CAIE P3 (Pure Mathematics 3) 2014 June

1 Solve the equation \(\log _ { 10 } ( x + 9 ) = 2 + \log _ { 10 } x\).

2 Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.

3

1. Show that the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ can be written in the form $$2 \tan ^ { 2 } x + ( \sqrt { } 3 ) \tan x - 1 = 0$$
2. Hence solve the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

4 The equation \(x = \frac { 10 } { \mathrm { e } ^ { 2 x } - 1 }\) has one positive real root, denoted by \(\alpha\).

1. Show that \(\alpha\) lies between \(x = 1\) and \(x = 2\).
2. Show that if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 1 + \frac { 10 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
3. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 The variables \(x\) and \(\theta\) satisfy the differential equation $$2 \cos ^ { 2 } \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = \sqrt { } ( 2 x + 1 )$$ and \(x = 0\) when \(\theta = \frac { 1 } { 4 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\theta\).

6
\includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_405_914_260_612} The diagram shows the curve \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 \left( x ^ { 2 } - y ^ { 2 } \right)\) and one of its maximum points \(M\). Find the coordinates of \(M\).

7

1. The complex number \(\frac { 3 - 5 \mathrm { i } } { 1 + 4 \mathrm { i } }\) is denoted by \(u\). Showing your working, express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 2 - \mathrm { i } | \leqslant 1\) and \(| z - \mathrm { i } | \leqslant | z - 2 |\).
   2. Calculate the maximum value of \(\arg z\) for points lying in the shaded region.

8 Let \(f ( x ) = \frac { 6 + 6 x } { ( 2 - x ) \left( 2 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B x + C } { 2 + x ^ { 2 } }\).
2. Show that \(\int _ { - 1 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 3 \ln 3\).

9
\includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_639_387_1749_879} The diagram shows the curve \(y = \mathrm { e } ^ { 2 \sin x } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Using the substitution \(u = \sin x\), find the exact value of the area of the shaded region bounded by the curve and the axes.
2. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } + \lambda ( 3 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\) and the plane \(p\) has equation \(2 x + 3 y - 5 z = 18\).

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