CAIE P2 (Pure Mathematics 2) 2002 June

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

2 The cubic polynomial \(3 x ^ { 3 } + a x ^ { 2 } - 2 x - 8\) is denoted by \(\mathrm { f } ( x )\).

1. Given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { f } ( x )\) completely.

3 Two variable quantities \(x\) and \(y\) are related by the equation $$y = A x ^ { n }$$ where \(A\) and \(n\) are constants.
\includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-2_422_697_977_740} When a graph is plotted showing values of \(\ln y\) on the vertical axis and values of \(\ln x\) on the horizontal axis, the points lie on a straight line. This line crosses the vertical axis at the point ( \(0,2.3\) ) and also passes through the point (4.0,1.7), as shown in the diagram. Find the values of \(A\) and \(n\).

4

1. Express \(3 \cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 1 decimal place.
2. Solve the equation $$3 \cos \theta + 2 \sin \theta = 3.5$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
3. The graph of \(y = 3 \cos \theta + 2 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), has one stationary point. State the coordinates of this point.
   \includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-3_421_823_299_662} The diagram shows the curve \(y = 2 x \mathrm { e } ^ { - x }\) and its maximum point \(P\). Each of the two points \(Q\) and \(R\) on the curve has \(y\)-coordinate equal to \(\frac { 1 } { 2 }\).

1. Find the exact coordinates of \(P\).
2. Show that the \(x\)-coordinates of \(Q\) and \(R\) satisfy the equation $$x = \frac { 1 } { 4 } e ^ { x } .$$
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 4 } e ^ { x _ { n } }$$ with initial value \(x _ { 1 } = 0\), to find the \(x\)-coordinate of \(Q\) correct to 2 decimal places, showing the value of each approximation that you calculate.

6

1. 1. Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
   2. By using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 2 } x \mathrm {~d} x\).
2. 1. Use the trapezium rule with 2 intervals to estimate the value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec x d x\), giving your answer correct to 2 significant figures.
   2. Determine, by sketching the appropriate part of the graph of \(y = \sec x\), whether the trapezium rule gives an under-estimate or an over-estimate of the true value.

7 The parametric equations of a curve are $$x = t + 2 \ln t , \quad y = 2 t - \ln t$$ where \(t\) takes all positive values.

1. Express \(\frac { d y } { d x }\) in terms of \(t\).
2. Find the equation of the tangent to the curve at the point where \(t = 1\).
3. The curve has one stationary point. Show that the \(y\)-coordinate of this point is \(1 + \ln 2\) and determine whether this point is a maximum or a minimum.