JEE Main 2025 · 2 Apr 2025 Shift 1

Let the focal chord PQ of the parabola y² = 4x make an angle of 60° with the positive x-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y-axis at the point (0, α), then 5α² is equal to :

1. A15
2. B20
3. C25
4. D30

Let z be a complex number such that |z| = 1. If (2 + k²z)/(k + z̄) = kz, k ∈ R, then the maximum distance of k + ik² from the circle |z − (1 + 2i)| = 1 is :

1. A√3 + 1
2. B2
3. C√5 + 1
4. D3

If θ ∈ [−2π, 2π], then the number of solutions of 2√2 cos² θ + (2 − √6)cos θ − √3 = 0, is equal to :

1. A8
2. B6
3. C10
4. D12

If S and S′ are the foci of the ellipse x²/18 + y²/9 = 1 and P be a point on the ellipse, then min(SP·S′P) + max(SP·S′P) is equal to :

1. A3(1 + √2)
2. B3(6 + √2)
3. C9
4. D27

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of the △BCD is equal to :

1. A7√3
2. B12
3. C√110
4. D√340

The largest n ∈ N such that 3ⁿ divides 50! is :

1. A23
2. B22
3. C21
4. D20

For α, β, γ ∈ R, if limₓ→₀ (x² sin αx + (γ − 1)eˣ²)/(sin 2x − βx) = 3, then β + γ − α is equal to :

1. A−1
2. B4
3. C6
4. D7

If the function f(x) = 2x³ − 9ax² + 12a²x + 1, where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that p² = q, then f(3) is equal to :

1. A37
2. B55
3. C10
4. D23