CAIE P3 (Pure Mathematics 3) 2010 June

1 Solve the equation $$\frac { 2 ^ { x } + 1 } { 2 ^ { x } - 1 } = 5$$ giving your answer correct to 3 significant figures.

2 Show that \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x = \pi ^ { 2 } - 4\).

3 It is given that \(\cos a = \frac { 3 } { 5 }\), where \(0 ^ { \circ } < a < 90 ^ { \circ }\). Showing your working and without using a calculator to evaluate \(a\),

1. find the exact value of \(\sin \left( a - 30 ^ { \circ } \right)\),
2. find the exact value of \(\tan 2 a\), and hence find the exact value of \(\tan 3 a\).

4 \includegraphics[max width=\textwidth, alt={}, center]{20de5ba6-9426-4431-99af-6e8e62607f3e-2_513_895_1055_625} The diagram shows the curve \(y = \frac { \sin x } { x }\) for \(0 < x \leqslant 2 \pi\), and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation $$x = \tan x$$
2. The iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } \right) + \pi$$ can be used to determine the \(x\)-coordinate of \(M\). Use this formula to determine the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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

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

6 The equation of a curve is $$x \ln y = 2 x + 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { y } { x ^ { 2 } }\).
2. Find the equation of the tangent to the curve at the point where \(y = 1\), giving your answer in the form \(a x + b y + c = 0\).

7 The variables \(x\) and \(t\) are related by the differential equation $$\mathrm { e } ^ { 2 t } \frac { \mathrm {~d} x } { \mathrm {~d} t } = \cos ^ { 2 } x$$ where \(t \geqslant 0\). When \(t = 0 , x = 0\).

1. Solve the differential equation, obtaining an expression for \(x\) in terms of \(t\).
2. State what happens to the value of \(x\) when \(t\) becomes very large.
3. Explain why \(x\) increases as \(t\) increases.

8 The variable complex number \(z\) is given by $$z = 1 + \cos 2 \theta + i \sin 2 \theta$$ where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
2. Prove that the real part of \(\frac { 1 } { z }\) is constant.

9 The plane \(p\) has equation \(3 x + 2 y + 4 z = 13\). A second plane \(q\) is perpendicular to \(p\) and has equation \(a x + y + z = 4\), where \(a\) is a constant.

1. Find the value of \(a\).
2. The line with equation \(\mathbf { r } = \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) meets the plane \(p\) at the point \(A\) and the plane \(q\) at the point \(B\). Find the length of \(A B\).

10

1. Find the values of the constants \(A , B , C\) and \(D\) such that $$\frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \equiv A + \frac { B } { x } + \frac { C } { x ^ { 2 } } + \frac { D } { 2 x - 1 }$$
2. Hence show that $$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \mathrm { d } x = \frac { 3 } { 2 } + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 27 } \right)$$