Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form cos θ + i sin θ, where –π < θ ≤ π. Simplify (x − [cos(π/5) + i sin(π/5)]) (x − [cos(π/5) − i sin(π/5)]). Hence find the real factors of x⁵ - 1. Express the six roots of the equation x⁶ - x³ + 1 = 0 as three conjugate pairs, in the form cos θ ± i sin θ. Hence find the real factors of x⁶ - x³ + 1.

Use de Moivre's theorem to show that tan 5θ = (5t - 10t³ + t⁵) / (1 - 10t² + 5t⁴), where t = tan θ. Deduce that the r...

By considering ∑_{r=1}^{n} z²ʳ⁻¹, where z = cos θ + i sin θ, show that, if sin θ ≠ 0, ∑_{r=1}^{n} sin (2r − 1)θ = sin...

Use de Moivre's theorem to express cot 7θ in terms of cot θ. Use the equation cot 7θ = 0 to show that the roots of th...

The equation x³ + x − 1 = 0 has roots α, β, γ. Use the relation x = √y to show that the equation y³ + 2y² + y − 1 = 0...

The lines l₁ and l₂ have vector equations r = 4i – 2j + λ(2i + j – 4k) and r = 4i – 5j + 2k + μ(i – j – k) respectively.

The matrix A is given by A = [[4, 1, -1], [-4, -1, 4], [0, -1, 5]].