JEE Main 2024 8 April Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

The value of k ∈ N for which the integral I_n = ∫₀¹ (1 - x^k)^n dx, n ∈ N, satisfies 147I₍₂₀₎ = 148I₍₂₁₎, is:

• (1) 10
• (2) 8
• (3) 14
• (4) 7

The sum of all solutions of the equation 8^(2x) - 16 · 8^x + 48 = 0 is:

• (1) 1 + log₆(8)
• (2) log₈(6)
• (3) 1 + log₈(6)
• (4) log₈(4)

Let the circles C₁: (x - a)² + (y - 3)² = r² and C₂: (x - 8)² + (y - 1)² = (r/2)² touch each other externally at the point (6,6). If the point (6,6) divides the line segment joining the centers of the circles C₁ and C₂ internally in the ratio 2:1, then:

(a + b) + 4.(r + r³) equals:

• (1) 110
• (2) 130
• (3) 125
• (4) 145

The distance of point P from the x-axis, given the conditions:

OP = γ; the angle between OQ and the positive x-axis is θ; and the angle between OP and the positive z-axis is ϕ.

• (1) γ√(1 - sin²(ϕ) cos²(θ))
• (2) γ√(1 + cos²(θ) sin²(ϕ))
• (3) γ√(1 - sin²(θ) cos²(ϕ))
• (4) γ√(1 + cos²(ϕ) sin²(θ))

The number of critical points of f(x) = (x - 2)^(2/3) (2x + 1) is:

• (1) 2
• (2) 0
• (3) 1
• (4) 3

The differential equation, whose general solution is y = c₁f(x) + c₂, where c₁ and c₂ are arbitrary constants, is:

• (1) (8e^x - 1) d²y/dx² + dy/dx = 0
• (2) (8e^x + 1) d²y/dx² - dy/dx = 0
• (3) (8e^x + 1) d²y/dx² + dy/dx = 0
• (4) (8e^x - 1) d²y/dx² - dy/dx = 0

The number of points of local maxima of f(x) = 4 cos³(x) + 3√3 cos²(x) - 10 in the interval (0, 2π) is:

• (1) 1
• (2) 2
• (3) 3
• (4) 4

If the variance of the frequency distribution is 160, then the value of c ∈ N is:

• (1) 5
• (2) 8
• (3) 7
• (4) 6

Let the range of the function f(x) = 1 / (2 + sin(3x) + cos(3x)), where x ∈ R and x ∈ [a, b]. If α and β are respectively the A.M. and G.M. of a and b, then α/β is equal to:

• (1) √2
• (2) 2
• (3) √π
• (4) π

Between the following two statements:

Statement-I: Let a = i + 2j − 3k and b = 2i + j − k. Then the vector r satisfying a × r = a × b and a · r = 0 is of magnitude √10.

Statement-II: In a triangle ABC, cos(2A) + cos(2B) + cos(2C) ≥ −3/2.

• (1) Both Statement-I and Statement-II are incorrect
• (2) Statement-I is incorrect but Statement-II is correct
• (3) Both Statement-I and Statement-II are correct
• (4) Statement-I is correct but Statement-II is incorrect

Evaluate the following limit:

lim x → π/2 ∫(π/2)^3 x^3 (sin(2t^(1/3)) + cos(t^(1/3))) dt / (x − π/2)^2

• (1) 9π²/8
• (2) 11π²/10
• (3) 3π²/2
• (4) 5π²/9

The sum of the coefficients of x^(2/3) and x^(-2/5) in the binomial expansion of (x^(2/3) + 1/(2x^(2/5)))^9 is:

• (1) 21/4
• (2) 69/16
• (3) 63/16
• (4) 19/4

Let B = [1 3; 1 5] and A be a 2 × 2 matrix such that AB⁻¹ = A⁻¹. If BCB⁻¹ = A and C⁴ + αC² + βI = O, then 2β − α is equal to:

• (1) 16
• (2) 2
• (3) 8
• (4) 10

If log(e) y = 3 sin^(-1) x, then (1 − x)^2 y′′ − x y′ at x = 1/2 is equal to:

• (1) 9e^(π/6)
• (2) 3e^(π/6)
• (3) 3e^(π/2)
• (4) 9e^(π/2)

The integral ∫(3/4 to 1/4) cos(2 cot^(-1)(√(1 − x) / (1 + x))) dx is equal to:

• (1) −1/2
• (2) 1/4
• (3) 1/2
• (4) −1/4

Let a, ar, ar², ... be an infinite G.P. If ∑(n=0 to ∞) ar^n = 57 and ∑(n=0 to ∞) a³r^(3n) = 9747, then a + 18r is equal to:

• (1) 27
• (2) 46
• (3) 38
• (4) 31

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the i-th roll than the number obtained in the (i−1)-th roll, i = 2, 3, is equal to:

• (1) 3/54
• (2) 2/54
• (3) 5/54
• (4) 1/54

The value of the integral ∫(2 to −1) log(e)(x + √(x² + 1)) dx is:

• (1) √5 − √2 + log(e)(9 + 4√5)/(1 + √2)
• (2) √2 − √5 + log(e)(9 + 4√5)/(1 + √2)
• (3) √5 − √2 + log(e)(7 + 4√5)/(1 + √2)
• (4) √2 − √5 + log(e)(7 + 4√5)/(1 + √2)

Let α, β; α > β, be the roots of the equation x² − √2x − √3 = 0. Let Pn = αⁿ − βⁿ, n ∈ N. Then (11√3 − 10√2)P₁₀ + (11√2 + 10)P₁₁ − 11P₁₂ is equal to:

• (1) 10√2P₉
• (2) 10√3P₉
• (3) 11√2P₉
• (4) 11√3P₉

Let a = 2î + αĵ + k̂, b = −î + k̂, c = βĵ − k̂, where α and β are integers and αβ = −6. Let the values of the ordered pair (α, β) for which the area of the parallelogram of diagonals a + b and b + c is √21/2, be (α₁, β₁) and (α₂, β₂). Then α₁ + β₁ − α₂β₂ is equal to:

• (1) 17
• (2) 24
• (3) 21
• (4) 19

Consider the circle C: x² + y² = 4 and the parabola P: y² = 8x. If the set of all values of α, for which three chords of the circle C on three distinct lines passing through the point (α, 0) are bisected by the parabola P, is the interval (p, q), then (2q − p)² is equal to:

• (1) 80

Let the set of all values of p, for which f(x) = (p² − 6p + 8)(sin²(2x) − cos²(2x)) + 2(2 − p)x + 7 does not have any critical point, be the interval (a, b). Then 16ab is equal to:

• (1) 252

For a differentiable function f : R → R, suppose f ′(x) = 3f(x) + α, where α ∈ R, f(0) = 1 and lim x→∞ f(x) = 7. Then, 9f(− log(3)) is equal to:

• (1) 61

The number of integers between 100 and 1000 having the sum of their digits equal to 14 is:

• (1) 70

Let A = {(x, y) : 2x + 3y = 23, x, y ∈ N} and B = {x : (x, y) ∈ A}. Then the number of one-one functions from A to B is equal to:

• (1) 24

Let A, B, and C be three points on the parabola y² = 6x, and let the line segment AB meet the line L through C, parallel to the x-axis, at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L. Then (AM · BN / CD)² is equal to:

• (1) 36

The square of the distance of the image of the point (6, 1, 5) in the line x−1/3 = y/2 = z−2/4, from the origin is:

• (1) 62
• (2) 58
• (3) 65
• (4) 60

If (1/alpha + 1 + 1/(alpha+2) + ... + 1/(alpha+1012)) − (1/2·1 + 1/(4·3) + 1/(6·5) + ... + 1/(2024·2023)) = 1/2024, then alpha is equal to:

• (1) 1011
• (2) 1009
• (3) 1010
• (4) 1012

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation 2sin⁻¹(x) + 3cos⁻¹(x) = 2π/5 is:

• (1) 0
• (2) 1
• (3) 2
• (4) Infinite

Consider the matrices A = [2, −5; 3, m], B = [20, m], and X = [x, y]. Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b). Then 8∫b a |A| dm is equal to:

• (1) 450
• (2) 400
• (3) 500
• (4) 350

A nucleus at rest disintegrates into two smaller nuclei with their masses in the ratio 2:1. After disintegration, they will move:

• (1) In opposite directions with speed in the ratio of 1:2 respectively
• (2) In opposite directions with speed in the ratio of 2:1 respectively
• (3) In the same direction with the same speed
• (4) In opposite directions with the same speed

The following figure represents two biconvex lenses L1 and L2 having focal lengths 10 cm and 15 cm, respectively. The distance between L1 and L2 is:

• (1) 10 cm
• (2) 15 cm
• (3) 25 cm
• (4) 35 cm

The temperature of a gas is −78°C, and the average translational kinetic energy of its molecules is K. The temperature at which the average translational kinetic energy of the molecules of the same gas becomes 2K is:

• (1) −39°C
• (2) 117°C
• (3) 127°C
• (4) −78°C

A hydrogen atom in ground state is given an energy of 10.2 eV. How many spectral lines will be emitted due to the transition of electrons?

• (1) 6
• (2) 3
• (3) 10
• (4) 1

The magnetic field in a plane electromagnetic wave is:

By = (3.5 × 10^-7) sin(1.5 × 10³ x + 0.5 × 10¹¹ t) T. The corresponding electric field will be:

• (1) Ey = 1.17 sin(1.5 × 10³ x + 0.5 × 10¹¹ t) V/m
• (2) Ez = 105 sin(1.5 × 10³ x + 0.5 × 10¹¹ t) V/m
• (3) Ez = 1.17 sin(1.5 × 10³ x + 0.5 × 10¹¹ t) V/m
• (4) Ey = 10.5 sin(1.5 × 10³ x + 0.5 × 10¹¹ t) V/m

A square loop of side 15 cm is being moved towards right at a constant speed of 2 cm/s. The front edge enters the 50 cm wide magnetic field at t = 0. The value of induced emf in the loop at t = 10 s will be:

• (1) 0.3 mV
• (2) 4.5 mV
• (3) zero
• (4) 3 mV

Two cars are travelling towards each other at a speed of 20 m/s each. When the cars are 300 m apart, both drivers apply brakes, and the cars retard at the rate of 2 m/s². The distance between them when they come to rest is:

• (1) 200 m
• (2) 50 m
• (3) 100 m
• (4) 25 m

The I-V characteristics of an electronic device shown in the figure. The device is:

• (1) Solar cell
• (2) Transistor which can be used as an amplifier
• (3) Zener diode which can be used as a voltage regulator
• (4) Diode which can be used as a rectifier

The excess pressure inside a soap bubble is three times the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is:

• (1) 1:9
• (2) 1:3
• (3) 1:81
• (4) 1:27

The de-Broglie wavelength associated with a particle of mass m and energy E is:

λ = h / √(2mE). The dimensional formula for Planck’s constant is:

• (1) [ML⁻¹T⁻²]
• (2) [ML²T⁻¹]
• (3) [MLT⁻²]
• (4) [ML²T⁻²]

A satellite of 103 kg mass is revolving in a circular orbit of radius 2R. If 10⁴R⁶ joules of energy is supplied to the satellite, it would revolve in a new circular orbit of radius:

• (1) 2.5R
• (2) 3R
• (3) 4R
• (4) 6R

The effective resistance between A and B, if the resistance of each resistor is R, will be:

• (1) 2/3R
• (2) 8/3R
• (3) 5/3R
• (4) 4/3R

Five charges +q, +5q, −2q, +3q, −4q are situated as shown in the figure. The electric flux due to this configuration through the surface S is:

• (1) 5q/ϵ₀
• (2) 4q/ϵ₀
• (3) 3q/ϵ₀
• (4) q/ϵ₀

A proton and a deuteron (q = +e, m = 2.0u) having the same kinetic energies enter a region of uniform magnetic field B, moving perpendicular to B. The ratio of the radius r_d of the deuteron path to the radius r_p of the proton path is:

• (1) 1:1
• (2) 1:√2
• (3) √2:1
• (4) 1:2

UV light of 4.13 eV is incident on a photosensitive metal surface having a work function of 3.13 eV. The maximum kinetic energy of the ejected photoelectrons will be:

• (1) 4.13 eV
• (2) 1 eV
• (3) 3.13 eV
• (4) 7.26 eV

The energy released in the fusion of 2 kg of hydrogen deep in the sun is E_H and the energy released in the fission of 2 kg of ²³⁵U is E_U. The ratio E_H/E_U is approximately:

• (1) 9.13
• (2) 15.04
• (3) 7.62
• (4) 25.6

A real gas within a closed chamber at 27°C undergoes the cyclic process as shown in the figure. The gas obeys the PV³ = RT equation for the path A to B. The net work done in the complete cycle is (assuming R = 8 J/mol·K):

• (1) 225 J
• (2) 205 J
• (3) 20 J
• (4) −20 J

A 1 kg mass is suspended from the ceiling by a rope of length 4 m. A horizontal force F is applied at the midpoint of the rope so that the rope makes an angle of 45° with respect to the vertical axis as shown in the figure. The magnitude of F is:

• (1) 10√2 N
• (2) 1 N
• (3) 1/10√2 N
• (4) 10 N

A spherical balloon of radius 1 m is inflated with air at constant temperature. The work done to increase the volume of the balloon by 1 m³ is:

• (1) 1 J
• (2) 2 J
• (3) 4 J
• (4) 3 J

In the truth table of the above circuit, the value of X and Y are:

• (1) 1, 1
• (2) 1, 0
• (3) 0, 1
• (4) 0, 0

A straight magnetic strip has a magnetic moment of 44 Am². If the strip is bent in a semicircular shape, its magnetic moment will be:

• (1) 22 Am²
• (2) 33 Am²
• (3) 28 Am²
• (4) 44 Am²

A particle of mass 0.5 kg executes simple harmonic motion under a force F = -50x (Nm⁻¹). The time period of oscillation is x/35 seconds. Find the value of x.

• (1) 18
• (2) 22
• (3) 24
• (4) 28

A capacitor of reactance 4√3 Ω and a resistor of resistance 4 Ω are connected in series with an AC source of peak value 8√2 V. The power dissipation in the circuit is:

• (1) 2 W
• (2) 3 W
• (3) 5 W
• (4) 4 W

An electric field E = 2x î N/C exists in space. A cube of side 2 m is placed in the space. The electric flux through the cube is:

• (1) 8 Nm²/C
• (2) 12 Nm²/C
• (3) 16 Nm²/C
• (4) 20 Nm²/C

A circular disc reaches from top to bottom of an inclined plane of length l. When it slips down, it takes t seconds. When it rolls down, it takes (α/2)^1/2 × t seconds. Find α.

• (1) 2
• (2) 3
• (3) 4
• (4) 5

To determine the resistance (R) of a wire, a circuit is designed. The value of R is:

• (1) 1500 Ω
• (2) 2000 Ω
• (3) 2500 Ω
• (4) 3000 Ω

The resultant of two vectors A and B is perpendicular to A and its magnitude is half that of B. The angle between A and B is:

• (1) 90°
• (2) 120°
• (3) 150°
• (4) 180°

Monochromatic light of wavelength 500 nm is used in Young's double-slit experiment. When one slit is covered with a glass plate (refractive index μ = 1.5), the central maximum shifts by 4 fringes. Find the thickness of the glass plate.

• (1) 2 µm
• (2) 3 µm
• (3) 4 µm
• (4) 5 µm

A force (3x² + 2x - 5) N displaces a body from x = 2 m to x = 4 m. The work done by this force is:

• (1) 50 J
• (2) 58 J
• (3) 62 J
• (4) 70 J

At room temperature (27°C), the resistance of a heating element is 50 Ω. If the temperature coefficient of the material is 2.4 × 10⁻⁴ °C⁻¹, find the temperature of the element when its resistance is 62 Ω:

• (1) 927°C
• (2) 1027°C
• (3) 1127°C
• (4) 1227°C

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency A×10¹² hertz and that has a radiant intensity in that direction of 1/B watt per steradian. ’A’ and ’B’ are respectively:

• (1) 540 and 1
• (2) 540 and 683
• (3) 450 and 1
• (4) 450 and 683

The correct stability order of the following resonance structures of CH₃CH=CHCHO is:

