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JEE Main 2025 — 04 April 2025 (Shift 1) Solutions
Step-by-step mathematical derivations, formula references, and interactive practice for Physics, Chemistry, and Mathematics.
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Question Paper & Solutions (75 Questions Displayed)
Click any option to test in Practice ModeQ1physicsKinematics & Vectors• Class 11
ModerateMCQ
A particle of mass m moves in a circular path of radius R with variable speed v = α√s, where s is the distance traversed along the circle and α is a positive constant. The magnitude of the net acceleration of the particle after covering one complete revolution is:
A
α² √(1 + (4π)²)
B
(α² / 2) √(1 + 16π²)
C
α² √(1/4 + 4π²)
D
(α² / 2) √(1 + (4πR)²)
Official Key:B
Q2physicsThermodynamics & Kinetic Theory (KTG)• Class 11
EasyMCQ
An ideal gas undergoes a thermodynamic cyclic process consisting of three steps: (i) Isobaric expansion at pressure P₀ from volume V₀ to 2V₀, (ii) Isochoric cooling to pressure P₀/2, and (iii) Isothermal compression back to initial state (P₀, V₀). The total work done by the gas in one complete cycle is:
A
P₀V₀ [ 1 - ln 2 ]
B
P₀V₀ [ 1 - 2 ln 2 ]
C
P₀V₀ [ 1/2 + ln 2 ]
D
P₀V₀ [ 1 - (1/2) ln 2 ]
Official Key:A
Q3physicsRotational Dynamics & COM• Class 11
EasyMCQ
A solid uniform sphere of mass M and radius R rolls without slipping down an inclined plane of inclination θ. The linear acceleration of the center of mass of the sphere is:
A
(5/7) g sin θ
B
(3/5) g sin θ
C
(2/3) g sin θ
D
(1/2) g sin θ
Official Key:A
Q4physicsOptics (Ray & Wave Optics)• Class 12
ModerateMCQ
In a Young’s Double Slit Experiment, monochromatic light of wavelength λ = 600 nm illuminates two slits separated by distance d = 1 mm. The screen is placed at distance D = 2 m. When a thin transparent mica sheet of thickness t = 1.8 μm and refractive index μ = 1.5 is placed in front of one slit, the central bright fringe shifts by a distance of:
A
1.8 mm
B
2.4 mm
C
0.9 mm
D
3.6 mm
Official Key:A
Q5physicsMagnetic Effects of Current & Magnetism• Class 12
EasyMCQ
Two long parallel straight wires carrying currents of I₁ = 10 A and I₂ = 20 A in the same direction are separated by a distance of 10 cm in air. The force per unit length between the wires is:
A
4 × 10⁻⁴ N/m (Attractive)
B
4 × 10⁻⁴ N/m (Repulsive)
C
2 × 10⁻⁴ N/m (Attractive)
D
8 × 10⁻⁴ N/m (Repulsive)
Official Key:A
Q6physicsElectromagnetic Induction & AC Circuits• Class 12
EasyMCQ
In a series LCR resonant circuit, the inductance L = 10 mH, capacitance C = 1 μF, and resistance R = 10 Ω. The Quality Factor (Q-factor) of the circuit is equal to:
A
10
B
100
C
1
D
0.1
Official Key:A
Q7physicsElectrostatics & Capacitors• Class 12
ModerateMCQ
Two concentric conducting spherical shells of radii R and 2R carry charges +Q and +2Q respectively. The electric potential at a distance r = 1.5R from the common center is:
A
k [ Q / (1.5R) + 2Q / (2R) ] = (5/3) kQ/R
B
k [ 3Q / (1.5R) ] = 2 kQ/R
C
k [ Q / R + 2Q / (1.5R) ]
D
k [ Q / (1.5R) ]
Official Key:A
Q8physicsLaws of Motion & Friction• Class 11
EasyMCQ
A block of mass m = 2 kg rests on a rough horizontal surface with static friction coefficient μ_s = 0.5. If a horizontal force F = 8 N is applied on the block, the frictional force acting on the block is: (Take g = 10 m/s²)
A
8 N (Static)
B
10 N (Limiting)
C
0 N
D
4 N
Official Key:A
Q9physicsOscillations (SHM) & Waves• Class 11
EasyMCQ
