CAIE P2 (Pure Mathematics 2) 2018 November

1 Solve the inequality \(| 3 x - 5 | < 2 | x |\).

2 Given that \(9 ^ { x } + 3 ^ { x } = 240\), find the value of \(3 ^ { x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.

3
\includegraphics[max width=\textwidth, alt={}, center]{cc7e798e-0817-405c-bae0-b24b9f451fbf-04_378_486_260_826} The diagram shows the curve with equation $$y = 5 \sin 2 x - 3 \tan 2 x$$ for values of \(x\) such that \(0 \leqslant x < \frac { 1 } { 4 } \pi\). Find the \(x\)-coordinate of the stationary point \(M\), giving your answer correct to 3 significant figures.

4 Find the gradient of the curve $$4 x + 3 y \mathrm { e } ^ { 2 x } + y ^ { 2 } = 10$$ at the point \(( 0,2 )\).

5 The curve with equation $$y = 5 \mathrm { e } ^ { 2 x } - 8 x ^ { 2 } - 20$$ crosses the \(x\)-axis at only one point. This point has coordinates \(( p , 0 )\).

1. Show that \(p\) satisfies the equation \(x = \frac { 1 } { 2 } \ln \left( 1.6 x ^ { 2 } + 4 \right)\).
2. Show by calculation that \(0.75 < p < 0.85\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.
4. Find the gradient of the curve at the point \(( p , 0 )\).

6

1. Show that \(\int _ { 1 } ^ { 6 } \frac { 12 } { 3 x + 2 } \mathrm {~d} x = \ln 256\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 8 \sin ^ { 2 } x + \tan ^ { 2 } 2 x \right) \mathrm { d } x\), showing all necessary working.

7

1. Use the factor theorem to show that ( \(2 x + 3\) ) is a factor of $$8 x ^ { 3 } + 4 x ^ { 2 } - 10 x + 3$$
2. Show that the equation \(2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }\) can be expressed as $$8 \cos ^ { 3 } \theta + 4 \cos ^ { 2 } \theta - 10 \cos \theta + 3 = 0 .$$
3. Solve the equation \(2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.