AQA FP2 2014 June — Question 6 12 marks

Exam BoardAQA
ModuleFP2 (Further Pure Mathematics 2)
Year2014
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeIntegration using De Moivre identities
DifficultyChallenging +1.2 This is a structured multi-part FP2 question that guides students through standard De Moivre applications. Parts (a) and (b)(i) are routine bookwork requiring direct application of known identities. Part (b)(ii) involves algebraic manipulation but follows naturally from (b)(i). Part (c) requires recognizing a trigonometric substitution and applying previous results, which is more challenging but still follows a well-signposted path. The question is harder than typical A-level due to being Further Maths content, but it's a standard FP2 exercise without requiring novel insight.
Spec1.08h Integration by substitution4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae
6
    1. Use De Moivre's Theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\), then $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$
    2. Write down a similar expression for \(z ^ { n } + \frac { 1 } { z ^ { n } }\).
    1. Expand \(\left( z - \frac { 1 } { z } \right) ^ { 2 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\) in terms of \(z\).
    2. Hence show that $$8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = A + B \cos 4 \theta$$ where \(A\) and \(B\) are integers.
  3. Hence, by means of the substitution \(x = 2 \sin \theta\), find the exact value of $$\int _ { 1 } ^ { 2 } x ^ { 2 } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x$$ \includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-14_1180_1707_1525_153}
(a)(i) Use De Moivre's Theorem to show that if \(z = \cos y + i\sin y\), then \(z^n - \frac{1}{z^n} = 2i\sin ny\)
[3 marks]
(ii) Write down a similar expression for \(z^n + \frac{1}{z^n}\).
[1 mark]
(b)(i) Expand \(\left(z - \frac{1}{z}\right)^2\left(z + \frac{1}{z}\right)^2\) in terms of \(z\).
[1 mark]
(ii) Hence show that \(8\sin^2 y \cos^2 y = A + B\cos 4y\) where \(A\) and \(B\) are integers.
[2 marks]
(c) Hence, by means of the substitution \(x = 2\sin y\), find the exact value of \(\int_1^2 \sqrt{x^2(4-x^2)} \, dx\)
[5 marks]
(a)(i) Use De Moivre's Theorem to show that if $z = \cos y + i\sin y$, then $z^n - \frac{1}{z^n} = 2i\sin ny$
[3 marks]

(ii) Write down a similar expression for $z^n + \frac{1}{z^n}$.
[1 mark]

(b)(i) Expand $\left(z - \frac{1}{z}\right)^2\left(z + \frac{1}{z}\right)^2$ in terms of $z$.
[1 mark]

(ii) Hence show that $8\sin^2 y \cos^2 y = A + B\cos 4y$ where $A$ and $B$ are integers.
[2 marks]

(c) Hence, by means of the substitution $x = 2\sin y$, find the exact value of $\int_1^2 \sqrt{x^2(4-x^2)} \, dx$
[5 marks]
6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use De Moivre's Theorem to show that if $z = \cos \theta + \mathrm { i } \sin \theta$, then

$$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$
\item Write down a similar expression for $z ^ { n } + \frac { 1 } { z ^ { n } }$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Expand $\left( z - \frac { 1 } { z } \right) ^ { 2 } \left( z + \frac { 1 } { z } \right) ^ { 2 }$ in terms of $z$.
\item Hence show that

$$8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = A + B \cos 4 \theta$$

where $A$ and $B$ are integers.
\end{enumerate}\item Hence, by means of the substitution $x = 2 \sin \theta$, find the exact value of

$$\int _ { 1 } ^ { 2 } x ^ { 2 } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x$$

\includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-14_1180_1707_1525_153}
\end{enumerate}

\hfill \mbox{\textit{AQA FP2 2014 Q6 [12]}}
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Q1 7 Q2 8 Q3 7 Q4 14 Q5 9 Q6 12 Q7 7 Q8 11