• (1) II > III > I
• (2) III > II > I
• (3) I > II > III
• (4) II > I > III

The total number of stereoisomers possible for the given structure is:

• (1) 8
• (2) 2
• (3) 4
• (4) 3

The correct increasing order for bond angles among BF3, PF3, and CF3 is:

• (1) PF3 < BF3 < CF3
• (2) BF3 < PF3 < CF3
• (3) CF3 < PF3 < BF3
• (4) BF3 = PF3 < CF3

Match List-I (Test) with List-II (Observation):

List-I (Test) | List-II (Observation)

A. Br2 water test | I. Yellow-orange or orange-red precipitate formed

B. Ceric ammonium nitrate test | II. Reddish orange color disappears

C. Ferric chloride test | III. Red color appears

D. 2,4-DNP test | IV. Blue, green, violet, or red color appears

• (1) A-III, B-II, C-I, D-IV
• (2) A-II, B-III, C-IV, D-I
• (3) A-III, B-IV, C-I, D-II
• (4) A-IV, B-I, C-II, D-III

Match List-I (Cell) with List-II (Use/Property/Reaction):

List-I (Cell) | List-II (Use/Property/Reaction)

A. Leclanche cell | I. Converts energy of combustion into electrical energy

B. Ni-Cd cell | II. Does not involve any ion in solution and is used in hearing aids

C. Fuel cell | III. Rechargeable

D. Mercury cell | IV. Reaction at anode: Zn → Zn2+ + 2e-

• (1) A-I, B-II, C-III, D-IV
• (2) A-III, B-I, C-IV, D-II
• (3) A-IV, B-III, C-I, D-II
• (4) A-II, B-III, C-IV, D-I

Match List-I (Complex) with List-II (Hybridization):

List-I (Complex) | List-II (Hybridization)

A. K2[Ni(CN)4] | I. sp3

B. [Ni(CO)4] | II. sp3d2

C. [Co(NH3)6]Cl3 | III. dsp2

D. Na3[CoF6] | IV. d2sp3

• (1) A-III, B-I, C-II, D-IV
• (2) A-III, B-II, C-IV, D-I
• (3) A-I, B-III, C-II, D-IV
• (4) A-III, B-I, C-IV, D-II

The coordination environment of Ca2+ ion in its complex with EDTA4- is:

• (1) Tetrahedral
• (2) Octahedral
• (3) Square planar
• (4) Trigonal prismatic

The incorrect statement about glucose is:

• (1) Glucose is soluble in water because of having an aldehyde functional group
• (2) Glucose remains in multiple isomeric forms in its aqueous solution
• (3) Glucose is an aldohexose
• (4) Glucose is one of the monomer units in sucrose

The number of oxygen atoms present in the chemical formula of fuming sulfuric acid is:

• (1) 6
• (2) 7
• (3) 8
• (4) 9

Which of the following compounds can give a positive iodoform test when treated with aqueous KOH solution followed by potassium hypoiodite?

• (1) Option 1
• (2) Option 2
• (3) Option 3
• (4) Option 4

For a sparingly soluble salt AB2, the equilibrium concentrations of A2+ ions and B- ions are 1.2 × 10^-4 M and 0.24 × 10^-3 M, respectively. The solubility product of AB2 is:

• (1) 0.069 × 10^-12
• (2) 6.91 × 10^-12
• (3) 0.276 × 10^-12
• (4) 27.65 × 10^-12

Major product of the following reaction is:

• (1) (A)
• (2) (B)
• (3) (C)
• (4) (D)

Given below are two statements:

Statement I: The higher oxidation states are more stable down the group among transition elements unlike p-block elements.

Statement II: Copper cannot liberate hydrogen from weak acids.

Choose the correct answer from the options given below:

• (1) Both Statement I and Statement II are false
• (2) Statement I is false but Statement II is true
• (3) Both Statement I and Statement II are true
• (4) Statement I is true but Statement II is false

The incorrect statement regarding ethyne is:

• (1) The C-C bond in ethyne is shorter than that in ethene
• (2) Both carbons are sp hybridized
• (3) Ethyne is linear
• (4) The C-C bond in ethyne is weaker than that in ethene

Match List-I with List-II:

List-I (Element) | List-II (Electronic Configuration)

A. N | I. [Ar] 3d10 4s2 4p5

B. S | II. [Ne] 3s2 3p4

C. Br | III. [He] 2s2 2p3

D. Kr | IV. [Ar] 3d10 4s2 4p6

• (1) A-IV, B-III, C-II, D-I
• (2) A-III, B-II, C-I, D-IV
• (3) A-I, B-IV, C-III, D-II
• (4) A-II, B-I, C-IV, D-III

Match List-I with List-II:

List-I | List-II

A. Melting point (K) | I. Tl > In > Ga > Al > B

B. Ionic Radius (M+3/pm) | II. B > Tl > Al ≈ Ga > In

C. ∆iH₁ (kJ mol⁻¹) | III. Tl > In > Al > Ga > B

D. Atomic Radius (pm) | IV. B > Al > Tl > In > Ga

• (1) A-III, B-IV, C-I, D-II
• (2) A-II, B-III, C-IV, D-I
• (3) A-IV, B-I, C-II, D-III
• (4) A-I, B-II, C-III, D-IV

Which of the following compounds will give a silver mirror with ammoniacal silver nitrate?

• (1) Formic Acid
• (2) Formaldehyde
• (3) Benzaldehyde
• (4) Acetone

Which of the following is a correct equation to show change in molar conductivity with respect to concentration for a weak electrolyte, if the symbols carry their usual meaning?

• (1) Λm²C − KaΛ◦m² + KaΛmΛ◦m = 0
• (2) Λm − Λ◦m + AC¹/² = 0
• (3) Λm − Λ◦m − AC¹/² = 0
• (4) Λm²C + KaΛ◦m − KaΛmΛ◦m = 0

The electronic configuration of Einsteinium is: (Given atomic number of Einsteinium = 99)

• (1) [Rn] 5f¹² 6d⁰ 7s²
• (2) [Rn] 5f¹¹ 6d⁰ 7s²
• (3) [Rn] 5f¹³ 6d⁰ 7s²
• (4) [Rn] 5f¹⁰ 6d⁰ 7s²

The number of oxygen atoms present in the chemical formula of fuming sulfuric acid is:

A transition metal 'M' among Sc, Ti, V, Cr, Mn, and Fe has the highest second ionisation enthalpy. The spin-only magnetic moment value of M⁺ ion is (Nearest integer):

The vapor pressure of pure benzene and methyl benzene at 27°C is given as 80 Torr and 24 Torr, respectively. The mole fraction of methyl benzene in the vapor phase, in equilibrium with an equimolar mixture of those two liquids (ideal solution) at the same temperature is ........ ×10⁻² (nearest integer).

Consider the following test for a group-IV cation:

M²⁺ + H₂S → A (Black precipitate) + byproduct

A + aqua regia → B + NOCl + S + H₂O

B + KNO₂ + CH₃COOH → C + byproduct

The spin-only magnetic moment value of the metal complex C is ........ BM (Nearest integer).

Consider the following first-order gas-phase reaction at constant temperature:

A(g) → 2B(g) + C(g)

If the total pressure of the gases is found to be 200 Torr after 23 sec, and 300 Torr upon the complete decomposition of A after a very long time, then the rate constant of the given reaction is ............. × 10⁻² s⁻¹ (nearest integer).

In the given TLC, the distance of spot A and B are 5 cm and 7 cm, from the bottom of the TLC plate, respectively. The Rf value of B is x × 10⁻¹ times more than A. The value of x is:

Based on Heisenberg’s uncertainty principle, the uncertainty in the velocity of the electron to be found within an atomic nucleus of diameter 10⁻¹⁵ m is ........ × 10⁹ ms⁻¹ (nearest integer).

Number of compounds from the following which cannot undergo Friedel-Crafts reactions is:

Toluene, nitrobenzene, xylene, cumene, aniline, chlorobenzene, m-nitroaniline, m-dinitrobenzene.

Total number of electrons present in (π*) molecular orbitals of O₂, O₂⁺, and O₂⁻ is:

When ∆Hvap = 30 kJ/mol and ∆Svap = 75 J mol⁻¹K⁻¹, then the temperature of vapor, at one atmosphere, is: