**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.

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

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

(1) 1 : 2

(2) 1 : 3

(3) 1 : 4

(4) 1 : 5

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

Truth table of the shown circuit is:

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

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

5. The equivalent resistance between A and B is

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

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)

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:

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

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

11. An electron accelerated through a potential difference V_{1} has a de-Broglie wavelength of 𝜆. When the potential is changed to V_{2}, its de-Broglie wavelength increases by 50%. The value of (V_{1}/V_{2}) is equal to

(1) 3

(2) 3/2

(3) 4

(4) 9/4

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

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

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

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

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 = R_{0}A^{1/3}, where R_{0} 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

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

18. A thin prism, P_{1} with an angle 6th and made of glass of refractive index 1.54 is combined with another prism P_{2} made from glass of refractive index 1.72 to produce dispersion without average deviation. The angle of prism P_{2} is

(1) 1.3°

(2) 6°

(3) 4.5°

(4) 7.8°

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

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 ω_{1}. When the mass block is 2 kg the angular frequency is ω_{2}. The ratio ω_{2}/ ω_{1} is

(1) 1/√2

(2) √2

(3) 2

(4) 1/2

**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/s^{2}).

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

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)

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 ________.

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°

26. In an ac generator, a rectangular coil of 100 turns each having area 14 × 10^{−2} m^{2} 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)

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 _______.

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 m^{3}).

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 I_{1} and I_{2} If I_{0} denotes the intensity produced by each one of the individual slits, then

30. The velocity of a particle executing SHM varies with displacement (x) as 4v^{2 }= 50 − 𝑥^{2}. The time period of oscillations is The value of x is _______. (Take π = 22/7)

**Chemistry**

**SECTION-A**

31. The Cl−Co−Cl bond angle values in a fac- [Co(NH_{3})_{3}Cl_{3}] complex is/are:

(1) 90°

(2) 90° & 120°

(3) 180°

(4) 90° & 180°

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

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 Al_{2}O_{3} is mixed with Na_{3}AlF_{6} 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

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

35. 1 L, 0.02M solution of [Co(NH_{3})_{5}SO_{4}]Br is mixed with 1 L, 0.02M solution of [Co(NH_{3})_{5}Br]SO_{4}. The resulting solution is divided into two equal parts (X) and treated with excess of AgNO_{3} solution and BaCl_{2} solution respectively as shown below:

1 L solution (X) + AgNO_{3} solution (excess) → Y

1 L Solution (X) + BaCl_{2} 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

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

37. Which of the following reaction is correct?

38. Boric acid is solid, whereas BF_{3} is gas at room temperature because of

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

(2) Strong covalent bond in BF_{3}

(3) Strong ionic bond in Boric acid

(4) Strong hydrogen bond in Boric acid

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

40. Formulae for Nessler’s reagent is:

(1) HgI_{2}

(2) K_{2}HgI4

(3) KHgI_{3}

(4) KHg_{2}I_{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 −CH_{2}− 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

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

(1) 16

(2) 32

(3) 72

(4) 50

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

At r = r_{0}, radial node is formed. Thus, r_{0} in terms of a_{0}

(1) r_{0} = 4a_{0}

(2) r_{0} = a_{0}/2

(3) r_{0} = a_{0}

(4) r_{0} = 2a_{0}

44.

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

(1) (i) Br_{2}(aq) (ii) LiAIH_{4} (iii) H_{3}O^{+}

(2) (i) Br_{2}, Fe (ii) Fe, H^{+} (iii) LiAIH_{4}

(3) (i) Fe, H^{+} (ii) Br_{2} (aq) (iii) HNO_{2} (iv) H_{3}PO_{2}

(4) (i) Fe, H^{+} (ii) Br_{2} (aq) (iii) HNO_{2} (iv) CuBr

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

46. The most stable carbocation for the following is:

(1) a

(2) b

(3) c

(4) c

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

(1) K

(2) Be

(3) Mg

(4) Ca

48. KMnO_{4} oxidises I^{–} in acidic and neutral/faintly alkaline solution, respectively, to

(1) IO_{3}^{−} & IO_{3}^{−}

(2) I_{2} & IO_{3}^{−}

(3) I_{2} & I_{2}

(4) IO_{3}^{−} & I_{2}

49. Bond dissociation energy of “E-H” bond of the “H_{2}E ” hydrides of group 16 elements (given below), follows order.

(A) O

(B) S

(C) Se

(D) Te

Choose the correct from the options given below:

(1) B > A > C > D

(2) A > B > D > C

(3) A > B > C > D

(4) D > C > B > A

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

**SECTION B**

51. Number of compounds from the following which will not dissolve in cold NaHCO_{3} and NaOH solutions but will dissolve in hot NaOH solution is

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}

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

54. Lead storage battery contains 38% by weight solution of H_{2}SO_{4}. 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}

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

Given:

Molar mass of H_{2}O_{2} is 34 g mol^{−1} Molar volume of gas at STP = 22.7 L.

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

X|X^{2+}(0.001M||Y^{2+}(0.01M)|Y is____________ × 10^{−2} V (Nearest integer).

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

58. Consider the following equation:

2SO_{2}(g) + O_{2}(g) ⇌ 2SO_{3}(g), Δ𝐻=−190 kJ

The number of factors which will increase the yield of SO_{3} at equilibrium from the following is

(A) Increasing temperature

(B) Increasing pressure

(C) Adding more SO_{2}

(D) Adding more O_{2}

(E) Addition of catalyst

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 × 10^{23} mol^{−1}

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)

**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

62. Let a, b, c > 1, a^{3}, b^{3} and c^{3} 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

63. Let a_{1} = 1, a_{2}, a_{3}, a_{4}, ….. be consecutive natural numbers. Then is equal to

64. Let λ ∈ ℝ,

If then is equal to

(1) 132

(2) 136

(3) 140

(4) 144

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

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}_{→}_{1}g(h(x – 1)) is

(1) −1

(2) 0

(3) sin(1)

(4) 1

67. Let S be the set of all values of a_{1} for which the mean deviation about the mean of 100 consecutive positive integers a_{1}, a_{2}, a_{3}, …. a_{100} is 25 . Then S is

(1) N

(2) ϕ

(3) {99}

(4) {9}

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) x^{2} + 14x + 24 = 0

(2) x^{2} + 18x + 56 = 0

(3) x^{2} – 18x + 56 = 0

(4) x^{2} – 10x + 16 = 0

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

(1) −24

(2) −84

(3) −48

(4) −60

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

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

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

73. is equal to

(1) 12

(2) 19/3

(3) 0

(4) 19

74. Let A be a point on the x-axis. Common tangents are drawn from A to the curves x^{2} + y^{2} = 8 and y^{2} = 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

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

76. The range of the function is:

(1) [2√2, √11]

(2) [√5, √13]

(3) [√2, √7]

(4) [√5, √10]

77. The solution of the differential equation is

78. The parabolas : ax^{2} + 2bx + cy = 0 and dx^{2} + 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.

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)

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

**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 3α^{2} is equal to ______.

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 ______.

83. Let P(a_{1}, b_{1}) and Q(a_{2}, b_{2}) 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 a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2} is equal to _______.

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

85. The 8th common term of the series

S_{1} = 3 + 7 + 11 + 15 + 19 + …

S_{2} = 1 + 6 + 11 + 16 + 21 + …

is _______.

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 ______.

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

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

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 ______.

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 ______.