A-LevelFurther MathematicsComplex numbers9231/12 · May/June 2013 · Q710 Marks
By considering the binomial expansion of (z - 1/z)^6, where z = cos θ + i sin θ, express sin⁶ θ in the form 1/32 (p + q cos 2θ + r cos 4θ + s cos 6θ), where p, q, r and s are integers to be determined. Hence find the exact value of ∫₀^(π/4) sin⁶ θ dθ.
- (a)By considering the binomial expansion of (z - 1/z)^6, where z = cos θ + i sin θ, express sin⁶ θ in the form 1/32 (p + q cos 2θ + r cos 4θ + s cos 6θ), where p, q, r and s are integers to be determined.[6]
- (b)Hence find the exact value of ∫₀^(π/4) sin⁶ θ dθ.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.
- M1A1 Expands and groups. (z - z⁻¹)^6 = (z⁶ + z⁻⁶) - 6(z⁴ + z⁻⁴) + 15(z² + z⁻²) - 20
- M1 Use of z - z⁻¹ and z + z⁻¹. Correctly. (2i sin θ)⁶ = 2cos 6θ -12cos4θ + 30 cos 2θ – 20
- A1A1 Obtains result. sin⁶ θ = 1/32 (10 - 15 cos 2θ + 6 cos 4θ – cos 6θ)
- A1 (Allow p = 10, q = -15, r = 6, s = -1)
- M1A1 Integrates correctly. [1/32 (10θ - 15/2 sin 2θ + 6/4 sin 4θ - 1/6 sin 6θ)]₀^(π/4)
- M1A1 Inserts limits and evaluates. (15π - 44)/192 or 5π/64 - 11/48
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