7. A curve has parametric equations
$$x = 2 \sin t , \quad y = \cos 2 t + 2 \sin t$$
for \(- \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - 2 \sin t\) and hence find the coordinates of the stationary point.
- Find the cartesian equation of the curve.
- State the set of values that \(x\) can take and hence sketch the curve.
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\section*{8. In this question you must show detailed reasoning.}
The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\).
Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).
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