JEE Advanced 2012 · Paper 2

The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x − y + z = 3 and at a distance 2/√3 from the point (3, 1, −1) is

1. A5x − 11y + z = 17
2. B√2x + y = 3√2 − 1
3. Cx + y + z = √3
4. Dx − √2y = 1 − √2

Let PQR be a triangle of area Δ with a = 2, b = 7/2 and c = 5/2, where a, b and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively. Then (2 sin P − sin 2P)/(2 sin P + sin 2P) equals

1. A3/(4Δ)
2. B45/(4Δ)
3. C(3/(4Δ))²
4. D(45/(4Δ))²

If a⃗ and b⃗ are vectors such that |a⃗ + b⃗| = √29 and a⃗ × (2î + 3ĵ + 4k̂) = (2î + 3ĵ + 4k̂) × b⃗, then a possible value of (a⃗ + b⃗)·(−7î + 2ĵ + 3k̂) is

1. A0
2. B3
3. C4
4. D8

If P is a 3 × 3 matrix such that Pᵀ = 2P + I, where Pᵀ is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix X = [x y z]ᵀ ≠ [0 0 0]ᵀ such that

1. APX = [0 0 0]ᵀ
2. BPX = X
3. CPX = 2X
4. DPX = −X

Let α(a) and β(a) be the roots of the equation (∛(1 + a) − 1)x² + (√(1 + a) − 1)x + (⁶√(1 + a) − 1) = 0 where a > −1. Then limₐ→₀⁺ α(a) and limₐ→₀⁺ β(a) are

1. A−5/2 and 1
2. B−1/2 and −1
3. C−7/2 and 2
4. D−9/2 and 3

Four fair dice D₁, D₂, D₃ and D₄, each having six faces numbered 1, 2, 3, 4, 5 and 6, are rolled simultaneously. The probability that D₄ shows a number appearing on one of D₁, D₂ and D₃ is

1. A91/216
2. B108/216
3. C125/216
4. D127/216

The value of the integral ∫₋π/₂^π/₂ (x² + ln((π + x)/(π − x))) cos x dx is

1. A0
2. Bπ²/2 − 4
3. Cπ²/2 + 4
4. Dπ²/2

Let a₁, a₂, a₃, ... be in harmonic progression with a₁ = 5 and a₂₀ = 25. The least positive integer n for which aₙ < 0 is

1. A22
2. B23
3. C24
4. D25