CAIE P2 (Pure Mathematics 2) 2009 November

1 Solve the inequality \(| x + 3 | > | 2 x |\).

2 It is given that \(\ln ( y + 5 ) - \ln y = 2 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.

3

1. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sec x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
2. Using a sketch of the graph of \(y = \sec x\) for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\), explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i).

4 The parametric equations of a curve are $$x = 1 - \mathrm { e } ^ { - t } , \quad y = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 t } - 1\).
2. Hence find the exact value of \(t\) at the point on the curve at which the gradient is 2 .

5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } - 5 x + 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) and \(( x - 2 )\) are factors of \(\mathrm { p } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the other linear factor of \(\mathrm { p } ( x )\).

6

1. Express \(3 \cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$3 \cos x + 4 \sin x = 4.5$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{729aa2f6-2b62-445f-a2aa-a63b45cb6b64-3_604_971_262_587} The diagram shows the curve \(y = x ^ { 2 } \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Show by differentiation that the \(x\)-coordinate of \(M\) satisfies the equation $$\tan x = \frac { 2 } { x }$$
2. Verify by calculation that this equation has a root (in radians) between 1 and 1.2.
3. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \sin 2 x + \sec ^ { 2 } x \right) \mathrm { d } x\).
2. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 1 } { 2 x } + \frac { 1 } { x + 1 } \right) \mathrm { d } x = \ln 5\).