JEE Main 2024 29 Jan Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

SECTION- A

PHYSICS

Question 1:

If in a G.P. of 64 terms, the sum of all terms is 7 times the sum of the odd terms, then the common ratio of the G.P. is equal to:

In an A.P., the sixth term a₆ = 2. If the product a₁ * a₄ * a₅ is maximized, the common difference of the A.P. is equal to:

Given functions f(x) and g(x), find the range of (f o g)(x).

A fair die is thrown until the number 2 appears. What is the probability that 2 appears in an even number of throws?

If z = 1/(2 - 2i) such that |z + 1| = αz + β(1 + i), where i is the imaginary unit and α, β are real numbers, then α + β is equal to:

Evaluate the limit
lim (x → π/2) [(1/x - π/2) * ∫(0 to π/2) cos(1/t³) dt], which is equal to:

In a △ABC, suppose y = x is the equation of the bisector of angle B, and the equation of side AC is 2x - y = 2. If 2AB = BC and the points A and B are (4, 6) and (α, β), then α + 2β is equal to:

Let a, b, and c be three non-zero vectors such that b and c are non-collinear. If a + 5b is collinear with c, b + 6c is collinear with a, and a + αb + βc = 0, then α + β is equal to:

Let (5, a/4) be the circumcenter of a triangle with vertices A(a, -2), B(a, 6), and C(a/4, -2). Let α denote the circumradius, β the area, and γ the perimeter of the triangle. Then α + β + γ is:

For x ∈ (-π/2, π/2), if y(x) = csc(x) + sin(x) - (1 + cos(2x)) csc(sec(x) + tan(x)), then y(π/4) is equal to:

If α, with -π/2 < α < π/2, is the solution of 4cos(θ) + 5sin(θ) = 1, then the value of tan(α) is:

A function y = f(x) satisfies f(x)sin(2x) + sin(x) - (1 + cos(2x))f'(x) = 0 with f(0) = 0. Then f(π/2) is equal to:

Let O be the origin and the position vectors of A and B be (2i + 2j + k) and (2i + 4j + 4k), respectively. If the internal bisector of ∠AOB meets the line AB at C, then the length of OC is:

Consider the function f: [1/2,1] → R defined by f(x) = √(2x³ - 3√(2x - 1)). Which of the following statements is correct?

Let A = [ [1, 0, 0], [0, α, β], [0, β, α] ] and |2A|³ = 221, where α, β ∈ Z. Then a value of α is:

Let PQR be a triangle with R(−1, 4, 2). Suppose M(2, 1, 2) is the midpoint of PQ. The distance of the centroid of △PQR from the point of intersection of the line (x-2)/0 = y/2 = (z+3)/-1 and (x-1)/1 = (y+3)/-3 = (z+1)/1 is:

Let R be a relation on Z × Z defined by (a, b)R(c, d) if and only if ad - bc is divisible by 5. Then R is:

If the value of the integral ∫ from -π/2 to π/2 of [(x² cos(x))/(1 + πx) + (1 + sin²(x))/(1 + e^(sin(x)))] dx equals π/4(π + a) - 2, then the value of a is:

Suppose f(x) = (2x + 2^(-x)) tan(x) √(tan⁻¹(x² - x + 1)). The value of f'(0) is equal to:

Let A be a square matrix such that AAᵀ = I. Then (1/2)A[(A + Aᵀ)² + (A - Aᵀ)²] is equal to:

Equation of two diameters of a circle are 2x - 3y = 5 and 3x - 4y = 7. The line joining the points (-22/7, -4) and (1/7, 3) intersects the circle at only one point P(α, β). Then 17β - α is equal to:

All the letters of the word “GTWENTY” are written in all possible ways with or without meaning, and these words are arranged as in a dictionary. The serial number of the word “GTWENTY” is:

Let f(x) = 2x - x², x ∈ R. If m and n are respectively the number of points at which the curves y = f(x) and y = f'(x) intersect the x-axis, then the value of m + n is:

Let f(x) = 2x - x², x ∈ R. If m and n are respectively the number of points at which the curves y = f(x) and y = f'(x) intersect the x-axis, then the value of m + n is:

If the points of intersection of two distinct conics x² + y² = 4b and x²/16 + y²/b² = 1 lie on the curve y² = 3x², then 3√3 times the area of the rectangle formed by the intersection points is:

The solution curve y = y(x) of the differential equation (1 + y²)(1 + log(x)) dx + x dy = 0, x > 0, passes through the point (1, 1). If y(e) = (α - tan(3/2))/(β + tan(3/2)), then α + 2β is:

If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, α, β, 60 where α > β are 56 and 66.2 respectively, then α² + β² is equal to:

The area (in sq. units) of the part of the circle x² + y² = 169 which is below the line 5x - y = 13 is (πα/2β) * 65² + (α/β) * sin⁻¹(12/13). The value of α + β is:

If (11C1)(11C2) + (11C2)(11C3) + ... + (11C9)(11C10) = n/m with gcd(n, m) = 1, then n + m is equal to:

A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the points P and Q. If the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l² is:

In the given circuit, the breakdown voltage of the Zener diode is 3.0 V. What is the value of IZ?

The electric current through a wire varies with time as I = I₀ + βt, where I₀ = 20 A and β = 3 A/s. The amount of electric charge that crosses through a section of the wire in 20 s is:

Given below are two statements:
Statement I: If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water.
Statement II: If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in cold water.

A convex mirror of radius of curvature 30 cm forms an image that is half the size of the object. The object distance is:

Two charges of 5Q and −2Q are situated at the points (3a, 0) and (−5a, 0) respectively. The electric flux through a sphere of radius 4a having its center at the origin is:

A body starts moving from rest with constant acceleration and covers displacement S1 in the first (p−1) seconds and S2 in the first p seconds. The displacement S1 + S2 will be made in time:

The potential energy function (in J) of a particle in a region of space is given as U = (2x² + 3y³ + 2z). Here x, y, and z are in meters. The magnitude of the x-component of force (in N) acting on the particle at point P (1, 2, 3) m is:

The resistance R = V/I where V = (200± 5)V and I = (20± 0.2)A. The percentage error in the measurement of R is:

A block of mass 100 kg slides over a distance of 10 m on a horizontal surface. If the coefficient of friction between the surfaces is 0.4, then the work done against friction (in J) is:

Match List I with List II:
List I
A. ∮ B· dl = µ₀ic + µ₀ε₀ dΦE/dt
B. ∮ E· dl = −dΦB/dt
C. ∮ E· dA = Q/ε₀
D. ∮ B· dA = 0
List II
I. Gauss' law for electricity
II. Gauss' law for magnetism
III. Faraday's law
IV. Ampere–Maxwell law

If the radius of curvature of the path of two particles of the same mass are in the ratio 3:4, then in order to have constant centripetal force, their velocities will be in the ratio of:

A galvanometer having coil resistance 10 Ω shows a full scale deflection for a current of 3 mA. For it to measure a current of 8 A, the value of the shunt should be:

The de-Broglie wavelength of an electron is the same as that of a photon. If the velocity of the electron is 25% of the velocity of light, then the ratio of the K.E. of the electron to the K.E. of the photon will be:

The deflection in a moving coil galvanometer falls from 25 divisions to 5 divisions when a shunt of 24Ω is applied. The resistance of the galvanometer coil will be:

A biconvex lens of refractive index 1.5 has a focal length of 20 cm in air. Its focal length when immersed in a liquid of refractive index 1.6 will be:

A thermodynamic system is taken from an original state A to an intermediate state B by a linear process as shown in the figure. Its volume is then reduced to the original value from B to C by an isobaric process. The total work done by the gas from A to B and B to C would be:

At what distance above and below the surface of the earth a body will have the same weight (take radius of earth as R)?

A capacitor of capacitance 100 µF is charged to a potential of 12 V and connected to a 6.4 mH inductor to produce oscillations. The maximum current in the circuit would be:

The explosive in a Hydrogen bomb is a mixture of 1H2, 1H3, and 3Li6 in some condensed form. The chain reaction is given by:
3Li6 + 0 n1 → 2 He4 + 1 H3
1H2 + 1H3 → 2 He4 + 0 n1
During the explosion, the energy released is approximately:

Two vessels A and B are of the same size and are at the same temperature. A contains 1 g of hydrogen and B contains 1 g of oxygen. PA and PB are the pressures of the gases in A and B respectively, then PA/PB is:

When a hydrogen atom going from n = 2 to n = 1 emits a photon, its recoil speed is x/5 m/s. Where x = . (Use: mass of hydrogen atom = 1.6 × 10⁻²⁷ kg)

A ball rolls off the top of a stairway with horizontal velocity u. The steps are 0.1 m high and 0.1 m wide. The minimum velocity u with which the ball just hits the step 5 of the stairway will be √x m/s, where x = . (Use g = 10 m/s²)

A square loop of side 10 cm and resistance 0.7 Ω is placed vertically in the east-west plane. A uniform magnetic field of 0.20 T is set up across the plane in the northeast direction. The magnetic field is decreased to zero in 1 s at a steady rate. Then, the magnitude of induced emf is √x × 10⁻³ V. The value of x is .

A cylinder is rolling down on an inclined plane of inclination 60°. Its acceleration during rolling down will be x√3 m/s², where x = . (Use g = 10 m/s²)

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is 1.5 × 10⁻⁵ Tm. The magnetic moment of the dipole is Am². Given: µ₀ / 4π = 10⁻⁷ Tm/A

In a double slit experiment shown in the figure, when light of wavelength 400 nm is used, a dark fringe is observed at P. If D = 0.2 m, the minimum distance between the slits S1 and S2 is mm.

A 16 Ω wire is bent to form a square loop. A 9 V battery with internal resistance 1 Ω is connected across one of its sides. If a 4 µF capacitor is connected across one of its diagonals, the energy stored by the capacitor will be x² µJ, where x = .

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is x/8, where x = .

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density +σ. The electron at t = 0 is at a distance of 1 m from S and has a speed of 1 m/s. The maximum value of σ if the electron strikes S at t = 1 s is α[mϵ₀/e] C/m². The value of α is .

In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wings are 70 m/s and 65 m/s respectively. If the wing area is 2 m², the lift of the wing is N. Given: density of air = 1.2 kg/m³

Given below are two statements: Assertion A: The first ionisation enthalpy decreases across a period. Reason R: The increasing nuclear charge outweighs the shielding across the period.
(1) Both A and R are true and R is the correct explanation of A
(2) A is true but R is false
(3) A is false but R is true
(4) Both A and R are true but R is NOT the correct explanation of A

Match List I with List II
LIST-I (Substances) | LIST-II (Element Present)
A. Ziegler catalyst | I. Rhodium
B. Blood Pigment | II. Cobalt
C. Wilkinson catalyst | III. Iron
D. Vitamin B12 | IV. Titanium

In chromyl chloride test for confirmation of Cl− ion, a yellow solution is obtained. Acidification of the solution and addition of amyl alcohol and 10% H2O2 turns organic layer blue indicating formation of chromium pentoxide. The oxidation state of chromium in that is:

The difference in energy between the actual structure and the lowest energy resonance structure for the given compound is:
(1) electromeric energy
(2) resonance energy
(3) ionization energy
(4) hyperconjugation energy

Given below are two statements: Statement I: The electronegativity of group 14 elements from Si to Pb gradually decreases. Statement II: Group 14 contains non-metallic, metallic, as well as metalloid elements.
(1) Statement I is false but Statement II is true
(2) Statement I is true but Statement II is false
(3) Both Statement I and Statement II are true
(4) Both Statement I and Statement II are false

The correct set of four quantum numbers for the valence electron of rubidium atom (Z = 37) is:
(1) 5, 0, 0, +1/2
(2) 5, 0, 1, +1/2
(3) 5, 1, 0, +1/2
(4) 5, 1, 1, +1/2

The major product (P) in the following reaction is:
(1) Structure 1
(2) Structure 2
(3) Structure 3
(4) Structure 4

The arenium ion which is not involved in the bromination of Aniline is:
(1) Structure 1
(2) Structure 2
(3) Structure 3
(4) Structure 4

Appearance of blood red colour on treatment of the sodium fusion extract of an organic compound with FeSO₄ in presence of concentrated H₂SO₄ indicates the presence of element(s):
(1) Br
(2) N
(3) N and S
(4) S

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R: Assertion A: Aryl halides cannot be prepared by replacement of hydroxyl group of phenol by halogen atom. Reason R: Phenols react with halogen acids violently.
(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) A is true but R is false
(4) Both A and R are true and R is the correct explanation of A

