Sketch the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), and state the coordinates of any point where the curve crosses an axis.
Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), the axes, and the line \(x = 2\).
The point \(P\) on the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) has \(y\)-coordinate equal to \(\frac { 1 } { 6 }\). Prove that the \(x\)-coordinate of \(P\) may be written as
$$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$
\section*{6.}
In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by
$$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$
respectively.
Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
Hence find, in the form \(x + \mathrm { i } y\), the complex number representing the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).