**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.

(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 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_{A}I_{A} = N_{B}I_{B}

(2) N_{A} = 2N_{B}

(3) N_{A}I_{A} = 4N_{B}I_{B}

(4) 2N_{A}I_{A} = N_{B}I_{B}

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 (t_{3} – t_{2}) < t_{1}

(1) c and b

(2) b and c

(3) a and b

(4) c and a

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

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

5. A small object at rest, absorbs a light pulse of power 20 mW and duration 300 ns. Assuming speed of light as 3 × 10^{8} 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

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

7. The pressure (P) and temperature (T) relationship of an ideal gas obeys the equation PT^{2} = constant. The volume expansion coefficient of the gas will be :

(1) 3/T^{3}

(2) 3/T^{2}

(3) 3 T^{2}

(4) 3/T

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.

(C) Internal energy of the gas will not change.

(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

9. If the gravitational field in the space is given as (−K/r^{2}). 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

10. In a series LR circuit with X_{L} = R, power factor is P_{1}. If a capacitor of capacitance C with X_{C} = X_{L} is added to the circuit the power factor becomes P_{2}. The ratio of P_{1} to P_{2} will be :

(1) 1 : 3

(2) 1 : 2

(3) 1 : √2

(4) 1 : 1

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

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/s^{2}) :

(1) 360 m/s

(2) 400 m/s

(3) 60 m/s

(4) 120 m/s

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

14. The charge flowing in a conductor changes with time as Q(t) = αt – βt^{2} + γt^{3}. Where α, β and γ are constants. Minimum value of current is :

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

16. Speed of an electron in Bohr’s 7th orbit for Hydrogen atom is 3.6 × 10^{6} m/s. The corresponding speed of the electron in 3rd orbit, in m/s is:

(1) (1.8 × 10^{6})

(2) (3.6 × 10^{6})

(3) (7.5 × 10^{6})

(4) (8.4 × 10^{6})

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

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

(1) Nm^{3}C^{–1}; Nm^{2}C^{–1}

(2) Nm^{2}C^{–1}; Nm^{3}C^{–1}

(3) Nm^{3}C; Nm^{2}C

(4) Nm^{2}C; Nm^{3}C

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

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

**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

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

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

24.

As per the given figure, if then the value of V_{AB} at this instant will be V.

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

26. In the following circuit, the magnitude of current I_{1}, is ______ A.

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

28. In Young’s double slit experiment, two slits S_{1} and S_{2} 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 S_{1} and S_{2} respectively. The central bright fringe spot will shift by number of fringes.

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

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

**Chemistry**

**SECTION-A**

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

(1) LiH and Al(OH)_{3}

(2) LiH and Al_{2}Cl_{6 }

(3) LiCl and Al_{2}H_{6}

(4) LiCl,Al and H_{2}

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

(1) Terfenadine

(2) Meprobamate

(3) Brompheniramine

(4) Ranitidine

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

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

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}

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.

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

38. During the qualitative analysis of SO_{3}^{2}^{−} using dilute H_{2}SO_{4}, SO_{2} gas is evolved which turns K_{2}Cr_{2}O_{7} solution (acidified with dilute H_{2}SO_{4} ):

(1) green

(2) blue

(3) red

(4) black

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) Ni^{2+}

(2) Zn^{2+}

(3) Co^{2+}

(4) Fe^{3+}

40. For OF_{2} molecule consider the following :

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

(B) FOF angle is less than 104.5∘.

(C) Oxidation state of O is −2.

(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

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

(1) Nylon 6, 6

(2) Nylon 6

(3) Teflon

(4) Dacron

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

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, N_{2}O & NO

(2) O, NO & NO_{3}^{−}

(3) NO, O & O_{3}

(4) N, O_{2} & O_{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)

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

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

(A) EDTA

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

(C) Cis – Platin

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

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

(A) BeSO_{4} (2) MgSO_{4}

(C) CaSO_{4} (4) SrSO_{4}

(E) BaSO_{4}

Choose the correct answer from the options given below:

(1) B only

(2) A and B

(3) B and C

(4) A only

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

(1) CO < en < NH_{3} < C_{2}O_{4}^{2}^{−} < S^{2}^{−}

(2) NH_{3} < en < CO < S^{2}^{−} < C_{2}O_{4}^{2}^{−}

(3) S^{2}^{−} < C_{2}O_{4}^{2}^{−} < NH_{3} < en < CO

(4) S^{2}^{−} < NH_{3} < en < CO < C_{2}O_{4}^{2}

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

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

(1) remove FeO as FeSiO_{3}

(2) decrease the temperature needed for roasting of Cu_{2}S

(3) separate CuO as CuSiO_{3}

(4) remove calcium as CaSiO_{3}

**SECTION-B**

51. 600 mL of 0.01MHCl is mixed with 400 mL of 0.01MH_{2}SO_{4}. 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]

52. The energy of one mole of photons of radiation of frequency 2 × 10^{12} Hz in J mol^{–1} is _____. (Nearest integer)

[Given : h = 6.626 × 10^{−}^{34} Js

N_{A} = 6.022 × 10^{23} mol^{−}^{1}]

53. Consider the cell Pt_{(s)} |H_{2}(g, 1 atm)| H^{+}(aq, 1M)||Fe^{3+} (aq), Fe^{2+}(aq)| Pt(s)

When the potential of the cell is 0.712 V at 298 K, the ratio [Fe^{2+}] / [Fe^{3+}] is______ (Nearest integer)

Given : Fe^{3+} + e^{−} = Fe^{2+}, E^{θ}Fe^{3+}, Fe^{2+} | Pt = 0.771

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

55. A 300 mL bottle of soft drink has 0.2MCO_{2} dissolved in it. Assuming CO2 behaves as an ideal gas, the volume of the dissolved CO_{2} at STP is _____ mL. (Nearest integer)

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

56. A trisubstituted compound ‘A’, C_{10}H_{12}O_{2} gives neutral FeCl_{3} test positive. Treatment of compound ‘A’ with NaOH and CH_{3}Br gives C_{11}H_{14}O_{2}, with hydroiodic acid gives methyl iodide and with hot conc. NaOH gives a compound B, C_{10}H_{12}O_{2}. Compound ‘A’ also decolorises alkaline KMnO_{4}. The number of π bond/s present in the compound ‘A’ is

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]

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}

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)

60. Some amount of dichloromethane (CH_{2}Cl_{2}) is added to 671.141 mL of chloroform (CHCl_{3}) to prepare 2.6 × 10^{−3} M solution of CH_{2}Cl_{2 }(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}

**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 a^{2} – b^{2} is equal to:

(1) 392/3

(2) 196/3

(3) 98

(4) 196

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

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

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

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

66. If P(h, k) be a point on the parabola x = 4y^{2}, which is nearest to the point Q(0, 33), then the distance of P from the directrix of the parabola y^{2} = 4(x + y) is equal to :

(1) 2

(2) 6

(3) 8

(4) 4

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 }= r^{2}.

Then h + k is equal to

(1) 5(1 + √2)

(2) 5√2

(3) 6

(4) 5

68. The number of points on the curve y = 54x^{5} − 135x^{4} − 70x^{3 }+ 180x^{2} + 210x at which the normal lines are parallel to x + 90y + 2 = 0 is :

(1) 4

(2) 2

(3) 0

(4) 3

69. If then a_{1} + a_{2} + … + a_{25} is equal to

(1) 52/147

(2) 49/138

(3) 50/141

(4) 51/144

70. If then the value of is :

(1) 2

(2) 4 – 2√3

(3)

(4) 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

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

73. The line l_{1} passes through the point (2, 6, 2) and is perpendicular to the plane 2x + y − 2z = 10. Then the shortest distance between the line l_{1} and the line is :

(1) 13/3

(2) 19/3

(3) 7

(4) 9

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

(1) 1 + d^{2} = m^{2} + q^{2}

(2) 1 + d^{2} = (m + q)^{2}

(3) (1 + d)^{2} = m^{2} + q^{2}

(4) (1 + d)^{2} = (m + q)^{2}

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

(1) e^{8} – 1

(2) e^{7} – 1

(3) e^{8} – e

(4) e^{9} – e

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 ?

77. The coefficient of x^{301} in (1+x)^{500} + x(1+x)^{499} + x^{2}(1 + x)^{498} +⋯….. x^{500} is :

(1) ^{500}C_{300}

(2) ^{501}C_{200}

(3) ^{501}C_{302}

(4) ^{500}C_{301}

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

79. If the coefficient of x^{15} 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

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

**SECTION B**

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

82. If λ_{1} < λ_{2} are two values of λ such that the angle between the planes and then the square of the length of perpendicular from the point (38λ_{1}, 10λ_{2}, 2) to the plane P_{1} is

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

84. Let where a, b, c ∈ ℤ and Then a^{2} – b + c is equal to

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

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

87. Let

For n ≥ 2, define f^{n}(x) = f^{1} of f^{n}^{−}^{1} (x)

If then a + b is equal to

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

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

90. is equal to

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