An open organ pipe of length L₁ and a closed organ pipe of length L₂ vibrate in their fundamental modes. If their fundamental frequencies are equal, the ratio L₁ / L₂ is:
A
2 : 1
B
1 : 2
C
4 : 1
D
1 : 1
Official Key:A
Q10physicsModern Physics & Semiconductors• Class 12
ModerateMCQ
In a hydrogen-like atom, an electron makes a transition from n = 3 to n = 1 level. The ratio of the de Broglie wavelength of the electron in orbit n = 3 to that in orbit n = 1 is:
A
3 : 1
B
9 : 1
C
1 : 3
D
1 : 9
Official Key:A
Q11physicsGravitation & Planetary Motion• Class 11
EasyMCQ
The escape velocity from the surface of Earth is v_e = 11.2 km/s. If a planet has twice the radius and 8 times the mass of Earth, the escape velocity from its surface is:
A
22.4 km/s
B
11.2 km/s
C
44.8 km/s
D
5.6 km/s
Official Key:A
Q12physicsOptics (Ray & Wave Optics)• Class 12
EasyMCQ
A convex lens of focal length f = 20 cm in air is immersed in water (refractive index μ_w = 4/3). If the refractive index of glass is μ_g = 3/2, the focal length of the lens in water is:
A
80 cm
B
40 cm
C
20 cm
D
10 cm
Official Key:A
Q13physicsWork, Energy & Power• Class 11
EasyMCQ
The work done by an external agent in slowly stretching a spring of force constant k = 200 N/m from an elongation of x₁ = 5 cm to x₂ = 15 cm is:
A
2.0 Joules
B
1.0 Joules
C
4.0 Joules
D
0.5 Joules
Official Key:A
Q14physicsModern Physics & Semiconductors• Class 12
EasyMCQ
In an intrinsic semiconductor at room temperature, the number of conduction electrons is n_e and holes is n_h. When doped with a trivalent impurity (acceptor):
A
n_h >> n_e (P-type semiconductor)
B
n_e >> n_h (N-type semiconductor)
C
n_e = n_h
D
n_e and n_h both become zero
Official Key:A
Q15physicsElectromagnetic Induction & AC Circuits• Class 12
EasyMCQ
A metal rod of length L = 1 m rotates with uniform angular velocity ω = 20 rad/s in a plane perpendicular to a uniform magnetic field B = 0.5 T about an axis passing through one of its ends. The induced EMF across the rod is:
A
5.0 Volts
B
10.0 Volts
C
2.5 Volts
D
20.0 Volts
Official Key:A
Q16physicsElectrostatics & Capacitors• Class 12
EasyMCQ
A parallel plate capacitor with air between the plates has capacitance C₀ = 10 μF. When a dielectric slab of dielectric constant K = 4 is inserted to completely fill the space between the plates while the capacitor remains connected to a battery of voltage V = 12 V, the charge stored in the capacitor increases by:
A
360 μC
B
480 μC
C
120 μC
D
240 μC
Official Key:A
Q17physicsCurrent Electricity & Circuits• Class 12
EasyMCQ
In a meter bridge experiment, the balance point is obtained at 40 cm from the left end when a resistance of 6 Ω is connected in the left gap. The unknown resistance X in the right gap is equal to:
A
9.0 Ω
B
4.0 Ω
C
6.0 Ω
D
12.0 Ω
Official Key:A
Q18physicsThermodynamics & Kinetic Theory (KTG)• Class 11
EasyMCQ
A thermodynamic engine operates between temperatures T₁ = 600 K (source) and T₂ = 300 K (sink). If the engine absorbs 1000 Joules of heat from the source in each cycle, the maximum possible mechanical work produced per cycle is:
A
500 Joules
B
600 Joules
C
300 Joules
D
1000 Joules
Official Key:A
Q19physicsOscillations (SHM) & Waves• Class 11
EasyMCQ
A particle executes Simple Harmonic Motion (SHM) with an amplitude A = 10 cm and time period T = 6 s. The time taken by the particle to travel from the mean position (x = 0) to half of its amplitude (x = 5 cm) is:
A
0.5 seconds
B
1.0 seconds
C
1.5 seconds
D
0.25 seconds
Official Key:A
Q20physicsKinematics & Vectors• Class 11
ModerateMCQ
A projectile is launched with speed u = 40 m/s at an angle of 30° with the horizontal. The radius of curvature of its trajectory at the highest point is: (Take g = 10 m/s²)
A
120 meters
B
160 meters
C
80 meters
D
240 meters
Official Key:A
Q21physicsModern Physics & Semiconductors• Class 12
EasyNumerical
An electron is accelerated from rest across a potential difference of V = 150 Volts. The de Broglie wavelength associated with the electron is equal to (in Angstroms, Å, rounded to two decimal places):
Official Key:1.00
Q22physicsWork, Energy & Power• Class 11
EasyNumerical
A body of mass m = 5 kg falls freely from rest under gravity from a height h = 20 m. The power delivered by the gravitational force on the body just before it hits the ground is (in Watts): (Take g = 10 m/s²)
Official Key:1000
Q23physicsMagnetic Effects of Current & Magnetism• Class 12
EasyNumerical
A circular coil of 50 turns and radius 10 cm carries a steady current of 2 A. The magnetic dipole moment of the coil is (in A·m², take π = 3.14):
Official Key:3.14
Q24physicsOptics (Ray & Wave Optics)• Class 12
EasyNumerical
A light ray passes through an equilateral triangular glass prism of refractive index μ = √3 with minimum deviation. The angle of minimum deviation (in degrees) is:
Official Key:60
Q25physicsModern Physics & Semiconductors• Class 12
EasyNumerical
A radioactive sample has a half-life of T_half = 10 days. If the initial activity is A₀ = 800 Bq, the time required (in days) for its activity to reduce to 100 Bq is:
Official Key:30
Q26chemistryCoordination Compounds• Class 12
EasyMCQ
According to Crystal Field Theory, the spin-only magnetic moments of [CoF₆]³⁻ and [Fe(CN)₆]³⁻ (in Bohr Magnetons, BM) are respectively: [Atomic numbers: Fe = 26, Co = 27]
A
4.90 BM and 1.73 BM
B
0 BM and 5.92 BM
C
4.90 BM and 5.92 BM
D
3.87 BM and 1.73 BM
Official Key:A
Q27chemistryAldehydes, Ketones & Carboxylic Acids• Class 12
EasyMCQ
When Benzaldehyde (C₆H₅CHO) is treated with concentrated NaOH solution (50%), the products formed are:
A
Benzyl alcohol and Sodium benzoate
B
Benzoic acid and Benzene
C
Cinnamic acid and Sodium phenoxide
D
Hydrobenzamide and Sodium formate
Official Key:A
Q28chemistryChemical Bonding & Molecular Structure• Class 11
EasyMCQ
According to Molecular Orbital Theory (MOT), which of the following species has a bond order of 2.5 and is paramagnetic?
A
N₂⁺
B
O₂²⁻
C
C₂
D
N₂
Official Key:A
Q29chemistryElectrochemistry & Redox Reactions• Class 12
EasyMCQ
The standard reduction potentials of three metals X, Y, and Z are E°(X²⁺/X) = -0.76 V, E°(Y²⁺/Y) = +0.34 V, and E°(Z²⁺/Z) = -2.37 V. The correct order of their reducing power is:
A
Z > X > Y
B
Y > X > Z
C
X > Y > Z
D
Z > Y > X
Official Key:A
Q30chemistryGeneral Organic Chemistry (GOC)• Class 11
EasyMCQ
Which of the following carbocations is the MOST stable due to resonance and hyperconjugation?
A
(C₆H₅)₃C⁺ (Triphenylmethyl carbocation)
B
(CH₃)₃C⁺ (tert-Butyl carbocation)
C
C₆H₅CH₂⁺ (Benzyl carbocation)
D
(CH₃)₂CH⁺ (Isopropyl carbocation)
Official Key:A
Q31chemistryChemical Bonding & Molecular Structure• Class 11
EasyMCQ
The shape and hybridisation of XeF₄ molecule according to VSEPR theory are respectively:
A
Square Planar, sp³d²
B
Tetrahedral, sp³
C
See-saw, sp³d
D
Octahedral, sp³d²
Official Key:A
Q32chemistryAldehydes, Ketones & Carboxylic Acids• Class 12
EasyMCQ
Which of the following compounds will give a positive Iodoform test upon treatment with I₂ and NaOH?
