The complex number \(u\) is given by
$$u = \frac { \left( \cos \frac { 1 } { 7 } \pi + i \sin \frac { 1 } { 7 } \pi \right) ^ { 4 } } { \cos \frac { 1 } { 7 } \pi - i \sin \frac { 1 } { 7 } \pi }$$
Find the exact value of \(\arg u\).
The complex numbers \(u\) and \(u ^ { * }\) are plotted on an Argand diagram.
Describe the single geometrical transformation that maps \(u\) onto \(u ^ { * }\) and state the exact value of \(\arg u ^ { * }\).
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\includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-07_588_869_255_603}
The variables \(x\) and \(y\) satisfy the equation \(a y = b ^ { x }\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0.50,2.24\) ) and ( \(3.40,8.27\) ), as shown in the diagram.
Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure.