Standard +0.8 This is a multi-part Further Maths question requiring: (1) standard roots of unity work with exact cartesian forms, (2) a 3×3 determinant calculation involving ω (moderately computational), (3) complex number addition in polar form requiring careful angle/magnitude work, and (4) applying De Moivre's theorem to find cube roots. While each component uses standard FP1 techniques, the combination of computational intensity, multiple steps, and the need to work fluently with roots of unity properties makes this notably harder than average A-level questions, though still within expected FP1 scope.
Show the cube roots of 1 on an Argand diagram.
Show that the two non-real cube roots can be expressed in the form \(\omega\) and \(\omega ^ { 2 }\), and find these cube roots in exact cartesian form \(x + i y\).
Evaluate the determinant
$$\left| \begin{array} { c c c }
1 & 3 \omega & 2 \omega ^ { 2 } \\
3 \omega ^ { 2 } & 2 & \omega \\
2 \omega & \omega ^ { 2 } & 3
\end{array} \right|$$
It is given that \(z = ( 4 \sqrt { } 3 ) \left( \cos \frac { 4 } { 3 } \pi + i \sin \frac { 4 } { 3 } \pi \right) - 4 \left( \cos \frac { 11 } { 6 } \pi + i \sin \frac { 11 } { 6 } \pi \right)\). Express \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), giving exact values for \(r\) and \(\theta\).
Hence find the cube roots of \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
Show the cube roots of 1 on an Argand diagram.
Show that the two non-real cube roots can be expressed in the form $\omega$ and $\omega ^ { 2 }$, and find these cube roots in exact cartesian form $x + i y$.
Evaluate the determinant
$$\left| \begin{array} { c c c }
1 & 3 \omega & 2 \omega ^ { 2 } \\
3 \omega ^ { 2 } & 2 & \omega \\
2 \omega & \omega ^ { 2 } & 3
\end{array} \right|$$
It is given that $z = ( 4 \sqrt { } 3 ) \left( \cos \frac { 4 } { 3 } \pi + i \sin \frac { 4 } { 3 } \pi \right) - 4 \left( \cos \frac { 11 } { 6 } \pi + i \sin \frac { 11 } { 6 } \pi \right)$. Express $z$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, giving exact values for $r$ and $\theta$.
Hence find the cube roots of $z$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$.
\hfill \mbox{\textit{CAIE FP1 2013 Q11 OR}}