Identify product A and product B:
(1) Product A is formed through electrophilic substitution, Product B is from radical substitution
(2) Product A is formed via nucleophilic addition, Product B through electrophilic addition
(3) Product A is a result of free radical addition, Product B via electrophilic substitution
(4) Product A is formed through a free radical mechanism, Product B through electrophilic addition

Identify the incorrect pair from the following:
(1) Fluorspar - BF₃
(2) Cryolite - Na₃AlF₆
(3) Fluoroapatite - 3Ca₃(PO₄)₂·CaF₂
(4) Carnallite - KCl·MgCl₂·6H₂O

The interaction between π bond and lone pair of electrons present on an adjacent atom is responsible for:
(1) Hyperconjugation
(2) Inductive effect
(3) Electromeric effect
(4) Resonance effect

KMnO₄ decomposes on heating at 513 K to form O₂ along with:
(1) MnO₂ & K₂O₂
(2) K₂MnO₄ & Mn
(3) Mn & KO₂
(4) K₂MnO₄ & MnO₂

In which one of the following metal carbonyls, CO forms a bridge between metal atoms?
(1) [Co₂(CO)₈]
(2) [Mn₂(CO)₁₀]
(3) [Os₃(CO)₁₂]
(4) [Ru₃(CO)₁₂]

Type of amino acids obtained by hydrolysis of proteins is:
(1) β
(2) α
(3) δ
(4) γ

The final product A formed in the following multistep reaction sequence is:

Which of the following is not correct?
(1) ∆G is negative for a spontaneous reaction
(2) ∆G is positive for a spontaneous reaction
(3) ∆G is zero for a reversible reaction
(4) ∆G is positive for a non-spontaneous reaction

Chlorine undergoes disproportionation in alkaline medium as shown below:
aCl₂(g) + bOH⁻(aq) → cClO⁻(aq) + dCl⁻(aq) + eH₂O(l)
The values of a, b, c, and d in a balanced redox reaction are respectively:

In alkaline medium, MnO₄⁻ oxidizes I⁻ to:
(1) IO₄⁻
(2) IO⁻
(3) I₂
(4) IO₃⁻

Number of compounds with one lone pair of electrons on central atom amongst following is:
O₃, H₂O, SF₄, ClF₃, NH₃, BrF₅, XeF₄

The mass of zinc produced by the electrolysis of zinc sulphate solution with a steady current of 0.015 A for 15 minutes is ×10⁻⁴ g.
(Atomic mass of zinc = 65.4 amu)

For a reaction taking place in three steps at the same temperature, the overall rate constant K = K₁K₂ / K₃. If Eₐ₁, Eₐ₂, and Eₐ₃ are 40, 50, and 60 kJ/mol respectively, the overall Eₐ is kJ/mol.

For the reaction N₂O₄(g) ⇌ 2NO₂(g), Kₚ = 0.492 atm at 300K. Kc for the reaction at the same temperature is ×10⁻².
(Given: R = 0.082 L atm mol⁻¹ K⁻¹)

A solution of H₂SO₄ is 31.4% H₂SO₄ by mass and has a density of 1.25 g/mL. The molarity of the H₂SO₄ solution is M (nearest integer).
(Given molar mass of H₂SO₄ = 98 g mol⁻¹)

The osmotic pressure of a dilute solution is 7 × 10⁵ Pa at 273 K. Osmotic pressure of the same solution at 283 K is ×10⁴ N/m².

Number of compounds among the following which contain sulfur as a heteroatom is:
Compounds: Furan, Thiophene, Pyridine, Pyrrole, Cysteine, Tyrosine

The number of species from the following which are paramagnetic and with bond order equal to one is:
Species: H₂, He₂⁺, O₂⁺, N₂⁻, O₂²⁻, F₂, Ne₂⁺, B₂

From the compounds given below, the number of compounds which give a positive Fehling's test is:
Compounds: Benzaldehyde, Acetaldehyde, Acetone, Acetophenone, Methanal, 4-nitrobenzaldehyde, Cyclohexane carbaldehyde

Consider the given reaction. The total number of oxygen atoms present per molecule of the product (P) is:
Reaction: CH₃ − CH = CH − CH₃ → (i) O₃, (ii) Zn/H₂O → 2CH₃ − C = O