CAIE P2 (Pure Mathematics 2) 2016 November

1 Solve the equation \(| 0.4 x - 0.8 | = 2\).

2

1. Given that \(\frac { 1 + 4 ^ { y } } { 3 + 2 ^ { y } } = 5\), find the value of \(2 ^ { y }\).
2. Use logarithms to find the value of \(y\) correct to 3 significant figures.

3 The definite integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } \left( 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3 \right) \mathrm { d } x\).

1. Show that \(I = 8 \mathrm { e } - 2\).
2. Sketch the curve \(y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3\) for \(0 \leqslant x \leqslant 2\).
3. State whether an estimate of \(I\) obtained by using the trapezium rule will be more than or less than \(8 \mathrm { e } - 2\). Justify your answer.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + a x + 4$$ where \(a\) is a constant.

1. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) for all values of \(a\).
2. Given that the remainder is - 42 when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ), find the value of \(a\).
3. When \(a\) has the value found in part (ii), factorise \(\mathrm { p } \left( x ^ { 2 } \right)\) completely.

5 \includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-2_556_844_1731_648} The diagram shows the curve \(y = \frac { 4 \ln x } { x ^ { 2 } + 1 }\) and its stationary point \(M\). The \(x\)-coordinate of \(M\) is \(m\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m = \mathrm { e } ^ { 0.5 \left( 1 + m ^ { - 2 } \right) }\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(m\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

6

1. Show that \(\frac { \cos 2 \theta } { 1 + \cos 2 \theta } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \theta\).
2. Solve the equation \(\frac { \cos 2 \alpha } { 1 + \cos 2 \alpha } = 13 + 5 \tan \alpha\) for \(0 < \alpha < \pi\).
3. Find the exact value of \(\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 + \cos x } \mathrm {~d} x\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-3_533_698_735_717} The diagram shows the curve with parametric equations $$x = 4 \sin \theta , \quad y = 1 + 3 \cos \left( \theta + \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant \theta < 2 \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(k ( 1 + ( \sqrt { } 3 ) \tan \theta )\) where the exact value of \(k\) is to be determined.
2. Find the equation of the normal to the curve at the point where the curve crosses the positive \(y\)-axis. Give your answer in the form \(y = m x + c\), where the constants \(m\) and \(c\) are exact.