JEE Advanced 2015 · Paper 2

For any integer k, let αₖ = cos(kπ/7) + i sin(kπ/7), where i = √−1. The value of the expression (∑ₖ₌₁¹² |αₖ₊₁ − αₖ|)/(∑ₖ₌₁³ |α₄ₖ₋₁ − α₄ₖ₋₂|) is

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

The coefficient of x⁹ in the expansion of (1 + x)(1 + x²)(1 + x³) … (1 + x¹⁰⁰) is

Suppose that the foci of the ellipse x²/9 + y²/5 = 1 are (f₁, 0) and (f₂, 0) where f₁ > 0 and f₂ < 0. Let P₁ and P₂ be two parabolas with a common vertex at (0, 0) and with foci at (f₁, 0) and (2f₂, 0), respectively. Let T₁ be a tangent to P₁ which passes through (2f₂, 0) and T₂ be a tangent to P₂ which passes through (f₁, 0). If m₁ is the slope of T₁ and m₂ is the slope of T₂, then the value of (1/m₁² + m₂²) is

Let m and n be two positive integers greater than 1. If limₐ→₀ ((eᶜᵒˢ⁽ᵅⁿ⁾ − e)/αᵐ) = −(e/2), then the value of m/n is

If α = ∫₀¹ (e⁹ˣ⁺³ᵗᵃⁿ⁻¹ˣ)(12 + 9x²)/(1 + x²) dx where tan⁻¹x takes only principal values, then the value of (logₑ|1 + α| − 3π/4) is

Let f : ℝ → ℝ be a continuous odd function, which vanishes exactly at one point and f(1) = 1/2. Suppose that F(x) = ∫₋₁ˣ f(t) dt for all x ∈ [−1, 2] and G(x) = ∫₋₁ˣ |t| |f(f(t))| dt for all x ∈ [−1, 2]. If limₓ→₁ F(x)/G(x) = 1/14, then the value of f(1/2) is

Suppose that p⃗, q⃗ and r⃗ are three non-coplanar vectors in ℝ³. Let the components of a vector s⃗ along p⃗, q⃗ and r⃗ be 4, 3 and 5, respectively. If the components of this vector s⃗ along (−p⃗ + q⃗ + r⃗), (p⃗ − q⃗ + r⃗) and (−p⃗ − q⃗ + r⃗) are x, y and z, respectively, then the value of 2x + y + z is