CAIE P2 (Pure Mathematics 2) 2016 June

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1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{f85c4010-17b1-441c-ae8a-e77573d1b0c3-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.

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\includegraphics[max width=\textwidth, alt={}, center]{f85c4010-17b1-441c-ae8a-e77573d1b0c3-3_423_837_1352_651} The diagram shows the curve with parametric equations $$x = 2 - \cos t , \quad y = 1 + 3 \cos 2 t$$ for \(0 < t < \pi\). The minimum point is \(M\) and the curve crosses the \(x\)-axis at points \(P\) and \(Q\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 12 \cos t\).
2. Find the coordinates of \(M\).
3. Find the gradient of the curve at \(P\) and at \(Q\).