Standard +0.8 This is a standard Further Maths FP2 question requiring de Moivre's theorem application and binomial expansion, followed by algebraic manipulation to solve an equation. While it involves multiple steps and careful algebra, it follows a well-established template that FP2 students practice extensively. The connection between parts (a) and (b) is straightforward once you recognize the factor of 2.
A1: 2 correct answers. A1: 3rd correct answer with no extras within range, ignore extras outside range. Must be radians. Degrees or decimals score A0A0
# Question 4:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(\cos\theta + i\sin\theta)^6 = c^6 + 6c^5is + 15c^4i^2s^2 + 20c^3i^3s^3 + 15c^2i^4s^4 + 6ci^5s^5 + i^6s^6$ | M1 | Attempt to expand correctly or show real terms. Often seen with powers of $i$ simplified |
| $\cos 6\theta = c^6 - 15c^4s^2 + 15c^2s^4 - s^6$ | M1A1 | M1: Attempt to identify real parts. A1: Correct expression |
| $= c^6 - 15c^4(1-c^2) + 15c^2(1-c^2)^2 - (1-c^2)^3$ | M1 | Correct use of $s^2 = 1 - c^2$ in all sine terms |
| $\cos 6\theta = 32\cos^6\theta - 48\cos^4\theta + 18\cos^2\theta - 1$ | A1cso (5) | $\cos 6\theta$ must be seen somewhere |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $64\cos^6\theta - 96\cos^4\theta + 36\cos^2\theta - 3 = 0 \Rightarrow 2\cos 6\theta - 1 = 0$ | M1A1 | M1: Uses part (a) to obtain equation in $\cos 6\theta$. A1: Correct underlined equation |
| $\cos 6\theta = \frac{1}{2} \Rightarrow 6\theta = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3}$ | M1 | Valid attempt to solve $\cos 6\theta = k,\ -1 \leq k \leq 1$ leading to $\theta = \ldots$ Can be degrees |
| $\theta = \frac{\pi}{18},\ \frac{5\pi}{18},\ \frac{7\pi}{18}$ | A1A1 (5) | A1: 2 correct answers. A1: 3rd correct answer with no extras within range, ignore extras outside range. Must be radians. Degrees or decimals score A0A0 |
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