| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 14 |
| Topic | Complex numbers 2 |
| Type | Integration using De Moivre identities |
| Difficulty | Challenging +1.2 This is a structured multi-part question that guides students through a standard De Moivre's theorem application. Part (i) is routine verification, part (ii) uses binomial expansion with clear scaffolding, part (iii) is direct integration after the setup, and part (iv) uses symmetry. While it requires multiple techniques (complex numbers, binomial theorem, integration, graph sketching), the question provides significant guidance at each step, making it moderately above average but not requiring novel insight. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae |
**(i)** $z^n + z^{-n} = \cos(n\theta) + i\sin(n\theta) + \cos(-n\theta) + i\sin(-n\theta)$ M1
$= \cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)$
$= 2\cos(n\theta)$ (AG) A1 **[2]**
**(ii)** $(z + z^{-1})^5 = z^5 + 5z^3 + 10z + 10z^{-1} + 5z^{-3} + z^{-5}$ M1A1
Grouping terms: $(2\cos\theta)^5 = (z^5 + z^{-5}) + 5(z^3 + z^{-3}) + 10(z + z^{-1})$ M1A1
Applying result of **(i)**: $32\cos^5\theta = 2\cos5\theta + 10\cos3\theta + 20\cos\theta$ M1
$16\cos^5\theta = \cos5\theta + 5\cos3\theta + 10\cos\theta$ (AG) **[5]**
**(iii)** $\int_0^{\frac{1}{2}\pi}\cos^5\theta\, d\theta = \frac{1}{16}\int_0^{\frac{1}{2}\pi}(\cos5\theta + 5\cos3\theta + 10\cos\theta)\, d\theta$ M1
$= \frac{1}{16}\left[\frac{1}{5}\sin5\theta + \frac{5}{3}\sin3\theta + 10\sin\theta\right]_0^{\frac{1}{2}\pi}$ M1A1
$= \frac{1}{16}\left[\frac{1}{5} - \frac{5}{3} + 10\right] = \frac{8}{15}$ A1 **[4]**
**(iv)** Sketches graphs of $f(\theta) = \cos^5\theta$ and $f(\theta) = \sin^5\theta$ for $0 \leq \theta \leq \frac{1}{2}\pi$ B1B1
States $\int_0^{\frac{1}{2}\pi}\sin^5\theta\, d\theta = \frac{8}{15}$ by symmetry depB1 **[3]**11 The complex number $z$ is defined as $z = \cos \theta + \mathrm { i } \sin \theta$.\\
(i) Show that $z ^ { n } + z ^ { - n } = 2 \cos n \theta$.\\
(ii) By expanding $\left( z + z ^ { - 1 } \right) ^ { 5 }$, show that $16 \cos ^ { 5 } \theta = \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta$.\\
(iii) Hence find $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 5 } \theta \mathrm {~d} \theta$.\\
(iv) Sketch the graphs of $\mathrm { f } ( \theta ) = \sin ^ { 5 } \theta$ and $\mathrm { f } ( \theta ) = \cos ^ { 5 } \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$, and hence give the value of
$$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 5 } \theta \mathrm {~d} \theta$$
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q11 [14]}}