CAIE P2 (Pure Mathematics 2) 2018 November

1

1. Solve the equation \(| 9 x - 2 | = | 3 x + 2 |\).
2. Hence, using logarithms, solve the equation \(\left| 3 ^ { y + 2 } - 2 \right| = \left| 3 ^ { y + 1 } + 2 \right|\), giving your answer correct to 3 significant figures.

2 Show that \(\int _ { 1 } ^ { 7 } \frac { 6 } { 2 x + 1 } \mathrm {~d} x = \ln 125\).

3 Solve the equation \(\sec ^ { 2 } \theta = 3 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-06_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.

5 A curve has parametric equations $$x = t + \ln ( t + 1 ) , \quad y = 3 t \mathrm { e } ^ { 2 t }$$

1. Find the equation of the tangent to the curve at the origin.
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

6
\includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-10_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.

7
\includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-12_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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