CAIE P3 (Pure Mathematics 3) 2010 June

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

2 The variables \(x\) and \(y\) satisfy the equation \(y ^ { 3 } = A \mathrm { e } ^ { 2 x }\), where \(A\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line.

1. Find the gradient of this line.
2. Given that the line intersects the axis of \(\ln y\) at the point where \(\ln y = 0.5\), find the value of \(A\) correct to 2 decimal places.

3 Solve the equation $$\tan \left( 45 ^ { \circ } - x \right) = 2 \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).

4 Given that \(x = 1\) when \(t = 0\), solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { x } - \frac { x } { 4 } ,$$ obtaining an expression for \(x ^ { 2 }\) in terms of \(t\).

5
\includegraphics[max width=\textwidth, alt={}, center]{c5ec981d-7ff7-4698-82c4-eb0506b635a3-2_515_1031_1384_555} The diagram shows the curve \(y = \mathrm { e } ^ { - x } - \mathrm { e } ^ { - 2 x }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).

1. Find the exact value of \(p\).
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac { 1 } { 8 }\).

6 The curve \(y = \frac { \ln x } { x + 1 }\) has one stationary point.

1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac { x + 1 } { \ln x }$$ and that this \(x\)-coordinate lies between 3 and 4 .
2. Use the iterative formula $$x _ { n + 1 } = \frac { x _ { n } + 1 } { \ln x _ { n } }$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. Prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
2. Using this result, find the exact value of $$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } \theta \mathrm {~d} \theta$$

8

1. The equation \(2 x ^ { 3 } - x ^ { 2 } + 2 x + 12 = 0\) has one real root and two complex roots. Showing your working, verify that \(1 + i \sqrt { } 3\) is one of the complex roots. State the other complex root.
2. On a sketch of an Argand diagram, show the point representing the complex number \(1 + \mathrm { i } \sqrt { } 3\). On the same diagram, shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 - i \sqrt { } 3 | \leqslant 1\) and \(\arg z \leqslant \frac { 1 } { 3 } \pi\).
   1. Express \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in partial fractions.
   2. Hence obtain the expansion of \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

10 The straight line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\). The plane \(p\) has equation \(3 x - y + 2 z = 9\). The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\).
2. Find the acute angle between \(l\) and \(p\).
3. Find an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = d\).