CAIE P3 (Pure Mathematics 3) 2019 March

1

1. Show that the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\) can be written as a quadratic equation in \(x\).
2. Hence solve the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\), giving your answer correct to 3 significant figures.

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 6 } + 12 x _ { n } } { 3 x _ { n } ^ { 5 } + 8 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use the formula to calculate \(\alpha\) correct to 4 decimal places. Give the result of each iteration to 6 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

3

1. Given that \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\), find the exact value of \(\tan \theta\) in a form involving surds. You need not simplify your answer.
2. Hence solve the equation \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

4 Show that \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 3 } { 2 } } \ln x \mathrm {~d} x = 2 - \ln 4\).

5 The variables \(x\) and \(y\) satisfy the relation \(\sin y = \tan x\), where \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\). Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \cos x \sqrt { } ( \cos 2 x ) } .$$

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k y ^ { 3 } \mathrm { e } ^ { - x }$$ where \(k\) is a constant. It is given that \(y = 1\) when \(x = 0\), and that \(y = \sqrt { } \mathrm { e }\) when \(x = 1\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

7

1. Showing all working and without using a calculator, solve the equation $$( 1 + \mathrm { i } ) z ^ { 2 } - ( 4 + 3 \mathrm { i } ) z + 5 + \mathrm { i } = 0$$ Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. The complex number \(u\) is given by $$u = - 1 - \mathrm { i }$$ On a sketch of an Argand diagram show the point representing \(u\). Shade the region whose points represent complex numbers satisfying the inequalities \(| z | < | z - 2 \mathrm { i } |\) and \(\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi\).

8 Let \(\mathrm { f } ( x ) = \frac { 12 + 12 x - 4 x ^ { 2 } } { ( 2 + x ) ( 3 - 2 x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

9 Two planes have equations \(2 x + 3 y - z = 1\) and \(x - 2 y + z = 3\).

1. Find the acute angle between the planes.
2. Find a vector equation for the line of intersection of the planes.

10
\includegraphics[max width=\textwidth, alt={}, center]{dcfbe7af-c212-42b1-8a90-8e0418cf0ffd-16_330_689_264_726} The diagram shows the curve \(y = \sin ^ { 3 } x \sqrt { } ( \cos x )\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
2. Showing all your working, find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.