JEE Main 2024 4 April Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

SECTION-A

PHYSICS
 

Question 1:

Let f : R → R be a function given by:

1 − cos(2x)/x^2, for x < 0

beta * sqrt(1 − cos(x))/x, for x > 0.

If f is continuous at x = 0, then alpha^2 + beta^2 is equal to:

• (1) 48
• (2) 12
• (3) 4
• (4) 6

Three urns A, B, and C contain 7 red, 5 black; 5 red, 7 black; and 6 red, 6 black balls, respectively. One of the urns is selected at random, and a ball is drawn. If the ball drawn is black, then the probability that it is drawn from urn A is:

• (1) 4/17
• (2) 5/18
• (3) 5/16
• (4) 6/17

The vertices of a triangle are A(-1, 3), B(-2, 2), and C(3, -1). A new triangle is formed by shifting the sides of the triangle one unit inwards. Then the equation of the side of the new triangle nearest to the origin is:

• (1) x − y − (2 + sqrt(2)) = 0
• (2) −x + y − (2 − sqrt(2)) = 0
• (3) x + y − (2 − sqrt(2)) = 0
• (4) x + y + (2 − sqrt(2)) = 0

If the solution y = y(x) of the differential equation (x^4 + 2x^3 + 3x^2 + 2x + 2) dy = (2x^2 + 2x + 3) dx satisfies y(-1) = -pi/4, then y(0) is equal to:

• (1) -pi/12
• (2) 0
• (3) pi/4
• (4) pi/2

Let the sum of the maximum and minimum values of the function f(x) = (2x^2 − 3x + 8)/(2x^2 + 3x + 8) be m/n, where gcd(m, n) = 1. Then m + n is equal to:

• (1) 182
• (2) 217
• (3) 195
• (4) 201

One of the points of intersection of the curves y = 1 + 3x − 2x^2 and y = 1/x is (1/2, 2). Let the area of the region enclosed by these curves be 1/24 (l * sqrt(5) + m) − n ln(1 + sqrt(5)), where l, m, n are natural numbers. Then l + m + n is equal to:

• (1) 32
• (2) 30
• (3) 29
• (4) 31

If the system of equations x + (sqrt(2) * sin(alpha)) * y + (sqrt(2) * cos(alpha)) * z = 0,

x + (cos(alpha)) * y + (sin(alpha)) * z = 0,

x + (sin(alpha)) * y − (cos(alpha)) * z = 0 has a non-trivial solution, then alpha in the interval (0, pi/2) is equal to:

• (1) 3pi/4
• (2) 7pi/24
• (3) 5pi/24
• (4) 11pi/24

There are 5 points P1, P2, P3, P4, P5 on the side AB, excluding A and B, of a triangle ABC. Similarly, there are 6 points P6, P7, ..., P11 on the side BC and 7 points P12, P13, ..., P18 on the side CA of the triangle. The number of triangles that can be formed using the points P1, P2, ..., P18 as vertices is:

• (1) 776
• (2) 751
• (3) 796
• (4) 771

Let f(x) be defined as:

f(x) = x - 2, for -2 < x < 0

f(x) = 1 - 2x, for 0 <= x <= 2

Let h(x) = f(x) + f(x). Then the integral of h(x) dx is equal to:

• (1) 2
• (2) 4
• (3) 1
• (4) 6

The sum of all rational terms in the expansion of (1/2^5 + 1/5^3)^15 is equal to:

• (1) 3133
• (2) 633
• (3) 931
• (4) 6131

The values of α for which the determinant |1 3 2; α+3 2 1; 1 1 3| = 0 lie in the interval:
(1) (-2,1)
(2) (-3,0)
(3) (-3/2,3/2)
(4) (0,3)

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and the second draw gives all black balls is:
(1) 5/256
(2) 5/715
(3) 3/715
(4) 3/256

The integral ∫(x⁸ - x²) dx / ((x¹² + 3x⁶ + 1) arctan(x³ + 1/x³)) is equal to:
(1) log(arctan(x³ + 1/x³)^(1/3) + C)
(2) log(arctan(x³ + 1/x³)^(1/2) + C)
(3) log(arctan(x³ + 1/x³) + C)
(4) log(arctan(x³ + 1/x³)³ + C)

If 2tan²(θ) - 5sec(θ) = 1 has exactly 7 solutions in the interval [0, nπ/2] for the least value of n ∈ N, then Σ(k=1 to n) k / 2k is equal to:
(1) 1/215(214 - 14)
(2) 1/214(215 - 15)
(3) 1 - 15/213
(4) 1/213(214 - 15)

The position vectors of the vertices A, B, and C of a triangle are A = 2i - 3j + 3k, B = 2i + 2j + 3k, and C = -i + j + 3k respectively. If ℓ is the length of the angle bisector AD of ∠BAC, then 2ℓ² equals:
(1) 49
(2) 42
(3) 50
(4) 45

If y = y(x) is the solution curve of the differential equation (x² - 4) dy - (y² - 3y) dx = 0, x > 2, y(4) = 3/2, and the slope of the curve is never zero, then y(10) equals:
(1) 3/(1 + 8^(1/4))
(2) 3/(1 + 2√2)
(3) 3/(1 - 2√2)
(4) 3/(1 - 8^(1/4))
 

If e₁ is the eccentricity of the hyperbola x²/16 - y²/9 = 1 and e₂ is the eccentricity of the ellipse x²/a² + y²/b² = 1, which passes through the foci of the hyperbola and satisfies e₁e₂ = 1, then the length of the chord of the ellipse parallel to the x-axis and passing through (0,2) is:
(1) 4√5
(2) 8√5/3
(3) 10√5/3
(4) 3√5

Let α = (4!)! / (4!)³! and β = (5!)! / (5!)⁴!. Then:
(1) α ∈ N and β ∉ N
(2) α ∉ N and β ∈ N
(3) α ∈ N and β ∈ N
(4) α ∉ N and β ∉ N

Let the position vectors of vertices A, B, and C of a triangle be 2i + 2j + k, i + 2j + 2k, and 2i + j + 2k respectively. Let l₁, l₂, and l₃ be lengths of perpendiculars from the orthocenter to sides AB, BC, and CA. Then l₁² + l₂² + l₃² equals:
(1) 1/5
(2) 1/2
(3) 1/4
(4) 1/3

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking, it was found that an observation was read as 10 instead of 12. If µ and σ² denote the mean and variance of the correct observations, then 15(µ + µ² + σ²) is equal to:
 

The values of α for which lines 2x - y + 3 = 0, 6x + 3y + 1 = 0, and ax + 2y - 2 = 0 do not form a triangle are:
(1) (-2,1)
(2) (-3,0)
(3) (-3/2, 3/2)
(4) (0,3)

If the area of the region {(x, y) : 0 ≤ y ≤ min(2x, 6x - x²)} is A, then 12A is equal to:
(1) 304
(2) 594
(3) 128
(4) 360

Let Λ be a 2×2 real matrix and I be the identity matrix of order 2. If the roots of the equation |Λ - xI| = 0 are -1 and 3, then the sum of the diagonal elements of the matrix Λ² is:
(1) 10
(2) 6
(3) 14
(4) 7

If the sum of squares of all real values of α, for which the lines 2x - y + 3 = 0, 6x + 3y + 1 = 0, and ax + 2y - 2 = 0 do not form a triangle is p, then the greatest integer less than or equal to p is:
(1) 32
(2) 64
(3) 29
(4) 41

The coefficient of x^2012 in the expansion of (1 - x)^2008(1 + x + x²)^2007 is:
(1) 0
(2) 1
(3) -1
(4) 2

If the solution curve of the differential equation dy/dx = (x + y - 2)/(x - y) passes through the point (2, 1), the value of y(10) equals:
(1) 3/(1 + (8)^(1/4))
(2) 3/(1 + (8)^(1/3))
(3) 1/(3 + (8)^(1/4))
(4) 2/(3 + (8)^(1/4))

Let f(x) = ∫₀ˣ g(t) log((1 - t)/(1 + t)) dt, where g is a continuous odd function. If ∫(-π/2 to π/2) (f(x) x² cos(x)/(1 + e^x)) dx = π/2 - α, then α is equal to:
(1) 2
(2) 4
(3) 3
(4) 5

