**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.

(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. 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 :

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

(C) 75 W ultra – violet 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

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

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

(1) q/12ε_{0}

(2) q/6ε_{0}

(3) q/3ε_{0}

(4) q/2ε_{0}

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

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:

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

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

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

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

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

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 :

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 × 10^{6} m

(1) 28 km

(2) 36 km

(3) 32 km

(4) 64 km

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}

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

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

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.

(C) The speed of light is independent of the motion of the source.

(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

18. In a Young’s double slit experiment, two slits are illuminated with a light of wavelength 800 nm. The line joining A_{1}P is perpendicular to A1A_{2} 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

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.

(C) It emits light only when it is reverse 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

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

(1) μ_{0}I/πa

(2) μ_{0}I/2πa

(3) 0

(4) μ_{0}I/4πa

**SECTION-B**

21. As shown in the figure, three identical polaroids P_{1}, P_{2} and P_{3} are placed one after another. The pass axis of P_{2} and P_{3} are inclined at angle of 60∘ and 90∘ with respect to axis of P_{1}. The source S has an intensity of The intensity of light at point O is –W/m^{2}.

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/s^{2})

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.

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} )

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})

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}

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.

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 ____

29. A point charge q_{1} = 4q_{0} is placed at origin. Another point charge q_{2} = −q_{0} is placed at = 12 cm. Charge of proton is q_{0} .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.

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

**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) N_{2}O_{3}, NO_{2}

(2) NO_{2}, N_{2}O_{4}

(3) NO_{2}, N_{2}O_{5}

(4) NO, NO_{2}

32. Correct statement about smog is:

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

(2) Both NO_{2} and SO_{2} are present in classical smog

(3) NO_{2} is present in classical smog

(4) Photochemical smog has high concentration of oxidizing agents

33. The standard electrode potential (M^{3+}/M^{2+}) 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 H_{2} from a dilute acid are

(1) Mn^{2+} and Co^{2+}

(2) Cr^{2+} and Co^{2+}

(3) V^{2+} and Cr^{2+}

(4) V^{2+} and Mn^{2+}

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

35. The bond dissociation energy is highest for

(1) F_{2}

(2) Br_{2}

(3) I_{2}

(4) Cl_{2}

36. The increasing order of pK_{a} for the following phenols is

(A) 2, 4-Dinitrophenol

(B) 4-Nitrophenol

(C) 2, 4,5 – Trimethylphenol

(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)

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?

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

39. During the borax bead test with CuSO_{4}, a blue green colour of the bead was observed in oxidising flame due to the formation of

(1) CuO

(2) Cu(BO_{2})_{2}

(3) Cu_{3}B_{2}

(4) Cu

40. Which of the following salt solution would coagulate the colloid solution formed when FeCl_{3} is added to NaOH solution, at the fastest rate?

(1) 10 mL of 0.1 mol dm^{–3} Na_{2}SO_{4}

(2) 10 mL of 0.2 mol dm^{–3} AlCl_{3}

(3) 10 mL of 0.1 mol dm^{–3} Ca_{3}(PO_{4})_{2}

(4) 10 mL of 0.15 mol dm–3 CaCl2

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

(1) 5

(2) 2

(3) 4

(4) 3

42. The correct order of hydration enthalpies is

(A) K^{+} (B) Rb^{+} (C) Mg^{2+}

(D) Cs^{+} (E) Ca^{2+}

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

43. Chiral complex from the following is:

Here en = ethylene diamine

(1) cis −[PtCl_{2}(en)_{2}]^{2+}

(2) trans−[PtCl_{2}(en)_{2}]^{2+}

(3) cis−[PtCl_{2}(NH_{3})_{2}]

(4) trans−[Co(NH_{3})_{4}Cl_{2}]^{+}

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

45. The magnetic behavior of Li_{2}O, Na_{2}O_{2} and KO_{2}, respectively, are

(1) Paramagnetic, paramagnetic and diamagnetic

(2) diamagnetic, paramagnetic and diamagnetic

(3) paramagnetic, diamagnetic and paramagnetic

(4) diamagnetic, diamagnetic and paramagnetic

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

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

(1) Di-isobutylaluminium hydride

(2) NaNi_{…….}

(3) Li/P_{4}

(4) SiH_{4}

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

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

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

(1) NH_{2}OH.HCl

(2) CH_{3}NH_{2}.HCl

(3) NH_{4}Cl

(4) N_{2}H_{4}.HCl

**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.

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.

53. The sum of bridging carbonyls in W(CO)_{6} and Mn_{2}(CO)_{10} is_______

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 PbCl_{2} 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.)

55. 17 mg of a hydrocarbon (M.F. C_{10}H_{16} ) takes up 8.40 mL of the H_{2} 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_______

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

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

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

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

(A) NO_{2}

(B) ICl_{4}^{−}

(C) BrF_{3}

(D) ClO_{2}

(E) NO_{2}^{+}

(F) NO

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}

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

(C) In region I and III, the reaction is of first and zero order respectively

(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.

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

(C) At infinite dilution, the value of degree of dissociation approaches zero for electrolyte B.

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

**Mathematics**

**SECTION-A**

61. Let α and β be real numbers. Consider a 3 × 3 matrix A such that A^{2 }= 3A + αI. If A^{4} = 21A + βI, then

(1) β = −8

(2) β = 8

(3) α = 4

(4) α = 1

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

where [·] denotes greatest integer function, is

(1) 0

(2) −1

(3) 2

(4) 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

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

65. Let y = f(x) be the solution of the differential equation y(x + 1)dx − x^{2}dy = 0, y(1) = e. Then is equal to

(1) 1/e^{2}

(2) e^{2}

(3) 0

(4) 1/e

66. The domain of is

(1) ℝ − {3}

(2) (−1, ∞) −{3}

(3) (2, ∞) −{3}

(4) ℝ −{−1,3}

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

68. Let [x] denote the greatest integer ≤ Consider the function f(x) = max{x^{2}, 1 + [x]}. Then the value of the integral is

69. For two non-zero complex numbers z_{1} and z_{2}, if Re(z_{1}z_{2})=0 and Re(z_{1} + z_{2}) = 0, then which of the following are possible?

(A) Im (z_{1}) > 0 and Im(z_{2}) > 0

(B) Im (z_{1}) < 0 and Im (z_{2}) > 0

(C) Im(z_{1}) > 0 and Im (z_{2}) < 0

(D) Im(z_{1}) < 0 and Im (z_{2}) < 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

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

71. Let − 2((1 – sin^{2} 2θ) and If then f(β) is equal to

(1) 5/4

(2) 3/2

(3) 9/8

(4) 11/8

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

73. Let Δ be the area of the region {(x, y) ∈ ℝ^{2} : x^{2} + y^{2} ≤ 21, y^{2} ≤ 4x, x ≥ 1}. Then is equal to

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

75. Let A = {(x, y) ∈ ℝ^{2} : 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

76. Let λ ≠ 0 be a real number. Let α, β be the roots of the equation 14x^{2} – 31x + 3λ = 0 and α, γ be the roots of the equation 35x^{2} – 53x + 4λ = 0. Then 3α/β and 4α/γ are the roots of the equation

(1) 49x^{2} – 245x + 250 = 0

(2) 7x^{2} + 245x – 250 = 0

(3) 7x^{2} – 245x + 250 = 0

(4) 49x^{2} + 245x + 250 = 0

77. Let the tangents at the points A(4,−11) and B(8,−5) on the circle x^{2} + y^{2} – 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

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)

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, ∞)

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 σ^{2} represent mean and variance of 𝑋, respectively, then 10(μ^{2} + σ^{2}) is equal to

(1) 250

(2) 25

(3) 30

(4) 20

**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 α^{2} is equal to

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

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

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

85. Let a_{1}, a_{2}, a_{3}, … 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 a_{1}a_{9} + a_{2}a_{4}a_{9} + a_{5} + a_{7} is equal to

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 a^{2} + b^{2} + c^{2} is equal to

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

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

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

90. If all the six digit numbers x_{1}x_{2}x_{3}x_{4}x_{5}x_{6} with 0 < x_{1} < x_{2} < x_{3} < x_{4} < x_{5} < x_{6} are arranged in the increasing order, then the sum of the digits in the 72th number is

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