A-LevelFurther MathematicsComplex numbers9231/12 · Oct/Nov 2013 · Q1114 Marks

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.

(a)State the fifth roots of unity in the form cos θ + i sin θ, where –π < θ ≤ π.[2]
(b)Simplify (x − [cos(π/5) + i sin(π/5)]) (x − [cos(π/5) − i sin(π/5)]).[2]
(c)Hence find the real factors of x⁵ - 1.[2]
(d)Express the six roots of the equation x⁶ - x³ + 1 = 0 as three conjugate pairs, in the form cos θ ± i sin θ.[6]
(e)Hence find the real factors of x⁶ - x³ + 1.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

[B1B1] Obtains all fifth roots. z = cos(2kπ/5) + i sin(2kπ/5), k = 0, ±1, ±2.
[M1A1] Simplifies expression. x² - 2 cos(2π/5) x + 1
[M1A1] Obtains factors. (x² - 2 cos(2π/5) + 1) (x² - 2 cos(4π/5) + 1) (x - 1)
[M1A1] Solves quadratic in x³. x³ = 1/2 ± i√3/2 = cos(π/3) ± i sin(π/3)
[A1] Expresses them in polar form. x = cos(π/9) ± i sin(π/9), cos(7π/9) ± i sin(7π/9), cos(13π/9) ± i sin(13π/9)
[M1A1] Finds factors. (x² - 2 cos(π/9) x + 1) (x² - 2 cos(7π/9) x + 1) (x² - 2 cos(13π/9) x + 1)

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