JEE Main 2024 5 April Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

SECTION-A

PHYSICS

Question 1:

Let d be the distance of the point of intersection of the lines

(x + 6)/3 = (y)/2 = (z + 1)/1 and (x - 7)/4 = (y - 9)/3 = (z - 4)/2 from the point (7, 8, 9). Then d² + 6 is equal to:

• (1) 72
• (2) 69
• (3) 75
• (4) 78

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a + b)² is equal to:

• (1) 72
• (2) 60
• (3) 80
• (4) 64

Let two straight lines drawn from the origin O intersect the line 3x + 4y = 12 at the points P and Q such that ΔOPQ is an isosceles triangle and ∠POQ = 90°. If ℓ = OP² + PQ² + OQ², then the greatest integer less than or equal to ℓ is:

• (1) 44
• (2) 48
• (3) 46
• (4) 42

If y = y(x) is the solution of the differential equation dy/dx + 2y = sin(2x), y(0) = 3/4, then y(π/8) is equal to:

• (1) e^(-π/8)
• (2) e^(-π/4)
• (3) e^(π/4)
• (4) e^(π/8)

For the function f(x) = sin x + 3x - (2/π)(x² + x), where x ∈ [0, π/2], consider the following two statements:

(I) f is increasing in (0, π/2).

(II) f is decreasing in (0, π/2).

Between the above two statements:

• (1) only (I) is true.
• (2) only (II) is true.
• (3) neither (I) nor (II) is true.
• (4) both (I) and (II) are true.

If the system of equations

11x + y + λz = -5

2x + 3y + 5z = 3

8x - 19y - 39z = µ

has infinitely many solutions, then λ⁴ - µ is equal to:

• (1) 49
• (2) 45
• (3) 47
• (4) 51

Let A = {1, 3, 7, 9, 11} and B = {2, 4, 5, 7, 8, 10, 12}. Then the total number of one-one maps f : A → B, such that f(1) + f(3) = 14, is:

• (1) 180
• (2) 120
• (3) 480
• (4) 240

If the function f(x) = (sin 3x + α sin x - β cos 3x) / x³, x ∈ ℝ, is continuous at x = 0, then f(0) is equal to:

• (1) 2
• (2) -2
• (3) 4
• (4) -4

The integral ∫₀ to π/4 of (136 sin x / (3 sin x + 5 cos x)) dx is equal to:

• (1) 3π - 50 ln 2 + 20 ln 5
• (2) 3π - 25 ln 2 + 10 ln 5
• (3) 3π - 10 ln(2√2) + 10 ln 5
• (4) 3π - 30 ln 2 + 20 ln 5

The coefficients a, b, and c in the quadratic equation ax² + bx + c = 0 are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is:

• (1) 3/256
• (2) 1/128
• (3) 1/64
• (4) 3/128

Two integers x and y are chosen with replacement from the set {0,1,2,...,10}. Then the probability that |x - y| > 5 is:

(1) 30/121
(2) 62/121
(3) 60/121
(4) 31/121

If the domain of the function f(x) = cos⁻¹(2 - |x|) / 4 is [−α, β) − {γ}, then α + β + γ is equal to:

(1) 12
(2) 9
(3) 11
(4) 8

Consider the system of linear equations x + y + z = 4μ, x + 2y + 2z = 10μ, x + 3y + 4λz = μ² + 15. Which one of the following statements is NOT correct?

(1) The system has a unique solution if λ = 1/2 and μ = 1
(2) The system is inconsistent if λ = 1/2 and μ = 1
(3) The system has an infinite number of solutions if λ = 1/2 and μ = 15
(4) The system is consistent if λ = 1/2

If the circles (x + 1)² + (y +2)² = r² and x² + y² - 4x - 4y + 4 = 0 intersect at exactly two distinct points, then:

(1) 5 < r < 9
(2) 0 < r < 7
(3) 3 < r < 7
(4) 1/2 < r < 7

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

(1) √5 / 3
(2) √3 / 2
(3) 1 / √3
(4) 2 / √5

Let M denote the median of the following frequency distribution.
Class    Frequency
0-4            3
4-8            9
8-12          10
12-16         8
16-20         6
Then 20M is equal to:

(1) 416
(2) 104
(3) 52
(4) 208

If f(x) = 2cos⁴x / (3 + 2cos⁴x) + 2sin⁴x / (3 + 2sin⁴x), then 1/5 f'(0) is equal to:

(1) 0
(2) 1
(3) 2
(4) 6

Let A(2,3,5) and C(−3,4,−2) be opposite vertices of a parallelogram ABCD. If the diagonal BD = i + 2j + 3k, then the area of the parallelogram is equal to:

(1) 1/2 √410
(2) 1/2 √474
(3) 1/2 √586
(4) 1/2 √306

If 2sin³x + sin²x cosx + 4sinⁿx − 4 = 0 has exactly 3 solutions in the interval (0, nπ/2), n ∈ N, then the roots of the equation x² + nx + (n−3) = 0 belong to:

(1) (0, ∞)
(2) (−∞, 0)
(3) −√17/2, √17/2
(4) Z

Let f : −π/2, π/2 → R be a differentiable function such that f(0) = 1/2. If the limit lim(x→0) ∫(0 to x) f(t)dt / (e^(x²) - 1) = α, then 8α² is equal to:

(1) 16
(2) 2
(3) 1
(4) 4

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics, Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 in Physics, and 16 in Chemistry. At most 11 students passed in both Mathematics and Physics, 15 in both Physics and Chemistry, and 10 in both Mathematics and Chemistry. The maximum number of students passed in all three subjects is:

If d1 is the shortest distance between the lines x+1/2 = y-1/-12 = z/1 and x-1/-7 = y+8/2 = z-4/5, and d2 is the shortest distance between the lines x-1/2 = y-2/1 = z-6/-3 and x/1 = y+2/1 = z-1/6, then the value of 32√3d1/d2 is:

Let the latus rectum of the hyperbola x²/9 - y²/b² = 1 subtend an angle of π/3 at the center of the hyperbola. If b² is equal to 1/m(1+√n), where l and m are co-prime numbers, then l²+m²+n² is equal to:

Let A = {1,2,3,...,7} and let P(1) denote the power set of A. If the number of functions f : A → P(A) such that a ∈ f(a), ∀a ∈ A is mn, and m and n are least, then m + n is equal to:

The value of ∫(0 to 9) 10x / (x+1) dx is:

Number of integral terms in the expansion of (√7z + 1/(6√z))^824 is equal to:

Let y = y(x) be the solution of the differential equation (1 − x²)dy = xy + x³ + 2√1−x² dx, with y(0) = 0. If y(1/2) = m/n, where m and n are co-prime numbers, then m + n is equal to:

Let α, β ∈ N be roots of the equation x² − 70x + λ = 0, where λ/2, λ/3 ∉ N. If λ assumes the minimum possible value, then √α−1+√β−1(λ+35)/|α−β| is equal to:

If the function f(x) = { 1/|x| , |x| ≥ 2
                                            ax² + 2b , |x| < 2 } is differentiable on R, then 48(a + b) is equal to:

Let α = 1² + 4² + 8² + 13² + 19² + 26² + ... up to 10 terms and β = Σ₁₀n=1 n⁴. If 4α - β = 55k + 40, then k is equal to:

Match List-I with List-II.
List-I | List-II
A. Coefficient of viscosity | I. [ML⁻¹T⁻¹]
B. Surface Tension | II. [ML⁰T⁻²]
C. Angular momentum | III. [ML²T⁻¹]
D. Rotational kinetic energy | IV. [ML²T⁻²]

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

All surfaces shown in the figure are frictionless, and the pulleys and the string are light. The acceleration of the block of mass 2 kg is:

(1) g
(2) g/3
(3) 2g/3
(4) g/4

A potential divider circuit is shown in the figure. The output voltage ( V₀ ) is:

(1) 4V
(2) 2 mV
(3) 0.5 V
(4) 12 mV

Young’s modulus of a material of a wire of length ( L ) and cross-sectional area ( A ) is ( Y ). If the length of the wire is doubled and cross-sectional area is halved, then Young’s modulus will be:
(1) Y/4
(2) Y
(3) 4Y
(4) 2Y

The work function of a substance is 3.0 eV. The longest wavelength of light that can cause the emission of photoelectrons from this substance is approximately:
(1) 215 nm
(2) 414 nm
(3) 400 nm
(4) 200 nm

The ratio of the magnitude of the kinetic energy (KE) to the potential energy (PE) of an electron in the 5th excited state of a hydrogen atom is:

A particle is placed at point A on a frictionless track ABC as shown. It is gently pushed to the right. The speed of the particle when it reaches point B is:
(Take g = 10 m/s²)

(1) 20 m/s
(2) √10 m/s
(3) 2√10 m/s
(4) 10 m/s

The electric field of an electromagnetic wave in free space is represented as E = E₀ cos(ωt - kx) ĩ. The corresponding magnetic induction vector will be:

(1) B = E₀ C cos(ωt − kx) ĵ
(2) B = (E₀/C) cos(ωt − kx) ĵ
(3) B = E₀ C cos(ωt + kx) ĵ
(4) B = (E₀/C) cos(ωt + kx) ĵ

