8. (a) Use de Moivre's theorem to show that
$$\cos ^ { 5 } \theta \equiv p \cos 5 \theta + q \cos 3 \theta + r \cos \theta$$
where \(p , q\) and \(r\) are rational numbers to be found.
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Question 8(a) — Way 1:
Answer Marks
Guidance
Answer/Working Mark
Guidance
\(\left(z + \frac{1}{z}\right)^5 = z^5 + 5z^3 + 10z + \frac{10}{z} + \frac{5}{z^3} + \frac{1}{z^5}\) M1A1
M1: Attempt to expand \(\left(z \pm \frac{1}{z}\right)^5\); A1: Correct expansion with correct powers of \(z\)
\(z = \cos\theta + i\sin\theta \Rightarrow z + \frac{1}{z} = 2\cos\theta\) B1
May be implied
\(z^5 + \frac{1}{z^5} + 5\left(z^3 + \frac{1}{z^3}\right) + 10\left(z + \frac{1}{z}\right) = 2\cos5\theta + 10\cos3\theta + 20\cos\theta\) M1
Uses at least one of \(z^5 + \frac{1}{z^5} = 2\cos5\theta\) or \(z^3 + \frac{1}{z^3} = 2\cos3\theta\)
\(\left(z + \frac{1}{z}\right)^5 = 32\cos^5\theta\) B1
\(\cos^5\theta = \frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta\) A1
Correct expression
Question 8(a) — Way 2 (Using \(e^{i\theta}\)):
Answer Marks
Guidance
Answer/Working Mark
Guidance
\(\left(e^{i\theta} + e^{-i\theta}\right)^5 = e^{5i\theta} + 5e^{3i\theta} + 10e^{i\theta} + 10e^{-i\theta} + 5e^{-3i\theta} + e^{-5i\theta}\) M1A1
M1: Attempt to expand \(\left(e^{i\theta} \pm e^{-i\theta}\right)^5\); A1: Correct expansion
\(2\cos\theta = e^{i\theta} + e^{-i\theta}\) B1
May be implied
Uses one of \(e^{5i\theta} + e^{-5i\theta} = 2\cos5\theta\) or \(e^{3i\theta} + e^{-3i\theta} = 2\cos3\theta\) M1
\(\left(e^{i\theta} + e^{-i\theta}\right)^5 = 32\cos^5\theta\) B1
\(\cos^5\theta = \frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta\) A1
Correct expression
Question 8(a) — Way 3 (De Moivre on \(\cos5\theta\)):
Answer Marks
Guidance
Answer/Working Mark
Guidance
\((\cos\theta + i\sin\theta)^5 = c^5 + 5ic^4s + 10c^3i^2s^2 + 10c^2i^3s^3 + 5ci^4s^4 + i^5s^5\) M1A1
M1: Attempt to expand; A1: Correct real terms (may include \(i\)'s)
\(\cos5\theta = \cos^5\theta - 10\cos^3\theta\sin^2\theta + 5\cos\theta\sin^4\theta\) B1
Correct real terms with no \(i\)'s
\(= \cos^5\theta - 10\cos^3\theta(1-\cos^2\theta) + 5\cos\theta(1-\cos^2\theta)^2\) M1
Uses \(\sin^2\theta = 1 - \cos^2\theta\) to eliminate \(\sin\theta\)
\(16\cos^5\theta = \cos5\theta + 20\cos^3\theta - 5\cos\theta\) \(\cos3\theta = 4\cos^3\theta - 3\cos\theta\) B1
Correct identity for \(\cos3\theta\)
\(16\cos^5\theta = \cos5\theta + 5\cos3\theta + 10\cos\theta\) \(\cos^5\theta = \frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta\) A1
Correct expression
Question 8(b):
Answer Marks
Guidance
Answer/Working Mark
Guidance
\(\int\left(\frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta\right)d\theta = \frac{1}{80}\sin5\theta + \frac{5}{48}\sin3\theta + \frac{5}{8}\sin\theta\) M1A1ft
M1: Evidence of \(\cos n\theta \to \pm\frac{1}{n}\sin n\theta\) where \(n=5\) or \(3\); A1ft: Correct integration (ft their \(p,q,r\))
\(\left[\frac{1}{80}\sin5\theta + \frac{5}{48}\sin3\theta + \frac{5}{8}\sin\theta\right]_{\pi/6}^{\pi/3}\) evaluated with correct subtraction M1
Must show evidence of substitution of \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\) for at least 2 terms
\(= \frac{49\sqrt{3}}{160} - \frac{203}{480}\) A1
Allow exact equivalents e.g. \(\frac{1}{16}\left(4.9\sqrt{3} - \frac{203}{30}\right)\)
*Note: If \(p\), \(q\), \(r\) or their values used even from no working, M marks available but not A marks.*
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# Question 8(a) — Way 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(z + \frac{1}{z}\right)^5 = z^5 + 5z^3 + 10z + \frac{10}{z} + \frac{5}{z^3} + \frac{1}{z^5}$ | M1A1 | M1: Attempt to expand $\left(z \pm \frac{1}{z}\right)^5$; A1: Correct expansion with correct powers of $z$ |
| $z = \cos\theta + i\sin\theta \Rightarrow z + \frac{1}{z} = 2\cos\theta$ | B1 | May be implied |
| $z^5 + \frac{1}{z^5} + 5\left(z^3 + \frac{1}{z^3}\right) + 10\left(z + \frac{1}{z}\right) = 2\cos5\theta + 10\cos3\theta + 20\cos\theta$ | M1 | Uses at least one of $z^5 + \frac{1}{z^5} = 2\cos5\theta$ or $z^3 + \frac{1}{z^3} = 2\cos3\theta$ |
| $\left(z + \frac{1}{z}\right)^5 = 32\cos^5\theta$ | B1 | |
| $\cos^5\theta = \frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta$ | A1 | Correct expression |
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# Question 8(a) — Way 2 (Using $e^{i\theta}$):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(e^{i\theta} + e^{-i\theta}\right)^5 = e^{5i\theta} + 5e^{3i\theta} + 10e^{i\theta} + 10e^{-i\theta} + 5e^{-3i\theta} + e^{-5i\theta}$ | M1A1 | M1: Attempt to expand $\left(e^{i\theta} \pm e^{-i\theta}\right)^5$; A1: Correct expansion |
| $2\cos\theta = e^{i\theta} + e^{-i\theta}$ | B1 | May be implied |
| Uses one of $e^{5i\theta} + e^{-5i\theta} = 2\cos5\theta$ or $e^{3i\theta} + e^{-3i\theta} = 2\cos3\theta$ | M1 | |
| $\left(e^{i\theta} + e^{-i\theta}\right)^5 = 32\cos^5\theta$ | B1 | |
| $\cos^5\theta = \frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta$ | A1 | Correct expression |
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# Question 8(a) — Way 3 (De Moivre on $\cos5\theta$):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\cos\theta + i\sin\theta)^5 = c^5 + 5ic^4s + 10c^3i^2s^2 + 10c^2i^3s^3 + 5ci^4s^4 + i^5s^5$ | M1A1 | M1: Attempt to expand; A1: Correct real terms (may include $i$'s) |
| $\cos5\theta = \cos^5\theta - 10\cos^3\theta\sin^2\theta + 5\cos\theta\sin^4\theta$ | B1 | Correct real terms with no $i$'s |
| $= \cos^5\theta - 10\cos^3\theta(1-\cos^2\theta) + 5\cos\theta(1-\cos^2\theta)^2$ | M1 | Uses $\sin^2\theta = 1 - \cos^2\theta$ to eliminate $\sin\theta$ |
| $16\cos^5\theta = \cos5\theta + 20\cos^3\theta - 5\cos\theta$ | | |
| $\cos3\theta = 4\cos^3\theta - 3\cos\theta$ | B1 | Correct identity for $\cos3\theta$ |
| $16\cos^5\theta = \cos5\theta + 5\cos3\theta + 10\cos\theta$ | | |
| $\cos^5\theta = \frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta$ | A1 | Correct expression |
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# Question 8(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\left(\frac{1}{16}\cos5\theta + \frac{5}{16}\cos3\theta + \frac{5}{8}\cos\theta\right)d\theta = \frac{1}{80}\sin5\theta + \frac{5}{48}\sin3\theta + \frac{5}{8}\sin\theta$ | M1A1ft | M1: Evidence of $\cos n\theta \to \pm\frac{1}{n}\sin n\theta$ where $n=5$ or $3$; A1ft: Correct integration (ft their $p,q,r$) |
| $\left[\frac{1}{80}\sin5\theta + \frac{5}{48}\sin3\theta + \frac{5}{8}\sin\theta\right]_{\pi/6}^{\pi/3}$ evaluated with correct subtraction | M1 | Must show evidence of substitution of $\frac{\pi}{3}$ and $\frac{\pi}{6}$ for at least 2 terms |
| $= \frac{49\sqrt{3}}{160} - \frac{203}{480}$ | A1 | Allow exact equivalents e.g. $\frac{1}{16}\left(4.9\sqrt{3} - \frac{203}{30}\right)$ |
*Note: If $p$, $q$, $r$ or their values used even from no working, M marks available but not A marks.*
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8. (a) Use de Moivre's theorem to show that
$$\cos ^ { 5 } \theta \equiv p \cos 5 \theta + q \cos 3 \theta + r \cos \theta$$
where $p , q$ and $r$ are rational numbers to be found.\\
(b) Hence, showing all your working, find the exact value of
$$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \cos ^ { 5 } \theta \mathrm {~d} \theta$$
\hfill \mbox{\textit{Edexcel F2 2016 Q8 [10]}}