A
Ethanol (CH₃CH₂OH)
B
Methanol (CH₃OH)
C
Benzophenone (C₆H₅COC₆H₅)
D
3-Pentanone (CH₃CH₂COCH₂CH₃)
Official Key:A
Q33chemistryd- and f-Block Elements• Class 12
EasyMCQ
Lanthanoid contraction is primarily caused by:
A
Poor shielding effect of 4f electrons
B
Effective shielding of 5d electrons
C
Decreasing nuclear charge
D
Inert pair effect
Official Key:A
Q34chemistryChemical & Ionic Equilibrium• Class 11
EasyMCQ
For the reaction N₂(g) + 3H₂(g) ⇌ 2NH₃(g), the relation between equilibrium constants K_p and K_c is:
A
K_p = K_c (RT)⁻²
B
K_p = K_c (RT)²
C
K_p = K_c (RT)⁻¹
D
K_p = K_c
Official Key:A
Q35chemistryClassification of Elements & Periodicity• Class 11
EasyMCQ
Which of the following oxides is AMPHOTERIC in nature?
A
Al₂O₃
B
Na₂O
C
SO₃
D
CaO
Official Key:A
Q36chemistryAmines & Nitrogen Containing Compounds• Class 12
ModerateMCQ
An organic compound reacts with Hinsberg’s reagent (Benzenesulphonyl chloride) to give a precipitate that is SOLUBLE in alkali (aqueous KOH). The compound is a:
A
Primary amine (1° amine)
B
Secondary amine (2° amine)
C
Tertiary amine (3° amine)
D
Quaternary ammonium salt
Official Key:A
Q37chemistryp-Block Elements• Class 12
EasyMCQ
The correct order of increasing thermal stability of Group 16 hydrides (H₂O, H₂S, H₂Se, H₂Te) is:
A
H₂Te < H₂Se < H₂S < H₂O
B
H₂O < H₂S < H₂Se < H₂Te
C
H₂S < H₂O < H₂Te < H₂Se
D
H₂Se < H₂Te < H₂S < H₂O
Official Key:A
Q38chemistryPolymers & Biomolecules• Class 12
EasyMCQ
Which of the following polymers is a BIODEGRADABLE polyamide?
A
Nylon-2-nylon-6
B
Nylon-6,6
C
Dacron (Terylene)
D
Bakelite
Official Key:A
Q39chemistrySolutions & Colligative Properties• Class 12
EasyMCQ
For an ideal solution obeying Raoult’s law, which of the following thermodynamic properties is ZERO?
A
Both ΔH_mix = 0 and ΔV_mix = 0
B
ΔS_mix = 0
C
ΔG_mix = 0
D
Only ΔH_mix = 0
Official Key:A
Q40chemistryAmines & Nitrogen Containing Compounds• Class 12
EasyMCQ
In the Gabriel Phthalimide Synthesis, which type of amine CANNOT be prepared?
A
Aromatic primary amines (Aniline)
B
Aliphatic primary amines
C
Methylamine
D
Ethylamine
Official Key:A
Q41chemistryElectrochemistry & Redox Reactions• Class 11
ModerateMCQ
The oxidation state of Chromium in CrO₅ (Chromium peroxide) and Potassium dichromate (K₂Cr₂O₇) are respectively:
A
+6 and +6
B
+10 and +6
C
+6 and +3
D
+4 and +6
Official Key:A
Q42chemistryCoordination Compounds• Class 12
ModerateMCQ
Which of the following compounds exhibits OPTICAL ISOMERISM (Chirality)?
A
[Co(en)₃]³⁺
B
trans-[Co(NH₃)₄Cl₂]⁺
C
[Pt(NH₃)₂Cl₂]
D
[Ni(CO)₄]
Official Key:A
Q43chemistryAlcohols, Phenols & Ethers• Class 12
EasyMCQ
Phenol upon distillation with Zinc dust (Zn dust) gives:
A
Benzene
B
Toluene
C
Benzoic acid
D
Diphenyl
Official Key:A
Q44chemistryChemical & Ionic Equilibrium• Class 11
ModerateMCQ
The pH of a 10⁻⁸ M aqueous solution of HCl at 25°C is:
A
Between 6.9 and 7.0 (Slightly acidic, ~6.96)
B
8.0
C
7.0
D
6.0
Official Key:A
Q45chemistryPolymers & Biomolecules• Class 12
EasyMCQ
Which of the following amino acids is an ESSENTIAL amino acid that cannot be synthesized by the human body?
