JEE Main 2025 · 23 Jan 2025 Shift 2

Let A = {(x, y) ∈ R × R : |x + y| ≥ 3} and B = {(x, y) ∈ R × R : |x| + |y| ≤ 3}. If C = {(x, y) ∈ A ∩ B : x = 0 or y = 0}, then ∑₍ₓ,ᵧ₎∈C |x + y| is :

1. A24
2. B18
3. C15
4. D12

Let X = R × R. Define a relation R on X as : (a₁, b₁) R (a₂, b₂) ⇔ b₁ = b₂. Statement I : R is an equivalence relation. Statement II : For some (a, b) ∈ X, the set S = {(x, y) ∈ X : (x, y) R (a, b)} represents a line parallel to y = x. In the light of the above statements, choose the correct answer from the options given below :

1. ABoth Statement I and Statement II are true
2. BBoth Statement I and Statement II are false
3. CStatement I is true but Statement II is false
4. DStatement I is false but Statement II is true

The number of complex numbers z, satisfying |z| = 1 and |z/z̄ + z̄/z| = 1, is :

1. A4
2. B6
3. C8
4. D10

The system of equations x + y + z = 6, x + 2y + 5z = 9, x + 5y + λz = μ, has no solution if

1. Aλ = 17, μ ≠ 18
2. Bλ = 15, μ ≠ 17
3. Cλ ≠ 17, μ ≠ 18
4. Dλ = 17, μ = 18

Let A = [aᵢⱼ] be a 3 × 3 matrix such that A [0, 1, 0]ᵀ = [0, 0, 1]ᵀ, A [4, 1, 3]ᵀ = [0, 1, 0]ᵀ and A [2, 1, 2]ᵀ = [1, 0, 0]ᵀ, then a₂₃ equals :

1. A0
2. B1
3. C−1
4. D2

If in the expansion of (1 + x)ᵖ (1 − x)ᑫ, the coefficients of x and x² are 1 and −2, respectively, then p² + q² is equal to :

1. A8
2. B13
3. C18
4. D20

A rod of length eight units moves such that its ends A and B always lie on the lines x − y + 2 = 0 and y + 2 = 0, respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2 : 1 is 9(x² + αy² + βxy + γx + 28y) − 76 = 0, then α − β − γ is equal to :

1. A21
2. B22
3. C23
4. D24

The length of the chord of the ellipse x²/4 + y²/2 = 1, whose mid-point is (1, 1/2), is :

1. A√15
2. B(1/3)√15
3. C(5/3)√15
4. D(2/3)√15