CAIE P2 (Pure Mathematics 2) 2005 June

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

2

1. Use logarithms to solve the equation \(3 ^ { X } = 8\), giving your answer correct to 2 decimal places.
2. It is given that $$\ln z = \ln ( y + 2 ) - 2 \ln y$$ where \(y > 0\). Express \(z\) in terms of \(y\) in a form not involving logarithms.

3 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 2 } { x _ { n } ^ { 3 } }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use this iteration to calculate \(\alpha\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
2. State an equation which is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

4 The polynomial \(x ^ { 3 } - x ^ { 2 } + a x + b\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .

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

5

1. By differentiating \(\frac { 1 } { \cos \theta }\), show that if \(y = \sec \theta\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta\).
2. The parametric equations of a curve are $$x = 1 + \tan \theta , \quad y = \sec \theta$$ for \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin \theta\).
3. Find the coordinates of the point on the curve at which the gradient of the curve is \(\frac { 1 } { 2 }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{08210e25-0f0e-405b-b72d-1bf989689b0a-3_641_865_264_641} The diagram shows the part of the curve \(y = \frac { \ln x } { x }\) for \(0 < x \leqslant 4\). The curve cuts the \(x\)-axis at \(A\) and its maximum point is \(M\).

1. Write down the coordinates of \(A\).
2. Show that the \(x\)-coordinate of \(M\) is e, and write down the \(y\)-coordinate of \(M\) in terms of e.
3. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 1 } ^ { 4 } \frac { \ln x } { x } \mathrm {~d} x$$ correct to 2 decimal places.
4. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (iii).

7

1. By expanding \(\sin ( 2 x + x )\) and using double-angle formulae, show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x d x = \frac { 5 } { 24 }$$