CAIE P3 (Pure Mathematics 3) 2023 June

1 Solve the equation \(\ln ( x + 5 ) = 5 + \ln x\). Give your answer correct to 3 decimal places.

2 Find the quotient and remainder when \(2 x ^ { 4 } - 27\) is divided by \(x ^ { 2 } + x + 3\).

3 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3\) and \(| z | \geqslant | z - 4 \mathrm { i } |\).

4 The parametric equations of a curve are $$x = \frac { \cos \theta } { 2 - \sin \theta } , \quad y = \theta + 2 \cos \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 - \sin \theta ) ^ { 2 }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{72042f09-3495-42e9-bee9-96ec5ac0bf0c-06_352_643_274_744} The diagram shows the part of the curve \(y = x ^ { 2 } \cos 3 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\), and its maximum point \(M\), where \(x = a\).

1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { 2 } { 3 a } \right)\).
2. Use an iterative formula based on the equation in (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. Express \(3 \cos x + 2 \cos \left( x - 60 ^ { \circ } \right)\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$3 \cos 2 \theta + 2 \cos \left( 2 \theta - 60 ^ { \circ } \right) = 2.5$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

7

1. Use the substitution \(u = \cos x\) to show that $$\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x = \int _ { - 1 } ^ { 1 } 2 u \mathrm { e } ^ { 2 u } \mathrm {~d} u$$
2. Hence find the exact value of \(\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x\).

8 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } + 4 } { x ( y + 4 ) }$$ for \(x > 0\). It is given that \(x = 4\) when \(y = 2 \sqrt { 3 }\).
Solve the differential equation to obtain the value of \(x\) when \(y = 2\).

9 The lines \(l\) and \(m\) have equations $$\begin{aligned} l : & \mathbf { r } = a \mathbf { i } + 3 \mathbf { j } + b \mathbf { k } + \lambda ( c \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } )
m : & \mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ Relative to the origin \(O\), the position vector of the point \(P\) is \(4 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\).

1. Given that \(l\) is perpendicular to \(m\) and that \(P\) lies on \(l\), find the values of the constants \(a , b\) and \(c\). [4]
2. The perpendicular from \(P\) meets line \(m\) at \(Q\). The point \(R\) lies on \(P Q\) extended, with \(P Q : Q R = 2 : 3\). Find the position vector of \(R\).

10 Let \(\mathrm { f } ( x ) = \frac { 21 - 8 x - 2 x ^ { 2 } } { ( 1 + 2 x ) ( 3 - x ) ^ { 2 } }\).

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 \(x ^ { 2 }\).

11 The complex number \(z\) is defined by \(z = \frac { 5 a - 2 \mathrm { i } } { 3 + a \mathrm { i } }\), where \(a\) is an integer. It is given that \(\arg z = - \frac { 1 } { 4 } \pi\).

1. Find the value of \(a\) and hence express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Express \(z ^ { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the simplified exact values of \(r\) and \(\theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.