH₂Cl₂) is added to 671.141 mL of chloroform (CHCl₃) to prepare 2.6 × 10⁻³ M solution of CH₂Cl₂(DCM). The concentration of DCM is ppm (by mass).

Given : atomic mass : C = 12 H = 1 Cl = 35.5 density of CHC_l3= 1.49 g cm⁻³

Answer: (148.322)

Mathematics

SECTION-A

61. A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x-axis and y axis respectively. If the perpendicular from origin O to this line makes an angle of π/6 with positive direction of y-axis and the area of △OAB is then a² – b² is equal to:

(1) 392/3

(2) 196/3

(3) 98

(4) 196

Answer: (1)

62. The minimum number of elements that must be added to the relation R = {(a, b), (b, c)} on the set {a, b, c} so that is becomes symmetric and transitive is :

(1) 3

(2) 4

(3) 5

(4) 7

Answer: (4)

63. If an unbiased die, marked with −2, −1, 0, 1, 2, 3 on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is :

(1) 881/2592

(2) 27/288

(3) 440/2592

(4) 521/2592

Answer: (4)

64. If are three non-zero vectors and is a unit vector perpendicular to and is equal to :

(1) 9

(2) 15

(3) 6

(4) 12

Answer: (4)

65. Among the statements :

(S1) ((p ∨ q) ⇒ r) ⇔ (p ⇒ r)

(S2) ((p ∨ q) ⇒ r) ⇔ ((p ⇒ r) ⋁ (q ⇒ r))

(1) only (S2) is a tautology

(2) only (S1) is a tautology

(3) neither (S1) nor (S2) is a tautology

(4) both (S1) and (S2) are tautologies

Answer: (3)

66. If P(h, k) be a point on the parabola x = 4y², which is nearest to the point Q(0, 33), then the distance of P from the directrix of the parabola y² = 4(x + y) is equal to :

(1) 2

(2) 6

(3) 8

(4) 4

Answer: (2)

67. Let y = x + 2, 4y = 3x + 6 and 3y = 4x + 1 be three tangent lines to the circle (x − h)² + (y − k)²= r².

Then h + k is equal to

(1) 5(1 + √2)

(2) 5√2

(3) 6

(4) 5

Answer: (4)

68. The number of points on the curve y = 54x⁵ − 135x⁴ − 70x³+ 180x² + 210x at which the normal lines are parallel to x + 90y + 2 = 0 is :

(1) 4

(2) 2

(3) 0

(4) 3

Answer: (1)

69. If then a₁ + a₂ + … + a₂₅ is equal to

(1) 52/147

(2) 49/138

(3) 50/141

(4) 51/144

Answer: (3)

70. If then the value of is :

(1) 2

(2) 4 – 2√3

(3)

(4) 4

Answer: (4)

71. If the solution of the equation log_cosx cot x + 4log_{sin x} tan x = 1, x ∈ (0, π/2), is where α and β are integers, then α + β is equal to :

(1) 5

(2) 6

(3) 1

(4) 3

Answer: (3)

72. Let the system of linear equations

x + y + kz = 2

2x + 3y – z = 1

3x + 4y + 2z = k

have infinitely many solutions. Then the system

(k + 1)x + (2k − 1) y = 7 (2k + 1) x + (k + 5)y = 10 has:

(1) infinitely many solution

(2) unique solution satisfying x – y = 1

(3) unique solution satisfying x – y = 1

(4) no solution

Answer: (3)

73. The line l₁ passes through the point (2, 6, 2) and is perpendicular to the plane 2x + y − 2z = 10. Then the shortest distance between the line l₁ and the line is :

(1) 13/3

(2) 19/3

(3) 7

(4) 9

Answer: (9)

74. Let , d = |A| ≠ 0 and |A – d(Aadj A)| = 0. Then

(1) 1 + d² = m² + q²

(2) 1 + d² = (m + q)²

(3) (1 + d)² = m² + q²

(4) (1 + d)² = (m + q)²

Answer: (4)

75. If [t] denotes the greatest integer ≤ t, then the value of is :

(1) e⁸ – 1

(2) e⁷ – 1

(3) e⁸ – e

(4) e⁹ – e

Answer: (3)

76. Let a unit vector make angles α,β,γ with the positive directions of the co-ordinate axes OX, OY,OZ respectively, where β ∈ (0, π/2). If is perpendicular to the plane through points (1, 2, 3), (2, 3, 4) and (1, 5, 7), then which one of the following is true ?

Answer: (3)

77. The coefficient of x³⁰¹ in (1+x)⁵⁰⁰ + x(1+x)⁴⁹⁹ + x²(1 + x)⁴⁹⁸ +⋯….. x⁵⁰⁰ is :

(1) ⁵⁰⁰C₃₀₀

(2) ⁵⁰¹C₂₀₀

(3) ⁵⁰¹C₃₀₂

(4) ⁵⁰⁰C₃₀₁

Answer: (2)

78. Let the solution curve y = y(x) of the differential equation pass through the origin. Then y(1) is equal to:

Answer: (4)

79. If the coefficient of x¹⁵ in the expansion of is equal to the coefficient of x⁻¹⁵ in the expansion of where a and b are positive real numbers, then for each such ordered pair (a, b) :

(1) ab = 3

(2) ab = 1

(3) a = b

(4) a = 3b

Answer: (2)

80. Suppose f : ℝ → (0, ∞) be a differentiable function such that 5f(x + y) = f(x) ∙ f(y), ∀x, y ∈ℝ. If f(3) = 320, then is equal to :

(1) 6875

(2) 6525

(3) 6825

(4) 6575

Answer: (3)

SECTION B

81. Let z = 1 + i and Then is equal to ____

Answer: (9)

82. If λ₁ < λ₂ are two values of λ such that the angle between the planes and then the square of the length of perpendicular from the point (38λ₁, 10λ₂, 2) to the plane P₁ is

Answer: (315)

83. Let α be the area of the larger region bounded by the curve y²= 8x and the lines y = x and x = 2, which lies in the first quadrant. Then the value of 3α is equal to

Answer: (22)

84. Let where a, b, c ∈ ℤ and Then a² – b + c is equal to

Answer: (26)

85. Let α be the area of the larger region bounded by the curve y² = 8x and the lines y = x and x = 2, which lies in the first quadrant. Then the value of 3α is equal to

Answer: (15)

86. Let α be the area of the larger region bounded by the curve y² = 8x and the lines y = x and x = 2, which lies in the first quadrant. Then the value of 3α is equal to

Answer: (21)

87. Let

For n ≥ 2, define fⁿ(x) = f¹ of fⁿ⁻¹ (x)

If then a + b is equal to

Answer: (3125)

88. The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and a and b are respectively mean and variance of remaining 6 observation, then a + 3b − 5 is equal to

Answer: (37)

89. Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions f: S → P(S), where P(S) denote the power set of S, such that f(n) ⊂ f(m) where n < m is

Answer: (3240)

90. is equal to

Answer: (12)

JEE Main Session 1 29th January 2023 Shift 1 Question Paper and Answer Key

May 10, 2023May 9, 2023 by suresh84

JEE MAIN 29^th January 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.

1. Find the mutual inductance in the arrangement, when a small circular loop of wire of radius ‘R′ is placed inside a large square loop of wire of side (L >> R). The loops are coplanar and their centers coincide :

Answer: (4)

2. The threshold wavelength for photoelectric emission from a material is 5500A. Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a

(A) 75 W infra –red lamp

(B) 10 W infra-red lamp

(D) 10 W ultra-violet lamp

Choose the correct answer from the options given below:

(1) B and C only

(2) A and D only

(3) C only

(4) C and D only

Answer: (4)

3. 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-II, B – III, C-IV, D-I

(3) A-III, B – II, C-IV, D-I

(4) A-III, B – II, C-I, D-IV

Answer: (3)

4. In a cuboid of dimension 2L × 2L × L, a charge q is placed at the center of the surface ‘ S ‘ having area of 4L². The flux through the opposite surface to ‘ S ‘ is given by

(1) q/12ε₀

(2) q/6ε₀

(3) q/3ε₀

(4) q/2ε₀

Answer: (2)

5. A person observes two moving trains, ‘A’ reaching the station and ‘B’ leaving the station with equal speed of 30 m/s.If both trains emit sounds with frequency 300 Hz,(Speed of sound: 330m/s) approximate difference of frequencies heard by the person will be:

(1) 55 Hz

(2) 80 Hz

(3) 33 Hz

(4) 10 Hz

Answer: (1)

6. A block of mass m slides down the plane inclined at angle 30° with an acceleration g/4. The value of coefficient of kinetic friction will be:

Answer: (1)

7. A bicycle tyre is filled with air having pressure of 270 kPa at 27°C. The approximate pressure of the air in the tyre when the temperature increases to 36° C is

(1) 270 kPa

(2) 262 kPa

(3) 360 kPa

(4) 278 kPa

Answer: (4)

8. A single current carrying loop of wire carrying current I flowing in anticlockwise direction seen from +ve z direction and lying in xy plane is shown in figure. The plot of component of magnetic field (By) at a distance ꞌaꞌ (less than radius of the coil) and on yz plane vs z coordinate looks like

Answer: (1)

9. Surface tension of a soap bubble is 2.0 × 10^–2 Nm^–1. Work done to increase the radius of soap bubble from 3.5 cm to 7 cm will be:

Take [π = 22/7]

(1) 9.24 × 10⁻⁴ J

(2) 5.76 × 10⁻⁴ J

(3) 0.72 × 10⁻⁴ J

(4) 18.48 × 10⁻⁴ J

Answer: (4)

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

Assertion A: If dQ and dW represent the heat supplied to the system and the work done on the system respectively. Then according to the first law of thermodynamics dQ = dU – dW

Reason R: First law of thermodynamics is based on law of conservation of energy.

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) 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)

11. If a radioactive element having half-life of 30 min is undergoing beta decay, the fraction of radioactive element remains undecayed after 90 min. will be

(1) 1/8

(2) 1/2

(3) 1/4

(4) 1/16

Answer: (1)

12. Two particles of equal mass ‘m’ move in a circle of radius ‘r’ under the action of their mutual gravitational attraction. The speed of each particle will be :

Answer: (2)

13. If the height of transmitting and receiving antennas are 80 m each, the maximum line of sight distance will be: Given: Earth’s radius = 6.4 × 10⁶ m

(1) 28 km

(2) 36 km

(3) 32 km

(4) 64 km

Answer: (4)

14. A car is moving on a horizontal curved road with radius 50 m. The approximate maximum speed of car will be, if friction between tyres and road is 0.34.[take g = 10 ms⁻²]

(1) 17 ms⁻¹

(2) 13 ms⁻¹

(3) 22.4 ms⁻¹

(4) 3.4 ms⁻¹

Answer: (2)

15. Ratio of thermal energy released in two resistors R and 3R connected in parallel in an electric circuit is :

(1) 1 : 27

(2) 1 : 1

(3) 1 : 3

(4) 3 : 1

Answer: (4)

16. A stone is projected at angle 30° to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be

(1) 1 : 2

(2) 1 : 4

(3) 4 : 1

(4) 4 : 3

Answer: (4)

17. Which of the following are true?

(A) Speed of light in vacuum is dependent on the direction of propagation.

(B) Speed of light in a medium is independent of the wavelength of light.

(D) The speed of light in a medium is independent of intensity.

Choose the correct answer from the options given below:

(1) C and D only

(2) B and C only

(3) A and C only

(4) B and D only

Answer: (1)

18. In a Young’s double slit experiment, two slits are illuminated with a light of wavelength 800 nm. The line joining A₁P is perpendicular to A1A₂ as shown in the figure. If the first minimum is detected at P, the value of slits separation ‘a’ will be:

The distance of screen from slits D = 5 cm

(1) 0.5 mm

(2) 0.1 mm

(3) 0.4 mm

(4) 0.2 mm

Answer: (4)

19. Which one of the following statement is not correct in the case of light emitting diodes?

(A) It is a heavily doped p-n junction.

(B) It emits light only when it is forward biased.

(D) The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.

Choose the correct answer from the options given below:

(1) A

(2) C and D

(3) C

(4) B

Answer: (3)

20. The magnitude of magnetic induction at mid point O due to current arrangement as shown in Fig will be

(1) μ₀I/πa

(2) μ₀I/2πa

(3) 0

(4) μ₀I/4πa

Answer: (1)

SECTION-B

21. As shown in the figure, three identical polaroids P₁, P₂ and P₃ are placed one after another. The pass axis of P₂ and P₃ are inclined at angle of 60∘ and 90∘ with respect to axis of P₁. The source S has an intensity of The intensity of light at point O is –W/m².

Answer: (24)

22. A 0.4 kg mass takes 8 s to reach ground when dropped from a certain height ꞌP’ above surface of earth. The loss of potential energy in the last second of fall is ______ J.

(Take g = 10 m/s²)

Answer: (300)

23. Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is _______degree.

Answer: (120)

24. A tennis ball is dropped on to the floor from a height of 9.8 m. It rebounds to a height 5.0 m. Ball comes in contact with the floor for 0.2 s. The average acceleration during contact is _____ ms⁻² (Given g=10 ms⁻² )

Answer: (120)

25. A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field B= 0.8 T. When released the radius of the loop starts shrinking at a constant rate of 2cms⁻¹. The induced emf in the loop at an instant when the radius of the loop is 10 cm will be ____ mV. (Given g = 10 ms^–2)

Answer: (10)

26. A solid sphere of mass 2 kg is making pure rolling on a horizontal surface with kinetic energy 2240 J. The velocity of centre of mass of the sphere will be ______ ms⁻¹

Answer: (40)

27. A body cools from 60°C to 40°C in 6 minutes. If, temperature of surroundings is 10° Then, after the next 6 minutes, its temperature will be ______ °C.

Answer: (28)

28. In a metre bridge experiment the balance point is obtained if the gaps are closed by 2Ω and 3Ω. A shunt of X Ω is added to 3Ω resistor to shift the balancing point by 22.5 cm. The value of X is ____

Answer: (2)

29. A point charge q₁ = 4q₀ is placed at origin. Another point charge q₂ = −q₀ is placed at = 12 cm. Charge of proton is q₀ .The proton is placed on 𝑥xaxis so that the electrostatic force on the proton is zero. In this situation, the position of the proton from the origin is ___________ cm.

Answer: (24)

30. A radioactive element emits two α-articles, one electron and two positrons. The product nucleus is represented by The value of P is

Answer: (87)

Chemistry

SECTION-A

31. “A” obtained by Ostwald’s method involving air oxidation of NH3, upon further air oxidation produces “B”. “B” on hydration forms an oxoacid of Nitrogen along with evolution of “A”. The oxoacid also produces “A” and gives positive brown ring test. Identify A and B, respectively.

(1) N₂O₃, NO₂

(2) NO₂, N₂O₄

(3) NO₂, N₂O₅

(4) NO, NO₂

Answer: (4)

32. Correct statement about smog is:

(1) Classical smog also has high concentration of oxidizing agents

(2) Both NO₂ and SO₂ are present in classical smog

(3) NO₂ is present in classical smog

(4) Photochemical smog has high concentration of oxidizing agents

Answer: (4)

33. The standard electrode potential (M³⁺/M²⁺) for V, Cr, Mn & Co are −0.26 V, −0.41 V,+1.57 V and +1.97 V, respectively. The metal ions which can liberate H₂ from a dilute acid are

(1) Mn²⁺ and Co²⁺

(2) Cr²⁺ and Co²⁺

(3) V²⁺ and Cr²⁺

(4) V²⁺ and Mn²⁺

Answer: (3)

34. The shortest wavelength of hydrogen atom in Lyman series is 𝜆. The longest wavelength in Balmer series of He⁺ is

(1) 36λ/5

(2) 9λ/5

(3) 5/9λ

(4) 5λ/9

Answer: (2)

35. The bond dissociation energy is highest for

(1) F₂

(2) Br₂

(3) I₂

(4) Cl₂

Answer: (4)

36. The increasing order of pK_a for the following phenols is

(A) 2, 4-Dinitrophenol

(B) 4-Nitrophenol

(D) Phenol

(E) 3-Chlorophenol

Choose the correct answer from the option given below:

(1) (A),(B),(E),(D),(C)

(2) (C), (D), (E), (B), (A)

(3) (A), (E), (B), (D), (C)

(4) (C), (E), (D), (B), (A)

Answer: (1)

37. For 1 mol of gas, the plot of pV p is shown below. p is the pressure and V is the volume of the gas

What is the value of compressibility factor at point?

Answer: (2)

38. 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)−II, (C)−IV, (D)−III

(3) (A)−II, (B)−I, (C)−IV, (D)−II

(4) (A) −III, (B)−I, (C)−II, (D)−IV

Answer: (4)

39. During the borax bead test with CuSO₄, a blue green colour of the bead was observed in oxidising flame due to the formation of

(1) CuO

(2) Cu(BO₂)₂

(3) Cu₃B₂

(4) Cu

Answer: (2)

40. Which of the following salt solution would coagulate the colloid solution formed when FeCl₃ is added to NaOH solution, at the fastest rate?

(1) 10 mL of 0.1 mol dm^–3 Na₂SO₄

(2) 10 mL of 0.2 mol dm^–3 AlCl₃

(3) 10 mL of 0.1 mol dm^–3 Ca₃(PO₄)₂

(4) 10 mL of 0.15 mol dm–3 CaCl2

Answer: (2)

41. Number of cyclic tripeptides formed with 2 amino acids A and B is:

(1) 5

(2) 2

(3) 4

(4) 3

Answer: (3)

42. The correct order of hydration enthalpies is

(A) K⁺ (B) Rb⁺ (C) Mg²⁺

(D) Cs⁺ (E) Ca²⁺

Choose the correct answer from the options given below:

(1) E > C > A > B > D

(2) C > A > E > B > D

(3) C > E > A > D > B

(4) C > E > A > B > D

Answer: (4)

43. Chiral complex from the following is:

Here en = ethylene diamine

(1) cis −[PtCl₂(en)₂]²⁺

(2) trans−[PtCl₂(en)₂]²⁺

(3) cis−[PtCl₂(NH₃)₂]

(4) trans−[Co(NH₃)₄Cl₂]⁺

Answer: (1)

44. Identify the correct order for the given property for following compounds.

Choose the correct answer from the option given below:

(1) (B), (C) and (D) only

(2) (A), (C) and (D) only

(3) (A), (B) and (E) only

(4) (A), (C) and (E) only

Answer: (4)

45. The magnetic behavior of Li₂O, Na₂O₂ and KO₂, respectively, are

(1) Paramagnetic, paramagnetic and diamagnetic

(2) diamagnetic, paramagnetic and diamagnetic

(3) paramagnetic, diamagnetic and paramagnetic

(4) diamagnetic, diamagnetic and paramagnetic

Answer: (4)

46. The reaction representing the Mond process for metal refining is__________

Answer: (4)

47. Which of the given compounds can enhance the efficiency of hydrogen storage tank?

(1) Di-isobutylaluminium hydride

(2) NaNi_…….

(3) Li/P₄

(4) SiH₄

Answer: (2)

48. 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) −II, (B)−IV, (C)−I, (D)−III

Answer: (1)

49. The major product ‘P’ for the following sequence of reactions is:

Answer: (4)

50. Compound that will give positive Lassaigne’s test for both nitrogen and halogen is:

(1) NH₂OH.HCl

(2) CH₃NH₂.HCl

(3) NH₄Cl

(4) N₂H₄.HCl

Answer: (2)

SECTION-B

51. Millimoles of calcium hydroxide required to produce 100 mL of the aqueous solution of pH 12 is x × 10⁻¹. The value of x is_______ (Nearest integer). Assume complete dissociation.

Answer: (5)

52. Water decomposes at 2300 K

The percent of water decomposing at 2300 K and 1 bar is_______ (Nearest integer). Equilibrium constant for the reaction is 2 × 10⁻³ at 2300 K.

Answer: (2)

53. The sum of bridging carbonyls in W(CO)₆ and Mn₂(CO)₁₀ is_______

Answer: (0)

54. Solid Lead nitrate is dissolved in 1 litre of water. The solution was found to boil at 100.15°C. When 0.2 mol of NaCl is added to the resulting solution, it was observed that the solution froze at −0.80C. The solubility product of PbCl₂ formed is_____×10⁻⁶ at 298 K. (Nearest integer)

(Given : Kb=0.5 K kg mol^–1 and K_f

=1.8 K kg mol⁻¹. Assume molality to be equal to molarity in all cases.)

Answer: (13)

55. 17 mg of a hydrocarbon (M.F. C₁₀H₁₆ ) takes up 8.40 mL of the H₂ gas measured at 0°C and 760 mm of Hg. Ozonolysis of the same hydrocarbon yields

The number of double bond/s present in the hydrocarbon is_______

Answer: (3)

56. Consider the following reaction approaching equilibrium at 27°C and 1 atm pressure

The standard Gibb’s energy change (∆_rG^θ) at 27°C is (−) _______ KJ mol⁻¹ (Nearest integer).

(Given : R = 8.3 J K⁻¹ mol⁻¹ and ln 10 = 2.3)

Answer: (6)

57. The number of molecules or ions from the following, which do not have odd number of electrons are______

(A) NO₂

(B) ICl₄⁻

(D) ClO₂

(E) NO₂⁺

(F) NO

Answer: (3)

58. Following chromatogram was developed by adsorption of compound ‘A’ on a 6 cm TLC glass plate. Retardation factor of the compound ‘A’ is ______ × 10⁻¹

Answer: (6)

59. For certain chemical reaction X→Y, the rate of formation of product is plotted against the time as shown in the figure. The number of correct statement/s from the following is_____

(A) Over all order of this reaction is one

(B) Order of this reaction can’t be determined

(D) In region-II, the reaction is of first order

(E) In region-II, the order of reaction is in the range of 0.1 to 0.9.

Answer: (2)

60. Following figure shows dependence of molar conductance of two electrolytes on concentration. is the limiting molar conductivity.

The number of incorrect statement(s) from the following is________

(A) for electrolyte A is obtained by extrapolation

(B) For electrolyte B, Λm vs √c graph is a straight line with intercept equal to

(D) for any electrolyte A or B can be calculated using λ° for individual ions

Answer: (2)

Mathematics

SECTION-A

61. Let α and β be real numbers. Consider a 3 × 3 matrix A such that A²= 3A + αI. If A⁴ = 21A + βI, then

(1) β = −8

(2) β = 8

(3) α = 4

(4) α = 1

Answer: (1)

62. Let x = 2 be a root of the equation x² + px + q = 0 and

where [·] denotes greatest integer function, is

(1) 0

(2) −1

(3) 2

(4) 1

Answer: (1)

63. Let B and C be the two points on the line y + x = 0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y – 2x = 2 such that △ABC is an equilateral triangle. Then, the area of the △ABC is

(1) 10/√3

(2) 3√3

(3) 2√3

(4) 8/√3

Answer: (4)

64. Consider the following system of equations

αx + 2y + z = 1

2αx + 3y + z = 1

3x + αy + 2z = β

for some α, β ∈ ℝ. Then which of the following is NOT correct.

(1) It has a solution if α = −1 and β ≠ 2

(2) It has a solution for all α ≠ −1 and β = 2

(3) It has no solution for α =3 and β ≠ 2

(4) It has no solution for α = −1 and β ∈ ℝ

Answer: (4)

65. Let y = f(x) be the solution of the differential equation y(x + 1)dx − x²dy = 0, y(1) = e. Then is equal to

(1) 1/e²

(2) e²

(3) 0

(4) 1/e

Answer: (3)

66. The domain of is

(1) ℝ − {3}

(2) (−1, ∞) −{3}

(3) (2, ∞) −{3}

(4) ℝ −{−1,3}

Answer: (3)

67. Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is

(1) 5/24

(2) 1/6

(3) 5/36

(4) 2/15

Answer: (2)

68. Let [x] denote the greatest integer ≤ Consider the function f(x) = max{x², 1 + [x]}. Then the value of the integral is

Answer: (1)

69. For two non-zero complex numbers z₁ and z₂, if Re⁡(z₁z₂)=0 and Re⁡(z₁ + z₂) = 0, then which of the following are possible?

(A) Im (z₁) > 0 and Im(z₂) > 0

(B) Im (z₁) < 0 and Im (z₂) > 0

(D) Im(z₁) < 0 and Im (z₂) < 0

Choose the correct answer from the options given below:

(1) B and D

(2) A and B

(3) B and C

(4) A and C

Answer: (3)

70. If the vectors and are coplanar and the projection of is √54 units, then the sum of all possible values of λ + μ is equal to

(1) 0

(2) 24

(3) 6

(4) 18

Answer: (2)

71. Let − 2((1 – sin² 2θ) and If then f(β) is equal to

(1) 5/4

(2) 3/2

(3) 9/8

(4) 11/8

Answer: (2)

72. If p, q and r three propositions, then which of the following combination of truth values of p, q and r makes the logical expression {(p ∨ q) ∧ ((~p) ∨ r)}→((~q) ∨ r) false?

(1) p = T, q = T, r = F

(2) p = T, q = F, r = T

(3) p = F, q = T, r = F

(4) p = T, q = F, r = F

Answer: (3)

73. Let Δ be the area of the region {(x, y) ∈ ℝ² : x² + y² ≤ 21, y² ≤ 4x, x ≥ 1}. Then is equal to

Answer: (2)

74. A light ray emits from the origin making an angle 30∘ with the positive x-axis. After getting reflected by the line x + y = 1, if this ray intersects x-axis at Q, then the abscissa of Q is

Answer: (2)

75. Let A = {(x, y) ∈ ℝ² : y ≥ 0, and B = {(x, y) ∈ ℝ × ℝ: 0 ≤ y ≤ min Then the ratio of the area of A to the area of B is

Answer: (3)

76. Let λ ≠ 0 be a real number. Let α, β be the roots of the equation 14x² – 31x + 3λ = 0 and α, γ be the roots of the equation 35x² – 53x + 4λ = 0. Then 3α/β and 4α/γ are the roots of the equation

(1) 49x² – 245x + 250 = 0

(2) 7x² + 245x – 250 = 0

(3) 7x² – 245x + 250 = 0

(4) 49x² + 245x + 250 = 0

Answer: (1)

77. Let the tangents at the points A(4,−11) and B(8,−5) on the circle x² + y² – 3x + 10y −15 = 0, intersect at the point C. Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to

(1) 2√13

(2) √13

(3) 3√3/4

(4) 2√13/3

Answer: (4)

78. Let x ∈ ℝ be a function which satisfies Then (a + b) is equal to

(1) −2π(π – 2)

(2) −2π(π + 2)

(3) −π(π – 2)

(4) −π(π + 2)

Answer: (2)

79. Let f : R → R be a function such that Then

(1) f(x) is one-one in [1, ∞) but not in (−∞, ∞)

(2) f(x) is one-one in (−∞, ∞)

(3) f(x) is many-one in ((−∞, −1)

(4) f(x) is many-one in (1, ∞)

Answer: (1)

80. Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable 𝑋 denote the number of rotten apples. If μ and σ² represent mean and variance of 𝑋, respectively, then 10(μ² + σ²) is equal to

(1) 250

(2) 25

(3) 30

(4) 20

Answer: (4)

Section B

81. Let the co-ordinates of one vertex of △ ABC be A(0, 2, α) and the other two vertices lie on the line For α ∈ ℤ if the area of △ABC is 21 sq. units and the line segment BC has length 2√21 units, then α² is equal to

Answer: (9)

82. Let f : ℝ → ℝ be a differentiable function that satisfies the relation f(x + y) = f(x) + f(y) – 1, ∀x, y ∈ ℝ. If fꞌ(0) = 2, then |f(−2)| is equal to

Answer: (3)

83. Suppose f is a function satisfying f(x + y) = f(x) + f(y) for all x, y ∈ ℕ and f(1) = 1/5. If then m is equal to

Answer: (10)

84. Let the coefficients of three consecutive terms in the binomial expansion of (1+2x)ⁿ be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of these three terms, is

Answer: (1120)

85. Let a₁, a₂, a₃, … be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a₁a₉ + a₂a₄a₉ + a₅ + a₇ is equal to

Answer: (60)

86. Let the equation of the plane P containing the line be ax + by + 3z = 2(a + b) and the distance of the plane P from the point (1, 27, 7) be c. Then a² + b² + c² is equal to

Answer: (355)

87. If the co-efficient of x⁹ in and the co-efficient of x⁻⁹ in are equal, then (αβ)² is equal to

Answer: (1)

88. Let be three non-zero non-coplanar vectors. Le the position vectors of four points, A, B, C and D be If are coplanar, then λ is equal to

Answer: (2)

89. Five digit numbers are formed using the digits 1, 2, 3,5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1 . Then the serial number of 35337 is

Answer: (1436)

90. If all the six digit numbers x₁x₂x₃x₄x₅x₆ with 0 < x₁ < x₂ < x₃ < x₄ < x₅ < x₆ are arranged in the increasing order, then the sum of the digits in the 72th number is

Answer: (32)

JEE Main Session 1 25th January 2023 Shift 1 Question Paper and Answer Key

May 10, 2023May 6, 2023 by suresh84

JEE MAIN 25^th January 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.

1. Match List I with List II

Choose the correct answer from the options given below:

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

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

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

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

Answer: (2)

2. The ratio of the density of oxygen nucleus (₈¹⁶O) and helium nucleus (₂⁴He) is

(1) 4:1

(2) 2:1

(3) 1:1

(4) 8:1

Answer: (3)

3. The root mean square velocity of molecules of gas is

(1) Inversely proportional to square root of temperature

(2) Proportional to square of temperature (T²)

(3) Proportional to temperature (T)

(4) Proportional to square root of temperature (√T)

Answer: (4)

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-I, B-III, C-IV, D-II

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

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

Answer: (1)

5. A message signal of frequency 5kHz is used to modulate a carrier signal of frequency 2MHz. The bandwidth for amplitude modulation is:

(1) 20 kHz

(2) 5 kHz

(3) 10 kHz

(4) 2.5 kHz

Answer: (3)

6. An object of mass 8 kg hanging from one end of a uniform rod CD of mass 2 kg and length 1m pivoted at its end C on a vertical walls as shown in figure. It is supported by a cable AB such that the system is in equilibrium. The tension in the cable is: (Take g = 10 m/s²)

(1) 90 N

(2) 30 N

(3) 300 N

(4) 240 N

Answer: (3)

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

Assertion A: Photodiodes are used in forward bias usually for measuring the light intensity.

Reason R: For a p-n junction diode, at applied voltage

V the current in the forward bias is more than the current in the reverse bias for |V₂| > ± V ≥ |V₀| where V₀ is the threshold voltage and V_z is the breakdown voltage.

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 correct explanation A

(2) A is false but R is true

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

(4) A is true but R is false

Answer: (2)

8. In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes x times its initial resonant frequency ω₀.The value of x is:

(1) 4

(2) 1/16

(3) 16

(4) 1/4

Answer: (4)

9. A uniform metallic wire carries a current 2A, when 3.4 V battery is connected across it. The mass of uniform metallic wires is 8.92 × 10^–3 kg density is 8.92 × 103 kg/m3 and resistivity is 1.7 × 10^–8 Ω – m. The length of wire is:

(1) l = 10 m

(2) l = 100 m

(3) l = 5 m

(4) l = 6.8 m

Answer: (1)

10. A car travels a distance of ‘x′ with speed ν₁ and then same distance ′x′ with speed ν₂ in the same direction. The average speed of the car is:

Answer: (1)

11. A car is moving with a constant speed of 20 m/s in a circular horizontal track of radius 40m. A bob is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be: (Take g = 10 m/s²)

(1) π/3

(2) π/2

(3) π/4

(4) π/6

Answer: (3)

12. A bowl filled with very hot soup cools from 98°C to 86°C in 2 minutes when the room temperature is 22°C. How long it will take to cool from 75°C to 69°C?

(1) 1 minute

(2) 1.4 minutes

(3) 0.5 minute

(4) 2 minutes

Answer: (2)

13. A solenoid of 1200 turns is wound uniformly in a single layer on a glass tube 2m long and 0.2m in diameter. The magnetic intensity at the center of the solenoid when a current of 2A flows through it is?

(1) 2.4 × 10³ Am⁻¹

(2) 1.2 × 10³ Am⁻¹

(3) 2.4 × 10⁻³ Am⁻¹

(4) 1 Am⁻¹

Answer: (2)

14. In Young’s double slits experiment, the position of 5th bright fringe from the central maximum is 5cm. The distance between slits and screen is 1m and wavelength of used monochromatic light is 600 nm. The separation between the slits is:

(1) 48 μm

(2) 36 μm

(3) 12 μm

(4) 60 μm

Answer: (4)

15. An electromagnetic wave is transporting energy in the negative z direction. At a certain point and certain time the direction of electric field of the wave is along positive y direction. What will be the direction of the magnetic field of the wave at the point and instant?

(1) Negative direction of y

(2) Positive direction of z

(3) Positive direction of x

(4) Negative direction of x

Answer: (3)

16. A parallel plate capacitor has plate area 40 cm² and plates separation 2mm. The space between the plates is filled with a dielectric medium of a thickness 1 mm and dielectric constant 5. The capacitance of the system is:

(1) 24ε₀ F

(2)

(3)

(4) 10ε₀ F

Answer: (2)

17. Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g. The time period of the motion of the particle will be (approximately)

(Take g = 10 ms⁻²,radius of earth = 6400 km )

(1) 12 hours

(2) 1 hour 40 minutes

(3) 24 hours

(4) 1 hour 24 minutes

Answer: (4)

18. Electron beam used in an electron microscope, when accelerated by a voltage of 20kV,has a de−Broglie wavelength of 𝜆₀.If the voltage is increased to 40kV, then the de-Broglie wavelength associated with the electron beam would be:

(1) 3λ₀

(2) λ₀/2

(3) λ₀/√2

(4) 9λ₀

Answer: (3)

19. A Carnot engine with efficiency 50% takes heat from a source at 600 K. In order to increase the efficiency to 70%, keeping the temperature of sink same, the new temperature of the source will be :

(1) 300 K

(2) 900 K

(3) 1000 K

(4) 360 K

Answer: (3)

20. T is the time period of simple pendulum on the earth’s surface. Its time Period becomes x T when taken to a height R (equal to earth’s radius) above the earth’s surface. Then, the value of x will be:

(1) 4

(2) 2

(3) 1/4

(4) 1/2

Answer: (2)

SECTION-B

21. A uniform electric field of 10 N/C is created between two parallel charged pates (as shown in figure). An electron enters the field symmetrically between the plates with a kinetic energy 0.5eV. The length of each pate is 10 cm. The angle (θ) of deviation of the path of electron as it comes out of the field is ______ (in degree).

Answer: (45)

22. The wavelength of the radiation emitted is 𝜆₀ when an electron jumps from the second excited state to the first excited state of hydrogen atom. If the electron jumps from the third excited state to the second orbit of the hydrogen atom, the wavelength of the radiation emitted will be . The value of x is _____.

Answer: (27)

23. As shown in the figure, in an experiment to determine Young’s modulus of a wire, the extension-load curve is plotted. The curve is a straight line passing through the origin and makes an angle of 45° with the load axis. The length of wire is 62.8cm and its diameter is 4 mm. The Young’s modulus is found to be x × 10⁴ Nm^–2. The value of x is __________

Answer: (5)

24. I_CM is the moment of inertia of a circular disc about an axis (CM)passing through its center and perpendicular. To the plane of disc. I_AB is it′s moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at a distance 2/3R from center.

Where R is the radius of the disc. The ratio of I_AB and I_CM is x : 9.

The value of x is ______

Answer: ()

25. An object of mass ‘m’ initially at rest on a smooth horizontal plane starts moving under the action of force F = 2N. In the process of its linear motion, the angle θ (as shown in figure) between the direction of force and horizontal varies as θ = kx, where k is constant and x is the distance covered by the object from the initial position. The expression of kinetic energy of the object will be

Answer: (2)

26. An LCR series circuit of capacitance 62.5nF and resistance of 50Ω, is connected to an A.C. source of frequency 2.0kHz. For maximum value of amplitude of current in circuit, the value of inductance is _______ mH. Take π² = 10)

Answer: (100)

27. The distance between two consecutive points with phase difference of 60∘ in a wave of frequency 500 Hz is 6.0 m. The velocity with which wave is traveling is ___________ km/s

Answer: (18)

28. In the given circuit, the equivalent resistance between the terminal A and B is Ω.

Answer: (10)

29. If and then, The unit vector in the direction of The value of x is

Answer: (4)

30. A ray of light is incident from air on a glass plate having thickness √3 cm and refractive index √2. The angle of incidence of a ray is equal to the critical angle for glass-air interface. The lateral displacement of the ray when it passes through the plate is _______ × 10^–2 cm. (given sin 15° = 0.26)

Answer: (52)

Chemistry

SECTION-A

31. In the cumene to phenol preparation in presence of air, the intermediate is

Answer: (3)

32. The compound which will have the lowest rate towards nucleophilic aromatic substitution on treatment with OH⁻ is

Answer: (3)

33. Match List I with List II

Choose the correct answer from the options given below:

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

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

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

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

Answer: (2)

34. Which of the following conformations will be the most stable ?

Answer: (2)

35. The variation of the rate of an enzyme catalyzed reaction with substrate concentration is correctly represented by graph

Answer: (4)

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

Assertion A : Acetal / Ketal is stable in basic medium.

Reason R : The high leaving tendency of alkoxide ion gives the stability to acetal/ ketal in basic medium.

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 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)

37. A cubic solid is made up of two elements X and Y. Atoms of X are present on every alternate corner and one at the center of cube. Y is at 1/3^rd of the total faces. The empirical formula of the compound is

(1) XY_2.5

(2) X₂Y_1.5

(3) X_2.5Y

(4) X_1.5Y₂

Answer: (4)

38. Match the List-I with List-II

Correct match is-

(1) A → iii, B → i, C → iv, D → ii

(2) A → i, B → iii, C → ii, D → iv

(3) A → iv, B → ii, C → iii, D → i

(4) A → i, B → iii, C → iv, D → ii

Answer: (4)

39. Which of the following statements is incorrect for antibiotics?

(1) An antibiotic must be a product of metabolism.

(2) An antibiotic should promote the growth or survival of microorganisms.

(3) An antibiotic is a synthetic substance produced as a structural analogue of naturally occurring antibiotic.

(4) An antibiotic should be effective in low concentrations.

Answer: (2)

40. The correct order in aqueous medium of basic strength in case of methyl substituted amines is :

(1) Me₃ N > Me₂NH > MeNH₂ > NH₃

(2) Me₂NH > MeNH₂ > Me₃ N > NH₃

(3) Me₂NH > Me₃ N > MeNH₂ > NH₃

(4) NH₃ > Me₃ N > MeNH₂ > Me₂NH

Answer: (2)

41. ’25 volume’ hydrogen peroxide means

(1) 1 L marketed solution contains 25 g of H₂O₂.

(2) 1 L marketed solution contains 75 g of H₂O₂.

(3) 1 L marketed solution contains 250 g of H₂O₂.

(4) 100 mL marketed solution contains 25 g of H₂O₂.

Answer: (2)

42. The radius of the 2nd orbit of Li²⁺ is x. The expected radius of the 3rd orbit of Be³⁺ is

Answer: (1)

43. Reaction of thionyl chloride with white phosphorus forms a compound [A], which on hydrolysis gives [B], a dibasic acid. [A] and [B] are respectively

(1) P₄O₆ and H₃PO₃

(2) PCl₅ and H₃PO₄

(3) POCl₃ and H₃PO₄

(4) PCl₃ and H₃PO₃

Answer: (4)

44. Inert gases have positive electron gain enthalpy. Its correct order is

(1) He < Kr < Xe <Ne

(2) He < Xe < Kr < Ne

(3) He < Ne < Kr < Xe

(4) Xe < Kr < Ne < He

Answer: (2)

45. Identify the product formed ( and E)

Answer: (3)

46. Match items of Row I with those of Row II.

Answer: (4)

47. Which one of the following reactions does not occur during extraction of copper ?

(1) 2Cu₂ S + 3O₂ → 2Cu₂O + 2SO₂

(2) FeO + SiO₂ → FeSiO₃

(3) 2FeS + 3O₂→ 2FeO+2SO₂

(4) CaO + SiO₂→ CaSiO₃

Answer: (4)

48. Some reactions of NO₂ relevant to photochemical smog formation are

Identify A, B, X and Y

(1) Y = NO₂, A = O₃, B = O₂

(2) X = [O], Y = NO, A = O₂, B = O₃

(3) X = N₂O, Y = [O], A = O₃, B = NO

(4) X = NO, Y = [O], A = O₂, B = N₂O₃

Answer: (2)

49.

The correct sequence of reagents for the preparation of Q and R is :

(1) (i) CrO₂Cl₂,H₃O⁺; (ii) Cr₂O₃,770 K, 20 atm; (iii) NaOH; (iv) H₃O⁺

(2) (i) KMnO₄,OH⁻; (ii) Mo₂O₃,Δ; (iii) NaOH; (iv) H₃O⁺

(3) (i) Cr₂O₃,770 K, 20 atm; (ii) CrO₂Cl₂,H₃O⁺; (iii) NaOH; (iv) H₃O⁺

(4) (i) Mo₂O₃, Δ; (ii) CrO₂Cl₂, H₃O⁺; (iii) NaOH; (iv) H₃O⁺

Answer: (3)

50. Compound A reacts with NH₄Cl and forms a compound B. Compound B reacts with H₂O and excess of CO₂ to form compound C which on passing through or reaction with saturated NaCl solution forms sodium hydrogen carbonate. Compound A, B and C, are respectively.

(1) CaCl₂, NH₃, NH₄HCO₃

(2) Ca(OH)₂, NH₄^⨁, (NH₄)₂CO₃

(3) CaCl₂, NH₄^⨁, (NH₄)₂CO₃

(4) Ca(OH)₂, NH₃, NH₄HCO₃

Answer: (4)

SECTION-B

51. For the first order reaction A→B, the half life is 30 min. The time taken for 75% completion of the reaction is ____ min. (Nearest integer)

Given : log 2 = 0.3010

log 3 = 0.4771

log 5 = 0.6989

Answer: (60)

52. How many of the following metal ions have similar value of spin only magnetic moment in gaseous state? (Given: Atomic number : V, 23; Cr, 24; Fe, 26; Ni, 28) V³⁺, Cr³⁺, Fe²⁺, Ni³⁺

Answer: (2)

53. In sulphur estimation, 0.471 g of an organic compound gave 1.4439 g of barium sulphate. The percentage of sulphur in the compound is______ (Nearest Integer)

(Given: Atomic mass Ba: 137u, S:32 u, O: 16u )

Answer: (42)

54. The osmotic pressure of solutions of PVC in cyclohexanone at 300 K are plotted on the graph. The molar mass of PVC is ____ gmol⁻¹ (Nearest integer)

(Given : R = 0.083 L atm K⁻¹ mol⁻¹)

Answer: (41500)

55. The density of a monobasic strong acid (Molar mass 24.2 g/mol) is 1.21 kg/L. The volume of its solution required for the complete neutralization of 25 mL of 0.24MNaOH is ___ × 10⁻² mL (Nearest integer)

Answer: (12)

56. An athlete is given 100 g of glucose (C₆H₁₂O₆) for energy. This is equivalent to 1800 kJ of energy. The 50% of this energy gained is utilized by the athlete for sports activities at the event. In order to avoid storage of energy, the weight of extra water he would need to perspire is____ g (Nearest integer) Assume that there is no other way of consuming stored energy.

Given : The enthalpy of evaporation of water is 45 kJ mol⁻¹

Molar mass of C, H & O are 12, 1 and 16 g mol⁻¹

Answer: (360)

57. The number of paramagnetic species from the following is

[Ni(CN)₄]²⁻, [Ni(CO)₄], [NiCl₄]²⁻

[Fe(CN)₆]⁴⁻, [Cu(NH₃)₄]²⁺

[Fe(CN)₆]³⁻ and [Fe(H₂O)₆]²⁺

Answer: (4)

58. Consider the cell

Pt(s) | H₂(g) (1 atm) | H⁺] = 1) || Fe³⁺ (aq), Fe²⁺ (aq) | Pt(s)

Given and T = 298 K

If the potential of the cell is 0.712 V, the ratio of concentration of Fe²⁺ to Fe³⁺ is (Nearest integer)

Answer: (10)

59. The total number of lone pairs of electrons on oxygen atoms of ozone is

Answer: (6)

60. A litre of buffer solution contains 0.1 mole of each of NH₃ and NH₄ On the addition of 0.02 mole of HCl by dissolving gaseous HCl, the pH of the solution is found to be____×10⁻³ (Nearest integer)

[Given : pK_b(NH₃) = 4.745

log 2 = 0.301

Log 3 = 0.477

T = 298 K]

Answer: (9)

Mathematics

SECTION-A

61. The points of intersection of the line ax + by = 0, (a ≠ b) and the circle x² + y² – 2x = 0 are A(α, 0) and B(1, β). The image of the circle with AB as a diameter in the line x + y + 2 = 0 is :

(1) x² + y² + 3x + 3y + 4 = 0

(2) x² + y² + 3x + 5y + 8 = 0

(3) x² + y² − 5x − 5y + 12 = 0

(4) x² + y² + 5x + 5y + 12 = 0

Answer: (4)

62. The distance of the point (6, −2√2) from the common tangent y = mx + c, m > 0, of the curves x = 2y² and x = 1 + y² is :

(1) 14/3

(2) 5√3

(3) 1/3

(4) 5

Answer: (4)

63. Let be three non zero vectors such that If be a vector such that then is equal to

(1) −1/4

(2) 1/4

(3) 3/4

(4) 1/2

Answer: (2)

64. The vector is rotated through a right angle, passing through the y-axis in its way and the resulting vector is Then the projection of is :

(1) 2√3

(2) 1

(3) 3√2

(4) √6

Answer: (3)

65. Let z₁= 2 + 3i and z₂ = 3 + 4i. The set S ={z ∈ C : |z − z₁|² − |z − z₂|² = |z₁ − z₂|²} represents a

(1) hyperbola with the length of the transverse axis 7

(2) hyperbola with eccentricity 2

(3) straight line with the sum of its intercepts on the coordinate axes equals −18

(4) straight line with the sum of its intercepts on the coordinate axes equals 14

Answer: (4)

66. The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12 . If the new mean of the marks is 10.2, then their new variance is equal to :

(1) 3.96

(2) 4.08

(3) 4.04

(4) 3.92

Answer: (1)

67. Let S₁ and S₂ be respectively the sets of all a ∈ ℝ − {0} for which the system of linear equations

ax + 2ay – 3az = 1

(2a + 1)x + (2a + 3)y + (a + 1)z = 2

(3a + 5)x + (a + 5)y + (a + 2)z = 3

has unique solution and infinitely many solutions. Then

(1) S₁ is an infinite set and n(S₂) = 2

(2) S₁ = Φ and S₂ = ℝ − {0}

(3) n(S₁) = 2 and S₂ is an infinite set

(4) S₁ = ℝ − {0} and S₂ = Φ

Answer: (4)

68. The value of is :

Answer: (1)

69. The statement (p ∧ (∼q)) ⇒ (p ⇒ (∼q)) is

(1) a tautology

(2) a contradiction

(3) equivalent to p ∨ q

(4) equivalent to (∼p) ∨ (∼q)

Answer: (1)

70. Consider the lines L₁ and L₂ given by

A line L₃ having direction ratios 1, −1, −2, intersects L₁ and L₂ at the points P and Q respectively. Then the length of line segment PQ is

(1) 3√2

(2) 4√3

(3) 4

(4) 2√6

Answer: (4)

71. Let If then f(4) is equal to

(1) log_e 19 – log_e 20

(2) log_e 17 – log_e 18

(3)

(4)

Answer: (4)

72. The minimum value of the function is :

(1) e(e – 1)

(2) 2(e – 1)

(3) 2

(4) 2e – 1

Answer: ()

73. Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space and the event A = {x ∈ S : x is a multiple of 3}. Then P(A) is equal to

(1) 7/22

(2) 1/5

(3) 15/44

(4) 1/3

Answer: (4)

74. Let x = 2 be a local minima of the function f(x) = 2x⁴ – 18x² + 8x + 12, x ∈ (−4, 4). If M is local maximum value of the function f in (−4, 4), then M=

Answer: (3)

75. Let f: (0, 1) → ℝ be a function defined by and g(x) = (f(−x) – f(x)). Consider two statements

(I) g is an increasing function in (0, 1)

(II) g is one-one in (0,1)

Then,

(1) Both (I) and (II) are true

(2) Neither (I) nor (II) is true

(3) Only (I) is true

(4) Only (II) is true

Answer: (1)

76. Let y(x) = (1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) (1 + x¹⁶). Then yꞌ − yꞌꞌ at x = −1 is equal to:

(1) 976

(2) 944

(3) 464

(4) 496

Answer: (4)

77. The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to :

(1) √14

(2) 3

(3) √6

(4) 2√3

Answer: (1)

78. Let x, y, z > 1 and Then |adj(adj A²)| is equal to

(1) 2⁸

(2) 4⁸

(3) 6⁴

(4) 2⁴

Answer: (1)

79. If a_r is the coefficient of x^{10− r} in the Binomial expansion of (1 + x)¹⁰, then is equal to

(1) 5445

(2) 3025

(3) 4895

(4) 1210

Answer: (4)

80. Let y = y(𝑥) be the solution curve of the differential equation x > 0, y(1) = 3. Then is equal to :

Answer: (4)

SECTION-B

81. The constant term in the expansion of

Answer: (1080)

82. For some a, b, c ∈ ℕ, let f(x) = ax – 3 and g(x) = x^b + c, x ∈ ℝ. If then (f ° g) (ac) + (g ° f) (b) is equal to

Answer: (2039)

83. Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of 𝑆 that have the sum of all elements a multiple of 3, is

Answer: (43)

84. Let the equation of the plane passing through the line x – 2y – z – 5 = 0 = x + y + 3z – 5 and parallel to the line x + y + 2z – 7 = 0 = 2x + 3y + z – 2 be ax + by + cz = 65. Then the distance of the point (a, b, c) from the plane 2x + 2y – z + 16 = 0 is

Answer: (9)

85. If the sum of all the solutions of −1 < x < 1, x ≠ 0, is then α is equal to

Answer: (2)

86. The vertices of a hyperbola H are (±6, 0) and its eccentricity is √5/2. Let N be the normal to H at a point in the first quadrant and parallel to the line √2x + y = 2√

If ds the length of the line segment of N between H and the y-axis then d² is equal to

Answer: (216)

87. Let x and y be distinct integers where 1 ≤ x ≤ 25 and 1 ≤ y ≤ 25. Then, the number of ways of choosing x and y, such that x + y is divisible by 5 , is

Answer: (120)

88. Let Then the maximum value of β for which the equation has real roots, is

Answer: (25)

89. It the area enclosed by the parabolas P₁ : 2y = 5x² and P₂ : x² – y + 6 = 0 is equal to the area enclosed by 𝑃1 and y = αx, α > 0, then α³ is equal to

Answer: (600)

90. Let A₁, A₂, A₃ be the three A.P. with the same common difference d and having their first terms as A, A +1, A+2, respectively. Let a, b, c be the 7th , 9th ,17th terms of A₁, A₂, A₃, respectively such that

If a = 29, then the sum of first 20 terms of an AP whose first term is c – a − b and common difference is d/12, is equal to

Answer: (495)

JEE Main Session 1 24th January 2023 Shift 1 Question Paper and Answer Key

May 10, 2023May 6, 2023 by suresh84

JEE MAIN 24^th January 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.

1. A circular loop of radius 𝑟 is carrying current I A. The ratio of magnetic field at the center of circular loop and at a distance 𝑟 from the center of the loop on its axis is:

(1) 2√2 : 1

(2) 1 : 3√2

(3) 1 : √2

(4) 3√2 : 2

Answer: (1)

2. The weight of a body at the surface of earth is 18 N. The weight of the body at an altitude of 3200 km above the earth’s surface is (given, radius of earth R_e = 6400 km ):

(1) 8 N

(2) 4.9 N

(3) 9.8 N

(4) 19.6 N

Answer: (1)

3. Two long straight wires P and Q carrying equal current 10 A each were kept parallel to each other at 5 cm distance. Magnitude of magnetic force experienced by 10 cm length of wire P is F₁ – If distance between wires is halved and currents on them are doubled, force F₂ on 10 cm length of wire P will be:

(1) F₁/8

(2) 8F₁

(3) 10F₁

(4) F₁/10

Answer: (2)

4. Given below are two statements :

Statement I : The temperature of a gas is −73°C. When the gas is heated to 527°C, the root mean square speed of the molecules is doubled.

Statement II : The product of pressure and volume of an ideal gas will be equal to translational kinetic energy of the molecules. 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: (3)

5. The maximum vertical height to which a man can throw a ball is 136 m. The maximum horizontal distance upto which he can throw the same ball is:

(1) 272 m

(2) 68 m

(3) 192 m

(4) 136 m

Answer: (1)

6. Given below are two statements :

Statement I : If the Brewster’s angle for the light propagating from air to glass is 𝜃B, then the Brewster’s angle for the light propagating from glass to air is

Statement II : The Brewster’s angle for the light propagating from glass to air is tan⁻¹(μg) where μg is the refractive index of glass.

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

(1) Both Statement I and Statement II are false

(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 true

Answer: (2)

7. A 100 m long wire having cross-sectional area 6.25 × 10⁻⁴ m² and Young’s modulus is 10¹⁰ Nm⁻² is subjected to a load of 250 N, then the elongation in the wire will be:

(1) 4 × 10⁻³ m

(2) 6.25 × 10⁻³ m

(3) 6.25 × 10⁻⁶ m

(4) 4 × 10⁻⁴ m

Answer: (1)

8. If two charges q₁ and q₂ are separated with distance ‘d’ and placed in a medium of dielectric constant K. What will be the equivalent distance between charges in air for the same electrostatic force?

(1) 2d√k

(2) 1.5d√k

(3) d√k

(4) k√k

Answer: (3)

9. Consider the following radioactive decay process

The mass number and the atomic number of A₆ are given by:

(1) 210 and 84

(2) 210 and 82

(3) 211 and 80

(4) 210 and 80

Answer: (4)

10. From the photoelectric effect experiment, following observations are made. Identify which of these are correct.

(A) The stopping potential depends only on the work function of the metal.

(B) The saturation current increases as the intensity of incident light increases.

(D) Photoelectric effect can be explained using wave theory of light.

Choose the correct answer from the options given below:

(1) A, C, D only

(2) B, C only

(3) B only

(4) A, B, D only

Answer: (3)

11. Given below are two statements:

Statement I: An elevator can go up or down with uniform speed when its weight is balanced with the tension of its cable.

Statement II: Force exerted by the floor of an elevator on the foot of a person standing on it is more than his/her weight when the elevator goes down with increasing speed.

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

(1) Both Statement I and Statement II are true

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

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

(4) Both Statement I and Statement II are false

Answer: (3)

12. 1 g of a liquid is converted to vapour at 3 × 10⁵ Pa pressure. If 10% of the heat supplied is used for increasing the volume by 1600 cm³ during this phase change, then the increase in internal energy in the process will be:

(1) 432000 J

(2) 4320 J

(3) 4800 J

(4) 4.32 × 10⁸ J

Answer: (2)

13. As shown in the figure, a network of resistors is connected to a battery of 24 V with an internal resistance of 3Ω. The currents through the resistors R₄ and R₅ are I₄ and I₅ The values of I₄ and I₅ are:

Answer: (1)

14. A modulating signal is a square wave, as shown in the figure.

If the carrier wave is given as c(t) = 2sin(8πt) volts, the modulation index is:

(1) 1/4

(2) 1/2

(3) 1

(4) 1/3

Answer: (2)

15. A conducting circular loop of radius 10/√π cm is placed perpendicular to a uniform magnetic field of 0.5 T. The magnetic field is decreased to zero in 0.5 s at a steady rate. The induced emf in the circular loop at 0.25 s is:

(1) emf = 1mV

(2) emf = 5mV

(3) emf = 100mV

(4) emf = 10mV

Answer: (4)

16. In represent electric field and propagation vectors of the EM waves in vacuum, then magnetic field vector is given by :

(ω – angular frequency):

Answer: (4)

17. 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-III, B-I, C-II, D-IV

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

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

Answer: (4)

18. A travelling wave is described by the equation

y(x, t) = [0.05sin (8x – 4t)]m

The velocity of the wave is : [all the quantities are in SI unit]

(1) 8 ms⁻¹

(2) 4 ms⁻¹

(3) 0.5 ms⁻¹

(4) 2 ms⁻¹

Answer: (3)

19. As per given figure, a weightless pulley P is attached on a double inclined frictionless surfaces. The tension in the string (massless) will be (if g =10 m/s² )

(1) (4√3 + 1)N

(2) 4(√3 + 1)N

(3) (4√3 – 1)N

(4) 4(√3 – 1)N

Answer: (2)

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

Assertion A: Photodiodes are preferably operated in reverse bias condition for light intensity measurement.

Reason : The current in the forward bias is more than the current in the reverse bias for a p − n junction diode.

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)

SECTION-B

21. Vectors are perpendicular to each other when 3a + 2b = 7, the ratio of a to b is x/2. The value of x is

Answer: (1)

22. Assume that protons and neutrons have equal masses. Mass of a nucleon is 1.6×10⁻²⁷ kg and radius of nucleus is 1.5 × 10⁻¹⁵ A^1/3 The approximate ratio of the nuclear density and water density is n × 10¹³. The value of n is

Answer: (11)

23. A hollow cylindrical conductor has length of 3.14 m, while its inner and outer diameters are 4 mm and 8 mm respectively. The resistance of the conductor is n × 10⁻³Ω. If the resistivity of the material is 2.4 × 10⁻⁸ Ωm. The value of n is

Answer: (2)

24. A stream of a positively charged particles having and velocity is deflected by an electric field The electric field exists in a region of 10 cm along x direction. Due to the electric field, the deflection of the charge particles in the 𝑦 direction is _____ mm

Answer: (2)

25. As shown in the figure, a combination of a thin plano concave lens and a thin plano convex lens is used to image an object placed at infinity. The radius of curvature of both the lenses is 30 cm and refraction index of the material for both the lenses is 1.75. Both the lenses are placed at distance of 40 cm from each other. Due to the combination, the image of the object is formed at distance = ____cm, from concave lens.

Answer: (120)

26. Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm then its radius of gyration about PQ will be √x cm. The value of x is ______

Answer: (110)

27. A block of a mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is π/√x in SI unit. The value of x is ________

Answer: (5)

28. A hole is drilled in a metal sheet. At 27°C, the diameter of hole is 5 cm. When the sheet is heated to 177°C, the change in the diameter of hole is d × 10⁻³ The value of d will be ________ if coefficient of linear expansion of the metal is 1.6 × 10⁻⁵/°C.

Answer: (12)

29. In the circuit shown in the figure, the ratio of the quality factor and the band width is ______ S.

Answer: (10)

30. A spherical body of mass 2 kg starting from rest acquires a kinetic energy of 10000 J at the end of 5th second. The force acted on the body is ______ N.

Answer: (40)

Chemistry

SECTION-A

31. ‘A’ and ‘ B ‘ formed in the following set of reactions are:

Answer: (2)

32. Decreasing order of the hydrogen bonding in following forms of water is correctly represented by

(A) Liquid water

(B) Ice

Choose the correct answer from the options given below:

(1) B > A > C

(2) A > B > C

(3) A = B > C

(4) C > B > A

Answer: (1)

33. Increasing order of stability of the resonance structures is:

Choose the correct answer from the options given below:

(1) D, C, A, B

(2) D, C, B, A

(3) C, D, A, B

(4) C, D, B, A

Answer: (BONUS)

34. ꞌRꞌ formed in the following sequence of reactions is:

Answer: (2)

35. The primary and secondary valencies of cobalt respectively in [Co(NH₃)₅Cl^ClCl₂ are:

(1) 3 and 6

(2) 2 and 6

(3) 3 and 5

(4) 2 and 8

Answer: (1)

36. An ammoniacal metal salt solution gives a brilliant red precipitate on addition of dimethylglyoxime. The metal ion is:

(1) Co²⁺

(2) Ni²⁺

(3) Fe²⁺

(4) Cu²⁺

Answer: (2)

37. Reaction of BeO with ammonia and hydrogen fluoride gives A which on thermal decomposition gives BeF₂ and NH₄ What is ‘A’ ?

(1) (NH₄)₂BeF₄

(2) H₃NBeF₃

(3) (NH₄)Be₂F₅

(4) (NH₄)BeF₃

Answer: (1)

38. 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-I, B-III, C-II, D-IV

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

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

Answer: (1)

39. Match List I with List II

Choose the correct answer from the options given below:

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

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

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

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

Answer: (2)

40. In the following given reaction, ‘ A ‘ is

Answer: (3)

41. It is observed that characteristic X-ray spectra of elements show regularity. When frequency to the power “n” i.e. vⁿ of X-rays emitted is plotted against atomic number “Z”, following graph is obtained.

The value of ꞌꞌnꞌꞌ is

(1) 3

(2) 2

(3) 1

(4) 1/2

Answer: (4)

42. Given below are two statements:

Statement I : Noradrenaline is a neurotransmitter.

Statement II : Low level of noradrenaline is not the cause of depression in human.

In the light of the above statements, choose the correct 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) Both Statement I and Statement II are incorrect

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

Answer: (1)

43. Which of the Phosphorus oxoacid can create silver mirror from AgNO₃ solution?

(1) (HPO₃)_n

(2) H₄P₂O₆

(3) H₄P₂O₅

(4) H₄P₂O₇

Answer: (3)

44. Compound (X) undergoes following sequence of reactions to give the Lactone (Y).

Compound (X) is

Answer: (4)

45. Order of Covalent bond:

(A) KF > KI; LiF > KF

(B) KF < KI; LiF > KF

(D) LiF > KF; CuCl < NaCl

(E) KF < KI; CuCl > NaCl

Choose the correct answer from the options given below:

(1) C, E only

(2) B, C, E only

(3) A, B only

(4) B, C only

Answer:

46. Which of the following is true about freons?

(1) These are radicals of chlorine and chlorine monoxide

(2) These are chemicals causing skin cancer

(3) These are chlorofluorocarbon compounds

(4) All radicals are called freons

Answer: (3)

47. In the depression of freezing point experiment

(A) Vapour pressure of the solution is less than that of pure solvent

(B) Vapour pressure of the solution is more than that of pure solvent

(D) Only solvent molecules solidify at the freezing point

Choose the most appropriate answer from the options given below:

(1) A and C only

(2) A only

(3) A and D only

(4) B and C only

Answer: (3)

48. Statement I : For colloidal particles, the values of colligative properties are of small order as compared to values shown by true solutions at same concentration.

Statement II : For colloidal particles, the potential difference between the fixed layer and the diffused layer of same charges is called the electrokinetic potential or zeta potential.

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)

49. Assertion A : Hydrolysis of an alkyl chloride is a slow reaction but in the presence of NaI, the rate of the hydrolysis increases.

Reason R : I⁻ is a good nucleophile as well as a good leaving group.

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: (3)

50. The magnetic moment of a transition metal compound has been calculated to be 3.87 B.M. The metal ion is

(1) Cr²⁺

(2) Ti²⁺

(3) V²⁺

(4) Mn²⁺

Answer: (3)

SECTION-B

51. When Fe₉₃O is heated in presence of oxygen, it converts to Fe₂O₃. The number of correct statement/s from the following is

(A) The equivalent weight of Fe_0.93O is

(B) The number of moles of Fe²⁺ and Fe³⁺ in 1 mole of Fe_0.93O is 0.79 and 0.14 respectively

(D) The % composition of Fe²⁺ and Fe³⁺ in Fe_0.93O is 85% and 15% respectively

Answer: (4)

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

(A) Larger the activation energy, smaller is the value of the rate constant.

(B) The higher is the activation energy, higher is the value of the temperature coefficient.

(D) A plot of is a straight line with slope equal to –E_a/R

Answer: (4)

53. For independent processes at 300 K

The number of non-spontaneous processes from the following is

Answer: (2)

54. 5 g of NaOH was dissolved in deionized water to prepare a 450 mL stock solution. What volume (in mL ) of this solution would be required to prepare 500 mL of 0.1M solution? Given: Molar Mass of Na,O and H is 23,16 and 1 g mol⁻¹ respectively

Answer: (180)

55. If wavelength of the first line of the Paschen series of hydrogen atom is 720 nm, then the wavelength of the second line of this series is nm. (Nearest integer)

Answer: (492)

56. Uracil is a base present in RNA with the following structure. % of N in uracil is

Answer: (25)

57. The dissociation constant of acetic acid is x × 10⁻⁵. When 25 mL of 0.2MCH₃COONa solution is mixed with 25 mL of 0.02MCH₃COOH solution, the pH of the resultant solution is found to be equal to 5 . The value of 𝑥 is

Answer: (10)

58. Number of moles of AgCl formed in the following reaction is _______

Answer: (2)

59. The d-electronic configuration of [CoCl₄]²⁻ in tetrahedral crystal field is e^mt₂ⁿ. Sum of ꞌꞌmꞌꞌ and ꞌꞌnumber of unpaired electronsꞌꞌ is

Answer: (*)

60. At 298 K, a 1 litre solution containing 10mmol of Cr₂O₇²⁻ and 100 mmol of Cr³⁺ shows a pH of 3.0.

Given : Cr₂O₇²⁻ → Cr³⁺; E° = 1.330 V and

The potential for the half cell reaction is x × 10⁻³ V. The value of x is

Answer: (917)

Mathematics

SECTION-A

61. Let Then is equal to

(1) 2

(2) 3/2

(3) 1

(4) −2/3

Answer: (3)

62. is equal to

(1) n²

(2)

(3) n

(4) n² + n

Answer: (3)

63. Let α be a root of the equation (a – c)x² + (b – a)x + (c – b) = 0 where a, b, c are distinct real numbers such that the matrix is singular. Then, the value of is

(1) 12

(2) 9

(3) 3

(4) 6

Answer: (3)

64. The area enclosed by the curves y² + 4x = 4 and y – 2x = 2 is

(1) 9

(2) 22/3

(3) 23/3

(4) 25/3

Answer: (1)

65. Let p, q ∈ ℝ and (1 – √3i)²⁰⁰ = 2¹⁹⁹(p + iq), i = √−1 Then p + q + q² and p – q + q² are roots of the equation

(1) x² – 4x – 1 = 0

(2) x² – 4x + 1 =0

(3) x² + 4x – 1 =0

(4) x² + 4x + 1 =0

Answer: (2)

66. Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations

x + y + z = 1

2x + Ny + 2z = 2

3x + 3y + Nz = 3

has unique solution is k/6, then the sum of value of k and all possible values of N is

(1) 21

(2) 18

(3) 20

(4) 19

Answer: (3)

67. For three positive integers and r = pq + 1 such that 3, 3log_yx, 3log_zy, 7log_xz are in A.P. with common difference 1/2. Then r – p – q is equal to

(1) −6

(2) 12

(3) 6

(4) 2

Answer: (4)

68. The relation R = {(a, b): gcd(a, b) = 1, 2a ≠ b, a, b ∈Z} is :

(1) reflexive but not symmetric

(2) transitive but not reflexive

(3) symmetric but not transitive

(4) neither symmetric nor transitive

Answer: (4)

69. Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that Then is equal to

(1) 4

(2) 3

(3) 1

(4) 2

Answer: (2)

70. Let y = y(x) be the solution of the differential equation x³dy + (xy – 1)dx = 0, x > 0, y(1/2) = 3 – e. Then y(1) is equal to

(1) 1

(2) e

(3) 3

(4) 2 – e

Answer: (1)

71. If A and B are two non-zero n×n matrics such that A² + B = A² B, then

(1) A² = I or B – I

(2) A²B = I

(3) AB = I

(4) A²B = BA²

Answer: (4)

72. The equation x²– 4x + [x] + 3 = x[x], where [x] denotes the greatest integer function, has :

(1) a unique solution in (−∞,1)

(2) no solution

(3) exactly two solutions in (−∞,∞)

(4) a unique solution in (−∞,∞)

Answer: (4)

73. Let a tangent to the curve y² = 24x meet the curve xy = 2 at the points A and B. Then the mid points of such line segments AB lie on a parabola with the

(1) Length of latus rectum 3/2

(2) directrix 4x = −3

(3) length of latus rectum 2

(4) directrix 4x = 3

Answer: (4)

74. Let Ω be the sample space and A ⊆ Ω be an event.

Given below are two statements:

(S1) : If P(A) = 0, then A = ∅

(S2) : If P(A) = 1, then A = Ω

Then

(1) both (S1) and (S2) are true

(2) only (S1) is true

(3) only (S2) is true

(4) both (S1) and (S2) are false

Answer: (4)

75. The value of is

(1) ⁴⁴C₂₃

(2) ⁴⁵C₂₃

(3) ⁴⁴C₂₂

(4) ⁴⁵C₂₄

Answer: (2)

76. The distance of the point (−1, 9, −16) from the plane 2x + 3y – z = 5 measured parallel to the line is

(1) 31

(2) 13√2

(3) 20√2

(4) 26

Answer: (4)

77. is equal to:

(1) π/3

(2) π/4

(3) π/6

(4) π/2

Answer: (1)

78. Let

Then at x = 0

(1) f is continuous but not differentiable

(2) f and fꞌ both are continuous

(3) fꞌ is continuous but not differentiable

(4) f is continuous but fꞌ is not continuous

Answer: (4)

79. The compound statement (∼(P ∧ Q)) ∨ ((∼P) ∧ Q) ⇒ ((∼P) ∧ (∼Q)) is equivalent to

(1) (~Q) ∨ P

(2) ((~P) ∨ Q) ∧ (~Q)

(3) (~P) ∨ Q

(4) ((~P) ∨ Q) ∧ ((~Q) ∨ P)

Answer: (4)

80. The distance of the point (7,−3,−4) from the plane passing through the points (2,−3,1),(−1,1,−2) and (3,−4,2) is :

(1) 5

(2) 4

(3) 5√2

(4) 4√2

Answer: (3)

SECTION-B

81. Let λ ∈ ℝ and let the equation E be |x|² − 2|x| + |λ − 3| = 0. Then the largest element in the set S= {x + λ : x is an integer solution of E} is

Answer: (5)

82. Let a tangent to the curve 9x² + 16y²= 144 intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is

Answer: (7)

83. The shortest distance between the lines and is equal to

Answer: (14)

84. Suppose Then the value of α is

Answer: (1012)

85. The value of is

Answer: (2)

86. The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is

Answer: (60)

87. A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

Answer: (546)

88. The 4^th term of GP is 500 and its common ratio is 1/m. m ∈ Let S_n denote the sum of the first n terms of this GP. If S₆ > S₅ + 1 and S₇ < S₆ + 1/2, then the number of possible values of m is

Answer: (12)

89. Let C be the largest circle centred at (2,0) and inscribed in the ellipse If (1, α) lies on C, then 10 α² is equal to

Answer: (118)

90. The value of is

Answer: (22)