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\includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-10_702_597_258_772}
The diagram shows the curves \(y = 4 \cos \frac { 1 } { 2 } x\) and \(y = \frac { 1 } { 4 - x }\), for \(0 \leqslant x < 4\). When \(x = a\), the tangents to the curves are perpendicular.
- Show that \(a = 4 - \sqrt { } \left( 2 \sin \frac { 1 } { 2 } a \right)\).
- Verify by calculation that \(a\) lies between 2 and 3 .
- Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.