Two insulated circular loops A and B of radius a, carrying a current I in anticlockwise direction, are arranged perpendicular to each other. The magnitude of the magnetic induction at the center will be:

(1) √2 μ₀I/a
(2) μ₀I/(2a)
(3) μ₀I√2/a
(4) 2μ₀I/a

The diffraction pattern of light of wavelength 400 nm diffracting from a slit of width 0.2 mm is focused on the focal plane of a convex lens of focal length 100 cm. The width of the 1st secondary maxima will be:

Primary coil of a transformer is connected to 220 V ac. Primary and secondary turns of the transformer are 100 and 10 respectively. The secondary coil of the transformer is connected to two series resistances shown in the figure. The output voltage V₀ is:

The gravitational potential at a point above the surface of Earth is −5.12 × 10⁷ J/kg and the acceleration due to gravity at that point is 6.4m/s². Assume that the mean radius of Earth to be 6400 km. The height of this point above the Earth’s surface is:

An electric toaster has resistance of 60 Ω at room temperature (27°C). The toaster is connected to a 220 V supply. If the current flowing through it reaches 2.75 A, the temperature attained by toaster is around: (if α = 2× 10⁻⁴ °C⁻¹)

A Zener diode of breakdown voltage 10V is used as a voltage regulator as shown in the figure. The current through the Zener diode is:

Two thermodynamical processes are shown in the figure. The molar heat capacity for process A and B are CA and CB. The molar heat capacity at constant pressure and constant volume are represented by CP and CV respectively. Choose the correct statement:

The electrostatic potential due to an electric dipole at a distance r varies as:

A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m. The impulse of force imparted by the ground to the body is given by: (given g = 9.8m/s²)

A particle of mass m is projected with a velocity u making an angle of 30° with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection when the particle is at its maximum height is:

At which temperature does the r.m.s. velocity of a hydrogen molecule equal that of an oxygen molecule at 47°C?

A series L-R circuit connected to an AC source E = 25 sin(1000 t) V has a power factor of 1/√2. If the source of emf is changed to E = 20 sin(2000 t) V, the new power factor of the circuit will be:

The horizontal component of Earth’s magnetic field at a place is 3.5 × 10⁻⁵ T. A very long straight conductor carrying a current of √2 A is placed from South East to North West. The force per unit length experienced by the conductor is:

Two cells are connected in opposition. Cell E₁ has 8 V emf and 2 Ω internal resistance. Cell E₂ has 2 V emf and 4 Ω internal resistance. The terminal potential difference of cell E₂ is:

An electron in a hydrogen atom has energy E_n = −0.85 eV in an excited state. The maximum number of allowed transitions to lower energy levels is:

Each of three blocks P, Q, and R (each 3 kg) is attached to a wire. Wires A and B each have a cross-sectional area of 0.005 cm² and Young’s modulus of 2 × 10¹¹ N/m². Neglecting friction, the longitudinal strain on wire B is ×10⁻⁴:

The distance between the object and its image (which is twice the size of the object) formed by a convex lens is 45 cm. The focal length of the lens is:

The displacement and the increase in the velocity of a moving particle in the time interval from t to (t+1) seconds are 125 m and 50 m/s, respectively. The distance travelled by the particle in the (t+2)th second is:

A capacitor of capacitance C and potential V has energy E. It is connected to another capacitor of capacitance 2C and potential 2V. Then the loss of energy is x/3E, where x is:

Consider a disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is gently placed over the rotating disc along the same axis. The energy dissipated so that both discs continue to rotate together without slipping is:

In a closed organ pipe, the frequency of the fundamental note is 30 Hz. A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to 110 Hz. If the organ pipe has a cross-sectional area of 2 cm², the amount of water poured in the organ pipe is x grams. (Take speed of sound in air as 330 m/s)

A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm. The magnetic field of Earth in that region is 0.5 G and the angle of dip is 30°. The emf induced across the blades is Nπ × 10⁻⁵ V. The value of N is:

Given below are two statements:
Statement-I: The gas liberated on warming a salt with dilute H₂SO₄, turns a piece of paper dipped in lead acetate into black; it is a confirmatory test for sulphide ion.
Statement-II: In statement-I the colour of paper turns black because of formation of lead sulphide.

Choose the most appropriate answer from the options below:

This reduction reaction is known as:

Sugar which does not give reddish brown precipitate with Fehling’s reagent is:

Given below are the two statements: one is labeled as Assertion (A) and the other is labeled as Reason (R).
Assertion (A): There is a considerable increase in covalent radius from N to P. However, from As to Bi only a small increase in covalent radius is observed.
Reason (R): Covalent and ionic radii in a particular oxidation state increase down the group.

Choose the most appropriate answer from the options below:

Which of the following molecule/species is most stable?

