**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.

(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. 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

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

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

4. The correct relation between γ = c_{p}/c_{v} and temperature T is :

(1) γαT^{0}

(2) γαT

(3)

(4)

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

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

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 I_{0} then intensity of transmitted light after passing through polaroid B will be:

(1) I_{0}/4

(2) I_{0}/2

(3) Zero

(4) I_{0}/8

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

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

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

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

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

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

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_{L}X_{C}

(2)

(3)

(4) RX_{L}X_{C}

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

16. If a source of electromagnetic radiation having power 15 kW produces 10^{16} photons per second, the radiation belongs to a part of spectrum is.

(Take Planck constant h=6 × 10^{−34}Js )

(1) Micro waves

(2) Ultraviolet rays

(3) Gamma rays

(4) Radio waves

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?

18. A bar magnet with a magnetic moment 5.0 Am^{2} 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

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

20. A rod with circular cross-section area 2 cm^{2} 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 μ_{0} = 4π × 10^{−7}NA^{−2})

(1) 5/16

(2) 12.5

(3) 125

(4) 32/5

**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 × 10^{8} ms^{−1} and for the given medium μ_{r} = 1)

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}

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.

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 ________.

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-

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}.

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

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.

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.

30. A thin rod having a length of 1 m and area of cross-section 3 × 10^{−6} m^{2} 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 × 10^{11} Nm^{−2} and 2 × 10^{−5} K^{−1}, respectively. The value of M is _____ kg. (Take g = 10 ms^{−2})

**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)

32.

Consider the above reaction and identify the product B. Options

33. An organic compound ‘A’ with emperical formula C_{6}H_{6}O gives sooty flame on burning. Its reaction with bromine solution in low polarity solvent results in high yield of B.B is

34. When Cu^{2+} 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=CuI_{2} Y=Na_{2} S_{4}O_{6}

(2) X=CuI_{2} Y=Na_{2} S_{2}O_{3}

(3) X=Cu_{2}I_{2} Y=Na_{2} S_{4}O_{5}

(4) X=Cu_{2}I_{2} Y=Na_{2} S_{4}O_{6}

35. Choose the correct set of reagents for the following conversion. trans(Ph – CH = CH − CH_{3}) → cis(Ph – CH = CH − CH_{3})

(1) Br_{2}, aq ⋅ KOH, NaNH_{2}, Na(LiqNH_{3})

(2) Br_{2}, alc ⋅ KOH, NaNH_{2}, H_{2} Lindlar Catalyst

(3) Br_{2}, aq⋅KOH, NaNH_{2}, H_{2} Lindlar Catalyst

(4) Br_{2}, alc ⋅ KOH, NaNH_{2}, Na(LiqNH_{3})

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 NaHCO_{3} solution

(3) optically active and adds one mole of bromine

(4) racemic mixture and is neutral

37. The methods NOT involved in concentration of ore are

(A) Liquation

(B) Leaching

(C) Electrolysis

(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

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

39. Nd^{2+} =

(1) 4f^{3}

(2) 4f^{4}6 s^{2}

(3) 4f^{4}

(4) 4f^{2}6 s^{2}

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

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

(1) X = ClONO_{2}, Y = HOCl, Z = HNO_{3}

(2) X = ClONO_{2}, Y = HOCl, Z = NO_{2}

(3) X = ClNO_{2}, Y = HCl, Z = HNO_{3}

(4) X = ClNO_{3}, Y = Cl_{2}, Z = NO_{2}

42. The correct increasing order of the ionic radii is

(1) S^{2−} < Cl^{−} < Ca^{2+} < K^{+}

(2) K^{+} < S^{2−} < Ca^{2+} < Cl^{−}

(3) Ca^{2+} < K^{+} < Cl^{−} < S^{2−}

(4) Cl^{−} < Ca^{2+} < K^{+} < S^{2−}

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(H_{2}O)_{6}]^{2+}, Y = [CoCl_{4}]^{2−}, Z = Tetrahedral

(2) X = [Co(H_{2}O)_{6}]^{2+}, Y = [CoCl_{6}]^{3−}, Z = Octahedral

(3) X = [Co(H_{2}O)_{4}Cl_{2}]^{+}, Y = [CoCl_{4}]^{2−}, Z = Tetrahedral

(4) X = [Co(H_{2}O)_{6}]^{3+}, Y = [CoCl_{6}]^{3−}, Z= Octahedral

44. H_{2}O_{2} acts as a reducing agent in

(1) 2NaOCl + H_{2}O_{2} → 2NaCl + H_{2}O + O_{2}

(2) Na_{2}S + 4H_{2}O_{2} → Na_{2}SO_{4} + 4H_{2}O

(3) 2Fe^{2+} + 2H^{+} + H_{2}O_{2} → 2Fe^{3+} + 2H_{2}O

(4) Mn^{2+} + 2H_{2}O_{2} → MnO_{2} + 2H_{2}O

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

(1) a

(2) d

(3) b

(4) c

46. The correct order of melting points of dichlorobenzenes is

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

(1) V_{2}O_{5} > V_{2}O_{4} > V_{2}O_{3}

(2) V_{2}O_{4} > V_{2}O_{3} > V_{2}O_{5}

(3) V_{2}O_{3} > V_{2}O_{5} > V_{2}O_{4}

(4) V_{2}O_{3} > V_{2}O_{4} > V_{2}O_{5}

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

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

(1) Cl_{2} is formed at cathode

(2) O_{2} is formed at cathode

(3) H_{2} is formed at anode

(4) OH^{−} is formed at cathode

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

**SECTION B**

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

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

Given :

53. The logarithm of equilibrium constant for the reaction Pd^{2+} + 4Cl^{−} ⇌ PdCl_{4}^{2}^{−} is (Nearest integer)

54. On complete combustion, 0.492 g of an organic compound gave 0.792 g of CO_{2}. The % of carbon in the organic compound is _____ (Nearest integer)

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 H_{2} at STP = 22.7 L )

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 : ln10 = 2.3 R = 8.3 J K^{−1} mol^{−1}

log 5 = 0.70

log 3 = 0.48

log 2 = 0.30)

57. For reaction :

K_{p} = 2 × 10^{12} at 27°C and 1 atm pressure. The K_{c} for the same reaction is × 10^{13}. (Nearest integer) (Given R=0.082 L atm K^{−1} mol^{−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} )

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

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})

**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

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

(1) 34

(2) 1

(3) 17

(4) 19

63. For all z ∈ C on the curve C_{1} : |z| = 4, let the locus of the point be the curve C_{2}. Then :

(1) the curve C_{1} lies inside C_{2}

(2) the curve C_{2} lies inside C_{1}

(3) the curves C_{1} and C_{2} intersect 4 points

(4) the curves C_{1} and C_{2} intersect at 2 points

64. Then, at x = 1,

(1) √2yʹ − 3π^{2}y = 0

(2) yʹ + 3π^{2}y = 0

(3) 2yʹ + 3π^{2}y = 0

(4) 2yʹ + √3π^{2}y = 0

65. A wire of length 20 m is to be cut into two pieces. A piece of length 𝑙_{1} is bent to make a square of area 𝐴1 and the other piece of length 𝑙_{2} is made into a circle of area A_{2}. If 2A_{1 }+ 3A_{2} is minimum then (π𝑙_{1}) : 𝑙_{2} is equal to :

(1) 1 : 6

(2) 6 : 1

(3) 3 : 1

(4) 4 : 1

66. Let a circle C_{1} be obtained on rolling the circle x^{2} + y^{2} – 4x – 6y + 11 = 0 upwards 4 units on the tangent 𝑇 to it at the point (3, 2). Let C_{2} be the image of C_{1} in T. Let A and B be the centers of circles C_{1} and C_{2} 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)

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

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

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

70. The value of is equal to

71. Let the shortest distance between the lines and L_{1} : 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

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

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

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

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

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

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

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

(1) 3

(2) 1

(3) 2

(4) 0

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

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

(1) β + P_{50}(α)

(2) P_{50}(α) – β

(3) β – P_{50}(α)

(4) −(β + P_{50}(α))

**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

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

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

84. If the variance of the frequency distribution

is 3, then α is equal to

85. Let θ be the angle between the planes and Let L be the line that meets P_{2} at the point (4, −2, 5) and makes an angle θ with the normal of P_{2}. If α is the angle between L and P_{2}, then (tan^{2} θ) (cot^{2} α) is equal to

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

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

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 α^{2} is equal to

89. Let a_{1}, a_{2}, …, a_{n} be in A.P. If a_{5} = 2a_{7} and a_{11}= 18, then is equal to

90. The remainder on dividing 5^{99} by 11 is :

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