4. The curve \(C\) has parametric equations
$$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\)
The line \(l\) is the normal to \(C\) at \(P\). - Show that an equation for \(l\) is
$$2 x - 2 \sqrt { 3 } y - 1 = 0$$
The line \(l\) intersects the curve \(C\) again at the point \(Q\).
- Find the exact coordinates of \(Q\).
You must show clearly how you obtained your answers.
[0pt]
[BLANK PAGE] - On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation
$$\arg ( z + \mathrm { i } ) = \frac { \pi } { 6 }$$
[2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{1d67c98c-e81c-4967-8a0b-a78afd95a0aa-12_1307_1351_516_463} - \(\quad z _ { 1 }\) is a point on \(L\) such that \(| z |\) is a minimum.
Find the exact value of \(z _ { 1 }\) in the form \(a + b \mathrm { i }\)
[0pt]
[4 marks]
[0pt]
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