JEE Main 2024 question paper pdf with solutions- Download Jan 30 Shift 1 Mathematics Question Paper pdf

1. A line passing through the point A(9,0) makes an angle of 30° with the positive direction of the x-axis. If this line is rotated about A through an angle of 15° clockwise, then its equation in the new position is:

(1) y√3 - 2 + x = 9
(2) x√3 - 2 + y = 9
(3) x√3 + 2 + y = 9
(4) y√3 + 2 + x = 9

2. Let S_n denote the sum of the first n terms in an arithmetic progression. If S₂₀ = 790 and S₁₀ = 145, then S₁₅ - S₅ is:

(1) 395
(2) 390
(3) 405
(4) 410

3. If z = x + iy, xy = 0, satisfies the equation z² + iz = 0, then |z|² is equal to:
 

(1) 9
(2) 1
(3) 4
(4) 1/4

4. Let a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k be two vectors such that |a| = 1, a × b = 2, and |b| = 4. If c = 2(a × b) - 3b, then the angle between b and c is equal to:

(1) cos^-1(2/√3)
(2) cos^-1(-1/√3)
(3) cos^-1(-√3/2)
(4) cos^-1(2/3)

5. The maximum area of a triangle whose one vertex is at (0,0) and the other two vertices lie on the curve y = -2x² + 54 at points (x, y) and (-x, y) where y > 0 is:

(1) 88
(2) 122
(3) 92
(4) 108

6. The value of lim_n→∞ (Σ_k=1ⁿ n³ / (n² + k²)(n² + 3k²)) is:

(1) (2√3 + 3)π / 24
(2) 13π / (8(4√3 + 3))
(3) 13(2√3 - 3)π / 8
(4) π / (8(2√3 + 3))

7. Let g: R→R be a non-constant twice differentiable function such that g' (1/2) = g' (3/2). If a real-valued function f is defined as f(x) = 1/2 [g(x) + g(2 - x)], then:

(1) f''(x) = 0 for at least two x in (0,2)
(2) f''(x) = 0 for exactly one x in (0,1)
(3) f''(x) = 0 for no x in (0,1)
(4) f' (3/2) + f' (1/2) = 1

8. The area (in square units) of the region bounded by the parabola y² = 4(x - 2) and the line y = 2x - 8 is:

(1) 8
(2) 9
(3) 6
(4) 7

9. Let y = y(x) be the solution of the differential equation sec(x) dy + {2(1 - x) tan(x) + x(2 - x)} dx = 0 such that y(0) = 2. Then y(2) is equal to:

(1) 2
(2) 2{1 - sin(2)}
(3) 2{sin(2) + 1}
(4) 1

10. Let (α, β, γ) be the foot of the perpendicular from the point (1,2,3) on the line x/5 = (y - 1)/2 = (z + 4)/3. Then 19(α + β + γ) is equal to:

(1) 102
(2) 101
(3) 99
(4) 100

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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:

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

(1) (2√3+3)π/24
(2) 13π/8(4√3+3)
(3) 13(2√3−3)π/8
(4) π/8(2√3+3)

29. If the function f(x) = { 1/|x| , |x| ≥ 2
                                            ax² + 2b , |x| < 2 }

is differentiable on R, then 48(a + b) is equal to: