12 The Argand diagram shows the solutions to the equation \(z ^ { 5 } = 1\)
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-22_1079_995_354_520}
12
- Solve the equation
$$z ^ { 5 } = 1$$
giving your answers in the form \(z = \cos \theta + \mathrm { i } \sin \theta\), where \(0 \leq \theta < 2 \pi\)
[0pt]
[2 marks]
12 - Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon.
[0pt]
[2 marks]
\includegraphics[max width=\textwidth, alt={}]{a889963c-266c-497e-b7fc-99a249ba9e58-23_2484_1726_219_141}
12 - The Argand diagram on page 22 is repeated below.
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-24_1079_1000_354_520}
Explain, with reference to the Argand diagram, why the expression
$$16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1$$
has a repeated quadratic factor.
12 - \(O\) is the centre of a regular pentagon \(A B C D E\) such that \(O A = O B = O C = O D =\) \(O E = 1\) unit.
The distance from \(O\) to \(A B\) is \(h\)
By solving the equation \(16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1 = 0\), show that
$$h = \frac { \sqrt { 5 } + 1 } { 4 }$$
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-26_2492_1721_217_150}