Consider a circle (x - α)² + (y - β)² = 50, where α, β > 0. If the circle touches the line y + x = 0 at point P, whose distance from the origin is 4√2, then (α + β)² is equal to:
(1) 100
(2) 150
(3) 50
(4) 200

The lines (x - 2)/1 = (y - 1)/(-1) = (z - 7)/8 and (x + 3)/4 = (y + 2)/3 = (z + 2)/1 intersect at the point P. If the distance of P from the line (x + 1)/2 = (y - 1)/3 = (z - 1)/1 is l, then 14l² is equal to:
(1) 108
(2) 120
(3) 100
(4) 150

Let the complex numbers α and 1/α lie on the circles |z - z₀| = 2 and |z - z₀| = 4 respectively, where z₀ = 1 + i. Then, the value of 100|α|² is:
(1) 20
(2) 10
(3) 30
(4) 15

The equation of state of a real gas is given by P + (a/V²)(V - b) = RT, where P, V, and T are pressure, volume, and temperature. The dimensions of a/b² are similar to:
(1) P
(2) PV
(3) RT
(4) R

The total kinetic energy of 1 mole of oxygen at 27°C is:
(1) 6845.5 J
(2) 5942.0 J
(3) 6232.5 J
(4) 5670.5 J

The primary side of a transformer is connected to a 230 V, 50 Hz supply. Turns ratio of primary to secondary winding is 10:1. Load resistance on the secondary side is 46 Ω. The power consumed in it is:
(1) 12.5 W
(2) 10.0 W
(3) 11.5 W
(4) 12.0 W

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of Cp/Cv for the gas is:
(1) 5/3
(2) 3/2
(3) 7/5
(4) 9/7

The threshold frequency of a metal with work function 6.63 eV is:
(1) 16 ×10¹⁵ Hz
(2) 16 ×10¹² Hz
(3) 1.6 × 10¹² Hz
(4) 1.6 × 10¹⁵ Hz

A current of 200 µA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through π/10 radians is:
(1) 30 µA
(2) 120 µA
(3) 60 µA
(4) 180 µA

The atomic mass of 6C¹² is 12.000000 u and that of 6C¹³ is 13.003354 u. The required energy to remove a neutron from 6C¹³, if the mass of the neutron is 1.008665 u, will be:
(1) 62.5 MeV
(2) 6.25 MeV
(3) 4.95 MeV
(4) 49.5 MeV

A ball suspended by a thread swings in a vertical plane so that its acceleration in the extreme and lowest position are equal. The angle (θ) of deflection in the extreme position is:
(1) tan⁻¹(√2)
(2) 2tan⁻¹(1/2)
(3) tan⁻¹(1/2)
(4) 2tan⁻¹(1/√5)

Three voltmeters are joined as shown. When a potential difference is applied across A and B, their readings are V₁, V₂, and V₃. Choose the correct option:
(1) V₁ = V₂
(2) V₁ = V₃ - V₂
(3) V₁ + V₂ > V₃
(4) V₁ + V₂ = V₃

The total kinetic energy of 1 mole of oxygen at 27°C is:
(1) 6845.5 J
(2) 5942.0 J
(3) 6232.5 J
(4) 5670.5 J

Given that the angular speed of the moon in its orbit about the earth is greater than that of the earth around the sun, identify the reason.
(1) Shorter period of orbit
(2) Larger radius
(3) Higher mass of the moon
(4) Higher gravitational pull

The primary side of a transformer is connected to 230 V, 50 Hz supply with a turns ratio of 10:1. Load resistance is 46 Ω. The power consumed is:
(1) 12.5 W
(2) 10 W
(3) 11.5 W
(4) 12 W

During an adiabatic process, the pressure of a gas is proportional to the cube of its absolute temperature. The ratio of Cp/Cv is:
(1) 5/3
(2) 3/2
(3) 7/5
(4) 9/7

The threshold frequency of a metal with work function 6.63 eV is:
(1) 16 ×10¹⁵ Hz
(2) 16 ×10¹² Hz
(3) 1.6 × 10¹² Hz
(4) 1.6 × 10¹⁵ Hz

A current of 200 µA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through π/10 radians is:
(1) 30 µA
(2) 120 µA
(3) 60 µA
(4) 180 µA

The atomic mass of 6C¹² is 12.000000 u and that of 6C¹³ is 13.003354 u. The required energy to remove a neutron from 6C¹³, if the mass of the neutron is 1.008665 u, will be:
(1) 62.5 MeV
(2) 6.25 MeV
(3) 4.95 MeV
(4) 49.5 MeV

A ball suspended by a thread swings in a vertical plane so that its acceleration in the extreme and lowest position are equal. The angle (θ) of deflection in the extreme position is:
(1) tan⁻¹(√2)
(2) 2tan⁻¹(1/2)
(3) tan⁻¹(1/2)
(4) 2tan⁻¹(1/√5)

Three voltmeters are joined as shown. When a potential difference is applied across A and B, their readings are V₁, V₂, and V₃. Choose the correct option.
(1) V₁ = V₂
(2) V₁ = V₃ - V₂
(3) V₁ + V₂ > V₃
(4) V₁ + V₂ = V₃

The total kinetic energy of 1 mole of oxygen at 27°C is:
(1) 6845.5 J
(2) 5942.0 J
(3) 6232.5 J
(4) 5670.5 J

The primary side of a transformer is connected to 230 V, 50 Hz supply with a turns ratio of 10:1. Load resistance is 46 Ω. The power consumed is:
(1) 12.5 W
(2) 10 W
(3) 11.5 W
(4) 12 W

The magnetic field at the centre of a wire loop formed by two semicircular wires of radii R1 = 2m and R2 = 4m carrying current I = 4A as per figure given below is α× 10⁻⁷ T. The value of α is:

Two charges of −4µC and +4µC are placed at the points A(1, 0, 4)m and B(2,−1, 5)m located in an electric field E⃗ = 0.20iV/cm. The magnitude of the torque acting on the dipole is 8√α× 10⁻⁵ Nm, where α = :

A closed organ pipe 150 cm long gives 7 beats per second with an open organ pipe of length 350 cm, both vibrating in fundamental mode. The velocity of sound is:

A body falling under gravity covers two points A and B separated by 80 m in 2s. The distance of upper point A from the starting point is:

The reading of a pressure meter attached with a closed pipe is 4.5×10⁴ N/m². On opening the valve, water starts flowing and the reading of pressure meter falls to 2.0×10⁴ N/m². The velocity of water is found to be √V m/s. The value of V is:

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is:

A parallel beam of monochromatic light of wavelength 5000 Å is incident normally on a single narrow slit of width 0.001 mm. The light is focused by a convex lens on a screen, placed on its focal plane. The first minima will be formed for the angle of diffraction of:

The electric potential at the surface of an atomic nucleus (Z = 50) of radius 9×10⁻¹³ cm is ×10⁶ V:

If Rydberg’s constant is R, the longest wavelength of radiation in Paschen series will be α×7R, where α = :

A series LCR circuit with L = 100π mH, C = 10⁻³ F, and R = 10Ω, is connected across an ac source of 220V, 50Hz supply. The power factor of the circuit would be:

The order of relative stability of the contributing structures is:

1. I > II > III
2. II > I > III
3. I = II = III
4. III > II > I

Which among the following halide(s) will not show SN1 reaction:
(A) H₂C = CH - CH₂Cl
(B) CH₃ - CH = CH - Cl
Choose the most appropriate answer from the options given below:

1. (A), (B), and (D) only
2. (A) and (B) only
3. (B) and (C) only
4. (B) only

Which of the following statements is not correct about rusting of iron?

1. Coating of iron surface by tin prevents rusting, even if the tin coating is peeled off.
2. When pH lies above 9 or 10, rusting of iron does not take place.
3. Dissolved acidic oxides SO₂, NO₂ in water act as catalysts in the process of rusting.
4. Rusting of iron is envisaged as setting up of electrochemical cell on the surface of iron object.

Given below are two statements:
Statement (I): In the Lanthanides, the formation of Ce⁴⁺ is favored by its noble gas configuration.
Statement (II): Ce⁴⁺ is a strong oxidant reverting to the common +3 state. Choose the correct option:

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are true
3. Statement I is true but Statement II is false
4. Both Statement I and Statement II are false

Choose the correct option having all the elements with d¹⁰ electronic configuration from the following:

1. Zn, Co²⁺, Ni, Fe²⁺, Cr
2. Cu, Zn, Ag, Cd
3. Pd, Ni, Fe²⁺, Cr
4. Ni, Zn, Fe²⁺, Cu

Phenolic group can be identified by a positive:

1. Phthalein dye test
2. Lucas test
3. Tollen’s test
4. Carbylamine test

The molecular formula of second homologue in the homologous series of mono carboxylic acids is:

1. C₃H₆O₂
2. C₂H₄O₂
3. CH₂O
4. C₂H₂O₂

The technique used for purification of steam volatile water immiscible substance is:

1. Fractional distillation
2. Fractional distillation under reduced pressure
3. Distillation
4. Steam distillation

The final product A, formed in the following reaction sequence is a mixture of:

Correct Answer: (4)

Match List-I with List-II:

Correct Matching: (1) (A)-(IV), (B)-(I), (C)-(III), (D)-(II)

Major product formed in the following reaction is a mixture of:
Correct Answer: (4)

The bond line formula of HOCH(CN)₂ is:
Correct Answer: (4)

Given below are two statements:
Statement (I): Oxygen being the first member of group 16 exhibits only -2 oxidation state.
Statement (II): Down the group 16, stability of +4 oxidation state decreases and +6 oxidation state increases. In 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 correct
3. Both Statement I and Statement II are incorrect
4. Statement I is incorrect but Statement II is correct

Identify from the following species in which d²sp³ hybridization is shown by the central atom:

1. [Co(NH₃)₆]³⁺
2. BrF₅
3. [PtCl₄]²⁻
4. SF₆

Identify the product formed in the following reaction: Cl-(CH₂)₄-Cl excess NH₃ → A, NaOH → B + H₂O + NaCl
Correct Answer: (2)

The quantity which changes with temperature is:

1. Molarity
2. Mass percentage
3. Molality
4. Mole fraction

Which structure of protein remains intact after coagulation of egg white on boiling?

1. Primary
2. Tertiary
3. Secondary
4. Quaternary

Which of the following cannot function as an oxidising agent?

1. N³⁻
2. SO₄²⁻
3. BrO₃⁻
4. MnO₄⁻

The incorrect statement regarding conformations of ethane is:

1. Ethane has an infinite number of conformations
2. The dihedral angle in staggered conformation is 60°
3. Eclipsed conformation is the most stable conformation
4. The conformations of ethane are inter-convertible to one another

Identify the incorrect pair from the following:

1. Photography - AgBr
2. Polythene preparation - TiCl₄, Al(CH₃)₃
3. Haber process - Iron
4. Wacker process - PtCl₂

Total number of ions from the following with noble gas configuration is:

The number of non-polar molecules from the following is:

Time required for completion of 99.9% of a first-order reaction is times of half life (t_1/2) of the reaction:

The spin-only magnetic moment value of square planar complex [Pt(NH₃)₂Cl(NH₂CH₃)]Cl is B.M. (Nearest integer):

For a certain thermochemical reaction M → N at T = 400 K, ∆H◦ = 77.2 kJ/mol, ∆S◦ = 122 J/K, log equilibrium constant (log K) is x × 10⁻¹.

Volume of 3 M NaOH (formula weight 40 g/mol) which can be prepared from 84 g of NaOH is x × 10⁻¹ dm³.

1 mole of PbS is oxidized by “X” moles of O₃ to get “Y” moles of O₂. X + Y = ?

The hydrogen electrode is dipped in a solution of pH = 3 at 25°C. The potential of the electrode will be - x × 10⁻² V.

9.3 g of aniline is subjected to reaction with excess of acetic anhydride to prepare acetanilide. The mass of acetanilide produced if the reaction is 100% completed is x × 10⁻¹ g.

Total number of compounds with chiral carbon atoms from the following is: