CAIE P2 (Pure Mathematics 2) 2018 November

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\includegraphics[max width=\textwidth, alt={}, center]{1b410c91-2fe9-46cf-8478-631b4165f98d-06_652_789_260_676} The diagram shows the curve with equation $$y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$ The curve crosses the \(x\)-axis at points with coordinates \(( \alpha , 0 )\) and \(( \beta , 0 )\).

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of $$x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$
2. Show that \(\beta\) satisfies an equation of the form \(x = \sqrt [ 3 ] { } ( p + q x )\), and state the values of \(p\) and \(q\). [3]
3. Use an iterative formula based on the equation in part (ii) to find the value of \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

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\includegraphics[max width=\textwidth, alt={}, center]{1b410c91-2fe9-46cf-8478-631b4165f98d-10_351_488_264_826} The diagram shows the curve with equation \(y = \sqrt { } \left( 1 + 3 \cos ^ { 2 } \left( \frac { 1 } { 2 } x \right) \right)\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).

1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures.
2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced.

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\includegraphics[max width=\textwidth, alt={}, center]{1b410c91-2fe9-46cf-8478-631b4165f98d-12_424_488_260_826} The diagram shows the curve with equation \(y = \sin 2 x + 3 \cos 2 x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3 .

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. By first expressing \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in the form \(R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 significant figures.
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