CAIE P3 (Pure Mathematics 3) 2022 November

1 Solve the equation \(\ln ( 2 x - 1 ) = 2 \ln ( x + 1 ) - \ln x\). Give your answer correct to 3 decimal places.

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

3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\).

4 The parametric equations of a curve are $$x = 2 t - \tan t , \quad y = \ln ( \sin 2 t )$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).

5

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 | \leqslant 2\) and \(\operatorname { Im } z \geqslant 1\).
2. Find the greatest value of \(\arg z\) for points in the shaded region.

6 Solve the quadratic equation \(( 1 - 3 \mathrm { i } ) z ^ { 2 } - ( 2 + \mathrm { i } ) z + \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

7

1. Show that the equation \(\sqrt { 5 } \sec x + \tan x = 4\) can be expressed as \(R \cos ( x + \alpha ) = \sqrt { 5 }\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [4]
2. Hence solve the equation \(\sqrt { 5 } \sec 2 x + \tan 2 x = 4\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

8 The curve with equation \(y = \frac { x ^ { 3 } } { \mathrm { e } ^ { x } - 1 }\) has a stationary point at \(x = p\), where \(p > 0\).

1. Show that \(p = 3 \left( 1 - \mathrm { e } ^ { - p } \right)\).
2. Verify by calculation that \(p\) lies between 2.5 and 3 .
3. Use an iterative formula based on the equation in part (a) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

9 With respect to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 0
5
2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1
0
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 4
- 3
- 2 \end{array} \right)$$ The midpoint of \(A C\) is \(M\) and the point \(N\) lies on \(B C\), between \(B\) and \(C\), and is such that \(B N = 2 N C\).

1. Find the position vectors of \(M\) and \(N\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Find the position vector of the point \(Q\) where the line through \(M\) and \(N\) intersects the line through \(A\) and \(B\).

10 A gardener is filling an ornamental pool with water, using a hose that delivers 30 litres of water per minute. Initially the pool is empty. At time \(t\) minutes after filling begins the volume of water in the pool is \(V\) litres. The pool has a small leak and loses water at a rate of \(0.01 V\) litres per minute. The differential equation satisfied by \(V\) and \(t\) is of the form \(\frac { \mathrm { d } V } { \mathrm {~d} t } = a - b V\).

1. Write down the values of the constants \(a\) and \(b\).
2. Solve the differential equation and find the value of \(t\) when \(V = 1000\).
3. Obtain an expression for \(V\) in terms of \(t\) and hence state what happens to \(V\) as \(t\) becomes large.

11 Let \(\mathrm { f } ( x ) = \frac { 5 - x + 6 x ^ { 2 } } { ( 3 - x ) \left( 1 + 3 x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\), simplifying your answer.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.