AQA FP2 2012 January — Question 5 7 marks

Exam BoardAQA
ModuleFP2 (Further Pure Mathematics 2)
Year2012
SessionJanuary
Marks7
PaperDownload PDF ↗
TopicComplex numbers 2
TypeExpress roots in trigonometric form
DifficultyStandard +0.8 This requires applying De Moivre's theorem to both numerator and denominator, converting the denominator's negative angle, setting the resulting argument equal to π/2 (for i), and solving a Diophantine equation with LCM considerations. While the techniques are standard FP2 content, finding the smallest positive integers requires systematic algebraic manipulation and number theory insight beyond routine application.
Spec4.02d Exponential form: re^(i*theta)4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)
5 Find the smallest positive integer values of \(p\) and \(q\) for which $$\frac { \left( \cos \frac { \pi } { 8 } + \mathrm { i } \sin \frac { \pi } { 8 } \right) ^ { p } } { \left( \cos \frac { \pi } { 12 } - \mathrm { i } \sin \frac { \pi } { 12 } \right) ^ { q } } = \mathrm { i }$$ 
5 Find the smallest positive integer values of $p$ and $q$ for which

$$\frac { \left( \cos \frac { \pi } { 8 } + \mathrm { i } \sin \frac { \pi } { 8 } \right) ^ { p } } { \left( \cos \frac { \pi } { 12 } - \mathrm { i } \sin \frac { \pi } { 12 } \right) ^ { q } } = \mathrm { i }$$

\hfill \mbox{\textit{AQA FP2 2012 Q5 [7]}}
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Q1 8 Q2 8 Q3 12 Q4 6 Q5 7 Q6 8 Q7 12 Q8 14