A
Valine
B
Glycine
C
Alanine
D
Glutamic acid
Official Key:A
Q46chemistryChemical Kinetics• Class 12
EasyNumerical
A first order reaction is 50% complete in 20 minutes at 300 K. The time required for 75% completion of the same reaction at 300 K (in minutes) is:
Official Key:40
Q47chemistrySolutions & Colligative Properties• Class 12
ModerateNumerical
The depression in freezing point for a 0.1 molal aqueous solution of a monobasic weak acid HA (degree of dissociation α = 0.20) is (in K, rounded to two decimal places): [Given K_f for water = 1.86 K·kg/mol]
Official Key:0.22
Q48chemistryCoordination Compounds• Class 12
EasyNumerical
The spin-only magnetic moment of [Mn(H₂O)₆]²⁺ complex ion is equal to (in Bohr Magnetons, BM, rounded to two decimal places): [Atomic number of Mn = 25]
Official Key:5.92
Q49chemistryChemical Thermodynamics & Energetics• Class 11
EasyNumerical
For the reaction 2NO₂(g) ⇌ N₂O₄(g), the value of ΔH° is -57.2 kJ/mol and ΔS° is -175 J/(K·mol). The temperature (in Kelvin, rounded to nearest integer) at which the reaction attains equilibrium (ΔG° = 0) is:
Official Key:327
Q50chemistryElectrochemistry & Redox Reactions• Class 12
EasyNumerical
How many FARADAYS of electricity are required for the complete reduction of 1 mole of MnO₄⁻ to Mn²⁺ in acidic medium?
Official Key:5
Q51mathematicsVector Algebra & 3D Geometry• Class 12
ModerateMCQ
The shortest distance between the two skew lines: L₁: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and L₂: (x - 2)/3 = (y - 4)/4 = (z - 5)/5 is equal to:
A
1 / √6
B
1 / √3
C
1 / 2√6
D
0 (Lines Intersect)
Official Key:A
Q52mathematicsMatrices & Determinants• Class 12
ModerateMCQ
Let A be a 3 × 3 non-singular matrix such that det(A) = 3. If det(2 adj(3A)) = 2^α · 3^β, where α and β are positive integers, then the value of (α + β) is equal to:
A
11
B
9
C
13
D
7
Official Key:A
Q53mathematicsComplex Numbers & Quadratic Equations• Class 11
EasyMCQ
Let α and β be the roots of the quadratic equation x² - 6x - 2 = 0. If a_n = α^n - β^n for n ≥ 1, then the value of (a₁₀ - 2 a₈) / (2 a₉) is equal to:
A
3
B
6
C
1/3
D
-3
Official Key:A
Q54mathematicsDifferential Equations• Class 12
ModerateMCQ
The solution of the linear differential equation (dy/dx) + y cot x = 2x csc x, with y(π/2) = 0, at x = π/6 gives y(π/6) equal to:
A
- 4π² / 9
B
- 2π² / 9
C
- π² / 9
D
- 8π² / 9
Official Key:A
Q55mathematicsProbability & Statistics• Class 12
ModerateMCQ
Box I contains 3 red and 2 black balls. Box II contains 2 red and 4 black balls. A fair die is rolled: if 1 or 2 appears, Box I is chosen; otherwise Box II is chosen. If a drawn ball is Red, the probability that it was from Box I is:
A
9 / 19
B
10 / 19
C
3 / 8
D
5 / 19
Official Key:A
Q56mathematicsSequences & Series• Class 11
EasyMCQ
The value of the infinite sum S = ∑[n=1 to ∞] 1 / (4n² - 1) is equal to:
A
1 / 2
B
1
C
1 / 4
D
3 / 4
Official Key:A
Q57mathematicsCoordinate Geometry & Conic Sections• Class 11
ModerateMCQ
The line y = mx + c is a common tangent to the circle x² + y² = 2 and the parabola y² = 8x. The positive value of m is:
A
1
B
2
C
1 / √2
D
√2
Official Key:A
Q58mathematicsVector Algebra & 3D Geometry• Class 12
EasyMCQ
If the vectors a = î + ĵ + k̂, b = 2î - ĵ + 3k̂, and c = î - 2ĵ + α k̂ are COPLANAR, then the value of α is:
A
2
B
-2
C
4
D
0
Official Key:A
Q59mathematicsFunctions, Limits & Calculus• Class 12
EasyMCQ
