7 By considering the binomial expansion of \(\left( z - \frac { 1 } { z } \right) ^ { 6 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(\sin ^ { 6 } \theta\) in the form
$$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$
where \(p , q , r\) and \(s\) are integers to be determined.
Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta\).
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Question 7:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\((z-z^{-1})^6 = (z^6+z^{-6})-6(z^4+z^{-4})+15(z^2+z^{-2})-20\) M1A1, M1
Expands and groups; use of \(z-z^{-1}\) and \(z+z^{-1}\); correctly
\((2i\sin\theta)^6 = 2\cos6\theta-12\cos4\theta+30\cos2\theta-20\) A1A1
Obtains result
\(\sin^6\theta = \frac{1}{32}(10-15\cos2\theta+6\cos4\theta-\cos6\theta)\) A1
(Allow \(p=10\), \(q=-15\), \(r=6\), \(s=-1\))
\(\left[\frac{5\theta}{16}-\frac{15\sin2\theta}{64}+\frac{3\sin4\theta}{64}-\frac{\sin6\theta}{192}\right]_0^{\pi/4}\) M1A1
Integrates correctly
\(\frac{5\pi}{64}-\frac{15}{64}+\frac{1}{192} = \frac{5\pi}{64}-\frac{11}{48}\) M1A1
Inserts limits and evaluates; SC: 1f power of 2 consistently wrong ¾ for 2nd part
Total: [10]
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## Question 7:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(z-z^{-1})^6 = (z^6+z^{-6})-6(z^4+z^{-4})+15(z^2+z^{-2})-20$ | M1A1, M1 | Expands and groups; use of $z-z^{-1}$ and $z+z^{-1}$; correctly |
| $(2i\sin\theta)^6 = 2\cos6\theta-12\cos4\theta+30\cos2\theta-20$ | A1A1 | Obtains result |
| $\sin^6\theta = \frac{1}{32}(10-15\cos2\theta+6\cos4\theta-\cos6\theta)$ | A1 | (Allow $p=10$, $q=-15$, $r=6$, $s=-1$) |
| $\left[\frac{5\theta}{16}-\frac{15\sin2\theta}{64}+\frac{3\sin4\theta}{64}-\frac{\sin6\theta}{192}\right]_0^{\pi/4}$ | M1A1 | Integrates correctly |
| $\frac{5\pi}{64}-\frac{15}{64}+\frac{1}{192} = \frac{5\pi}{64}-\frac{11}{48}$ | M1A1 | Inserts limits and evaluates; SC: 1f power of 2 consistently wrong ¾ for 2nd part |
**Total: [10]**
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7 By considering the binomial expansion of $\left( z - \frac { 1 } { z } \right) ^ { 6 }$, where $z = \cos \theta + \mathrm { i } \sin \theta$, express $\sin ^ { 6 } \theta$ in the form
$$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$
where $p , q , r$ and $s$ are integers to be determined.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta$.
\hfill \mbox{\textit{CAIE FP1 2013 Q7 [10]}}