CAIE P2 (Pure Mathematics 2) 2013 June

1 Solve the equation \(\left| 2 ^ { x } - 7 \right| = 1\), giving answers correct to 2 decimal places where appropriate.

2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).

3

1. Show that \(12 \sin ^ { 2 } x \cos ^ { 2 } x \equiv \frac { 3 } { 2 } ( 1 - \cos 4 x )\).
2. Hence show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 12 \sin ^ { 2 } x \cos ^ { 2 } x d x = \frac { \pi } { 8 } + \frac { 3 \sqrt { } 3 } { 16 }$$

4 The polynomial \(a x ^ { 3 } - 5 x ^ { 2 } + b x + 9\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 3 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 8 .

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

5 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t }$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }\).
2. Find the equation of the normal to the curve at the point where \(t = 0\).

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 4 x - 2$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
3. Show that this root also satisfies the equation $$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
4. Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. Express \(5 \sin 2 \theta + 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta + \alpha )\), 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. Hence
2. solve the equation $$5 \sin 2 \theta + 2 \cos 2 \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
3. determine the least value of \(\frac { 1 } { ( 10 \sin 2 \theta + 4 \cos 2 \theta ) ^ { 2 } }\) as \(\theta\) varies.