CAIE P2 (Pure Mathematics 2) 2006 November

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

2

1. Express \(4 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\).
2. Hence find the values of \(x\) that satisfy the equation $$3 \left( 4 ^ { x } \right) - 10 \left( 2 ^ { x } \right) + 3 = 0 ,$$ giving your answers correct to 2 decimal places.

3 The polynomial \(4 x ^ { 3 } - 7 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x - 3\) ) is a factor of \(\mathrm { p } ( x )\).

1. Show that \(a = - 3\).
2. Hence, or otherwise, solve the equation \(\mathrm { p } ( x ) = 0\).

4

1. Prove the identity $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \tan 2 x .$$
2. Hence solve the equation $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) = 2 ,$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-2_396_392_1603_879} The diagram shows a chord joining two points, \(A\) and \(B\), on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of the shaded segment is one sixth of the area of the circle.

1. Show that \(\alpha\) satisfies the equation $$x = \frac { 1 } { 3 } \pi + \sin x .$$
2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \pi + \sin x _ { n } ,$$ with initial value \(x _ { 1 } = 2\), to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6
\includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-3_501_497_269_826} The diagram shows the part of the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { x }\) for \(x > 0\), and its minimum point \(M\).

1. Find the coordinates of \(M\).
2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { \mathrm { e } ^ { 2 x } } { x } \mathrm {~d} x$$ giving your answer correct to 1 decimal place.
3. 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 (ii).
4. Given that \(y = \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
5. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sec ^ { 2 } 2 x \mathrm {~d} x = \frac { 1 } { 2 } \sqrt { } 3$$ and, by using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan ^ { 2 } 2 x \mathrm {~d} x\).
6. Use the identity \(\cos 4 x \equiv 2 \cos ^ { 2 } 2 x - 1\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \frac { 1 } { 1 + \cos 4 x } \mathrm {~d} x$$