CAIE P3 (Pure Mathematics 3) 2020 November

1 Solve the equation $$\ln \left( 1 + \mathrm { e } ^ { - 3 x } \right) = 2$$ Give the answer correct to 3 decimal places.

2

1. Expand \(\sqrt [ 3 ] { 1 + 6 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

3 The variables \(x\) and \(y\) satisfy the relation \(2 ^ { y } = 3 ^ { 1 - 2 x }\).

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
2. Find the exact \(x\)-coordinate of the point of intersection of this line with the line \(y = 3 x\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.

4

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

5
\includegraphics[max width=\textwidth, alt={}, center]{77a45360-8e1d-4f4f-9830-075d832a14cf-08_334_895_258_625} The diagram shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta$$ for \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(- 2 \sin \theta \cos ^ { 3 } \theta\).
   The gradient of the curve has its maximum value at the point \(P\).
2. Find the exact value of the \(x\)-coordinate of \(P\).

6 The complex number \(u\) is defined by $$u = \frac { 7 + \mathrm { i } } { 1 - \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing \(u , 7 + \mathrm { i }\) and \(1 - \mathrm { i }\) respectively.
3. By considering the arguments of \(7 + \mathrm { i }\) and \(1 - \mathrm { i }\), show that $$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \frac { 1 } { 4 } \pi$$

7 The variables \(x\) and \(t\) satisfy the differential equation $$\mathrm { e } ^ { 3 t } \frac { \mathrm {~d} x } { \mathrm {~d} t } = \cos ^ { 2 } 2 x$$ for \(t \geqslant 0\). It is given that \(x = 0\) when \(t = 0\).

1. Solve the differential equation and obtain an expression for \(x\) in terms of \(t\).
2. State what happens to the value of \(x\) when \(t\) tends to infinity.

8 With respect to the origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2
1
5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 4
- 1
1 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 1
1
2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { l } 3
2
3 \end{array} \right)$$

1. Show that \(A B = 2 C D\).
2. Find the angle between the directions of \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\).
3. Show that the line through \(A\) and \(B\) does not intersect the line through \(C\) and \(D\).

9 Let \(\mathrm { f } ( x ) = \frac { 7 x + 18 } { ( 3 x + 2 ) \left( x ^ { 2 } + 4 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find the exact value of \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).

10
\includegraphics[max width=\textwidth, alt={}, center]{77a45360-8e1d-4f4f-9830-075d832a14cf-18_549_933_260_605} The diagram shows the curve \(y = \sqrt { x } \cos x\), for \(0 \leqslant x \leqslant \frac { 3 } { 2 } \pi\), and its minimum point \(M\), where \(x = a\). The shaded region between the curve and the \(x\)-axis is denoted by \(R\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 1 } { 2 a }\).
2. The sequence of values given by the iterative formula \(a _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 2 a _ { n } } \right)\), with initial value \(x _ { 1 } = 3\), converges to \(a\). Use this formula to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Find the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.