JEE Advanced 2013 · Paper 2

Let w = (√3 + i)/2 and P = {wⁿ : n = 1, 2, 3, …}. Further H₁ = {z ∈ C : Re z > 1/2} and H₂ = {z ∈ C : Re z < −1/2}, where C is the set of all complex numbers. If z₁ ∈ P ∩ H₁, z₂ ∈ P ∩ H₂ and O represents the origin, then ∠z₁Oz₂ =

1. Aπ/2
2. Bπ/6
3. C2π/3
4. D5π/6

If 3ˣ = 4ˣ⁻¹, then x =

1. A2 log₃ 2/(2 log₃ 2 − 1)
2. B2/(2 − log₂ 3)
3. C1/(1 − log₄ 3)
4. D2 log₂ 3/(2 log₂ 3 − 1)

Let ω be a complex cube root of unity with ω ≠ 1 and P = [pᵢⱼ] be a n × n matrix with pᵢⱼ = ωⁱ⁺ʲ. Then P² ≠ 0, when n =

1. A57
2. B55
3. C58
4. D56

The function f(x) = 2|x| + |x + 2| − ||x + 2| − 2|x|| has a local minimum or a local maximum at x =

1. A−2
2. B−2/3
3. C2
4. D2/3

For a ∈ R (the set of all real numbers), a ≠ −1, limₙ→∞ (1ᵃ + 2ᵃ + … + nᵃ)/[(n + 1)ᵃ⁻¹ ((na + 1) + (na + 2) + … + (na + n))] = 1/60. Then a =

1. A5
2. B7
3. C−15/2
4. D−17/2

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 2√7 on y-axis is (are)

1. Ax² + y² − 6x + 8y + 9 = 0
2. Bx² + y² − 6x + 7y + 9 = 0
3. Cx² + y² − 6x − 8y + 9 = 0
4. Dx² + y² − 6x − 7y + 9 = 0

Two lines L₁ : x = 5, y/(3 − α) = z/−2 and L₂ : x = α, y/−1 = z/(2 − α) are coplanar. Then α can take value(s)

1. A1
2. B2
3. C3
4. D4

In a triangle PQR, P is the largest angle and cos P = 1/3. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

1. A16
2. B18
3. C24
4. D22