CAIE P2 (Pure Mathematics 2) 2008 June

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

2 Use logarithms to solve the equation \(4 ^ { x } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.

3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } ( \cos 2 x + \sin x ) \mathrm { d } x\).

4 The polynomial \(2 x ^ { 3 } + 7 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, 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 5 . Find the values of \(a\) and \(b\).

5

1. Express \(5 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$5 \cos \theta - \sin \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 It is given that the curve \(y = ( x - 2 ) \mathrm { e } ^ { x }\) has one stationary point.

1. Find the exact coordinates of this point.
2. Determine whether this point is a maximum or a minimum point.

7 The equation of a curve is $$x ^ { 2 } + y ^ { 2 } - 4 x y + 3 = 0$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y - x } { y - 2 x }\).
2. Find the coordinates of each of the points on the curve where the tangent is parallel to the \(x\)-axis.

8 The constant \(a\), where \(a > 1\), is such that \(\int _ { 1 } ^ { a } \left( x + \frac { 1 } { x } \right) \mathrm { d } x = 6\).

1. Find an equation satisfied by \(a\), and show that it can be written in the form $$a = \sqrt { } ( 13 - 2 \ln a )$$
2. Verify, by calculation, that the equation \(a = \sqrt { } ( 13 - 2 \ln a )\) has a root between 3 and 3.5.
3. Use the iterative formula $$a _ { n + 1 } = \sqrt { } \left( 13 - 2 \ln a _ { n } \right)$$ with \(a _ { 1 } = 3.2\), to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.