The value of the limit L = lim_{x ⟶ 0} (1 - cos 2x) / (x sin x) is equal to:
A
2
B
1
C
4
D
1/2
Official Key:A
Q60mathematicsMatrices & Determinants• Class 12
EasyMCQ
If the system of equations x + y + z = 6, x + 2y + 3z = 10, and x + 2y + λ z = μ has INFINITELY MANY solutions, then (λ, μ) is:
A
(3, 10)
B
(3, 8)
C
(2, 10)
D
(2, 6)
Official Key:A
Q61mathematicsTrigonometric Functions & Equations• Class 12
EasyMCQ
The value of tan⁻¹(1/2) + tan⁻¹(1/3) is equal to:
A
π / 4
B
π / 2
C
π / 3
D
π / 6
Official Key:A
Q62mathematicsCoordinate Geometry & Conic Sections• Class 11
EasyMCQ
The eccentricity of the hyperbola x² / 16 - y² / 9 = 1 is:
A
5 / 4
B
4 / 3
C
5 / 3
D
√7 / 4
Official Key:A
Q63mathematicsFunctions, Limits & Calculus• Class 12
EasyMCQ
If f(x) = |x - 1| + |x - 2|, the number of points where f(x) is NOT differentiable is:
A
2 (at x = 1 and x = 2)
B
1
C
0
D
Infinitely many
Official Key:A
Q64mathematicsBinomial Theorem & Permutations• Class 11
ModerateMCQ
The number of all 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 without repetition which are DIVISIBLE BY 4 is:
A
24
B
32
C
16
D
48
Official Key:A
Q65mathematicsProbability & Statistics• Class 11
ModerateMCQ
If the mean of 5 observations is 5 and their variance is 9.2, and three of the observations are 1, 3, and 8, then the other two observations are:
A
4 and 9
B
5 and 8
C
6 and 7
D
2 and 11
Official Key:A
Q66mathematicsTrigonometric Functions & Equations• Class 11
EasyMCQ
The maximum value of f(x) = 3 sin x + 4 cos x + 5 for all x ∈ ℝ is:
A
10
B
12
C
7
D
5
Official Key:A
Q67mathematicsComplex Numbers & Quadratic Equations• Class 11
EasyMCQ
If z is a complex number such that |z - 2| = |z + 2i|, the locus of z in the complex plane is the straight line:
A
x + y = 0
B
x - y = 0
C
x + 2y = 0
D
2x - y = 0
Official Key:A
Q68mathematicsDefinite Integration & Area Under Curves• Class 12
EasyMCQ
The value of the definite integral I = ∫[0 to 1] x (1 - x)⁹ dx is equal to:
A
1 / 110
B
1 / 90
C
1 / 100
D
1 / 120
Official Key:A
Q69mathematicsFunctions, Limits & Calculus• Class 12
EasyMCQ
The derivative of tan⁻¹((2x) / (1 - x²)) with respect to sin⁻¹((2x) / (1 + x²)) for x ∈ (-1, 1) is equal to:
A
1
B
2
C
1 / 2
D
1 / (1 + x²)
Official Key:A
Q70mathematicsCoordinate Geometry & Conic Sections• Class 11
EasyMCQ
The radius of the circle passing through the points (0, 0), (0, 4), and (6, 0) is:
A
√13
B
5
C
√52
D
2√13
Official Key:A
Q71mathematicsDefinite Integration & Area Under Curves• Class 12
ModerateNumerical
If the value of the definite integral I = ∫[0 to π] (x sin x) / (1 + cos² x) dx is equal to π² / k, where k is a positive integer, then the value of k is equal to:
Official Key:4
Q72mathematicsDefinite Integration & Area Under Curves• Class 12
ModerateNumerical
The area (in square units) bounded between the parabola y² = 4x and the line y = 2x - 4 is equal to:
Official Key:9
Q73mathematicsBinomial Theorem & Permutations• Class 11
EasyNumerical
The remainder when 7¹⁰³ is divided by 25 is equal to:
Official Key:18
Q74mathematicsFunctions, Limits & Calculus• Class 12
ModerateNumerical
Evaluate the limit: L = lim_{x ⟶ 0} (cos(sin x) - cos x) / x⁴. The value of 12 L is equal to:
Official Key:2
Q75mathematicsVector Algebra & 3D Geometry• Class 12
ModerateNumerical
If the lines (x - 1)/2 = (y + 1)/3 = (z - 1)/4 and (x - 3)/1 = (y - k)/2 = z/1 intersect at a unique point in 3D space, then the value of k is equal to:
Official Key:4