Mathematics Verified Solutions
JEE Main 2025 — 04 April 2025 (Mathematics) Solutions
Step-by-step Mathematics explanations, formulas used, alternative methods, and practice mode for 04 April 2025 (Shift 1).
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Mathematics Questions (25 Questions)
Practice Mode InteractiveQ51mathematicsVector 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