Diamagnetic Lanthanoid ions are:

Aluminium chloride in acidified aqueous solution forms an ion having geometry:

Given below are two statements:
Statement-I: The orbitals having the same energy are called as degenerate orbitals.
Statement-II: In hydrogen atom, 3p and 3d orbitals are not degenerate orbitals.

Choose the most appropriate answer from the options below:

Example of vinylic halide is:

Structure of 4-Methylpent-2-enal is:

Match List-I with List-II:
List-I (Molecule) | List-II (Shape)
(A) BrF₅ | (I) T-shape
(B) H₂O | (II) See-saw
(C) ClF₃ | (III) Bent
(D) SF₄ | (IV) Square pyramidal

Choose the correct answer from the options below:

The final product A, formed in the following multistep reaction sequence is:
CH₃ - C≡CH + Na → A
B → CH₃ - C≡C - CH₂ - CH₂ - CH₃ + NaBr

Choose the correct answer from the options below:

In the given reactions, identify the reagent A and reagent B:
CH₃ - C≡CH + Na → A
B → CH₃ - C≡C - CH₂ - CH₂ - CH₃ + NaBr

(1) A-CrO₃, B-CrO₃
(2) A-CrO₃, B-CrO₂Cl₂
(3) A-CrO₂Cl₂, B-CrO₂Cl₂
(4) A-CrO₂Cl₂, B-CrO₃

Given below are two statements: one is labeled as Assertion (A) and the other is labeled as Reason (R).
Assertion (A): CH₂=CH−CH₂−Cl is an example of allyl halide.
Reason (R): Allyl halides are the compounds in which the halogen atom is attached to an sp² hybridised carbon atom.

Choose the most appropriate answer from the options below:

What happens to the freezing point of benzene when a small quantity of naphthalene is added to benzene?

Match List-I with List-II:
List-I (Species) | List-II (Electronic Distribution)
(A) Cr²⁺ | (I) 3d⁸
(B) Mn⁺ | (II) 3d⁵4s¹
(C) Ni²⁺ | (III) 3d⁴
(D) V⁺ | (IV) 3d³4s¹

Choose the correct answer from the options below:

Compound A formed in the following reaction reacts with B, giving the product C.
CH₃−C≡CH + Na → A
B → CH₃−C≡C−CH₂−CH₂−CH₃ + NaBr
Find out A and B.

Following is a confirmatory test for aromatic primary amines. Identify reagent (A) and (B).
A: NaNO₂ + HCl, 0–5°C
B: Phenol
Find out A and B.

The Lassaigne’s extract is boiled with dilute HNO₃ before testing for halogens because:

Choose the correct statements from the following:
(A) Ethane-1,2-diamine is a chelating ligand.
(B) Metallic aluminium is produced by electrolysis of aluminium oxide in presence of cryolite.
(C) Cyanide ion is used as ligand for leaching of silver.
(D) Phosphine acts as a ligand in Wilkinson catalyst.
(E) The stability constants of Ca²⁺ and Mg²⁺ are similar with EDTA complexes.

The rate of a first-order reaction is 0.04 mol·L⁻¹·s⁻¹ at 10 minutes and 0.03 mol·L⁻¹·s⁻¹ at 20 minutes after initiation. The half-life of the reaction is:

The pH at which Mg(OH)₂ [Ksp = 1 × 10⁻¹¹] begins to precipitate from a solution containing 0.10 M Mg²⁺ ions is:
 

An ideal gas undergoes a cyclic transformation starting from the point A and coming back to the same point by tracing the path A → B → C → A as shown in the diagram. The total work done in the process is:
 

If the IUPAC name of an element is "Unununnium", then the element belongs to nth group of the periodic table. The value of n is:
 

The total number of molecular orbitals formed from 2s and 2p atomic orbitals of a diatomic molecule is:
 

On a thin layer chromatographic plate, an organic compound moved by 3.5 cm, while the solvent moved by 5 cm. The retardation factor of the organic compound is ×10⁻¹:
 

The compound formed by the reaction of ethanol with semicarbazide contains number of nitrogen atoms.
 

A 0.05 cm thick coating of silver is deposited on a plate of 0.05 m² area. The number of silver atoms deposited on the plate is ×10²³ (At. mass Ag = 108, d = 7.9 g/cm³):
 

2MnO₄⁻ + 6I⁻ + 4H₂O → 3I₂ + 2MnO₂ + 8OH⁻. If the above equation is balanced with integer coefficients, the value of z is:
 

The mass of sodium acetate (CH₃COONa) required to prepare 250 mL of 0.35 M aqueous solution is (in grams). (Molar mass of CH₃COONa is 82.02 g/mol)