CAIE P2 (Pure Mathematics 2) 2017 June

1 Given that \(5 ^ { x } = 3 ^ { 4 y }\), use logarithms to show that \(y = m x\) and find the value of the constant \(m\) correct to 3 significant figures.

2 Solve the inequality \(| 4 - x | \leqslant | 3 - 2 x |\).

3 Given that \(\int _ { 0 } ^ { a } 4 \mathrm { e } ^ { \frac { 1 } { 2 } x + 3 } \mathrm {~d} x = 835\), find the value of the constant \(a\) correct to 3 significant figures. [5]

4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + x _ { n } + 9 } { \left( x _ { n } + 1 \right) ^ { 2 } }$$ with \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Find the value of \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. Determine the exact value of \(\alpha\).

5

1. Express \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta = 1\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-06_561_542_260_799} The diagram shows the curve \(y = \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).

1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region, giving your answer correct to 3 significant figures.
2. Find the exact volume of the solid formed when the shaded region is rotated completely about the \(x\)-axis.

7 The parametric equations of a curve are $$x = t ^ { 3 } + 6 t + 1 , \quad y = t ^ { 4 } - 2 t ^ { 3 } + 4 t ^ { 2 } - 12 t + 5$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and use division to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be written in the form \(a t + b\), where \(a\) and \(b\) are constants to be found.
2. The straight line \(x - 2 y + 9 = 0\) is the normal to the curve at the point \(P\). Find the coordinates of \(P\).

8
\includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-10_643_414_260_863} The diagram shows the curve with equation $$y = 3 x ^ { 2 } \ln \left( \frac { 1 } { 6 } x \right) .$$ The curve crosses the \(x\)-axis at the point \(P\) and has a minimum point \(M\).

1. Find the gradient of the curve at the point \(P\).
2. Find the exact coordinates of the point \(M\).