CAIE P2 (Pure Mathematics 2) 2021 June

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\includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-08_451_1086_260_525} The diagram shows the curve with equation $$y = ( \ln x ) ^ { 2 } - 2 \ln x$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a minimum point \(M\).

1. Find the exact value of the gradient of the curve at each of the points \(A\) and \(B\).
2. Find the exact \(x\)-coordinate of \(M\).

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\includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-10_465_785_260_680} The diagram shows the curve with parametric equations $$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$ for \(0 \leqslant t \leqslant 1\). The point \(P\) on the curve has parameter \(p\) and \(y\)-coordinate 3 .

1. Show that \(p = \frac { 1 } { 2 \sin 2 p }\).
2. Show by calculation that the value of \(p\) lies between 0.5 and 0.6 .
3. Use an iterative formula, based on the equation in part (a), to find the value of \(p\) correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
4. Find the gradient of the curve at \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.