CAIE P3 (Pure Mathematics 3) 2023 March

1 It is given that \(x = \ln ( 2 y - 3 ) - \ln ( y + 4 )\).
Express \(y\) in terms of \(x\).

2

1. On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(- \frac { 1 } { 3 } \pi \leqslant \arg ( z - 1 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 3 } \pi\) and \(\operatorname { Re } z \leqslant 3\).
2. Calculate the least value of \(\arg z\) for points in the region from (a). Give your answer in radians correct to 3 decimal places.

3 The polynomial \(2 x ^ { 4 } + a x ^ { 3 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - x + 1\) the remainder is \(3 x + 2\). Find the values of \(a\) and \(b\).

4 Solve the equation $$\frac { 5 z } { 1 + 2 \mathrm { i } } - z z ^ { * } + 30 + 10 \mathrm { i } = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

5 The parametric equations of a curve are $$x = t \mathrm { e } ^ { 2 t } , \quad y = t ^ { 2 } + t + 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 t }\).
2. Hence show that the normal to the curve, where \(t = - 1\), passes through the point \(\left( 0,3 - \frac { 1 } { \mathrm { e } ^ { 4 } } \right)\).

6

1. Express \(5 \sin \theta + 12 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. Hence solve the equation \(5 \sin 2 x + 12 \cos 2 x = 6\) for \(0 \leqslant x \leqslant \pi\).

7
\includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-10_627_611_255_767} The diagram shows a circle with centre \(O\) and radius \(r\). The angle of the minor sector \(A O B\) of the circle is \(x\) radians. The area of the major sector of the circle is 3 times the area of the shaded region.

1. Show that \(x = \frac { 3 } { 4 } \sin x + \frac { 1 } { 2 } \pi\).
2. Show by calculation that the root of the equation in (a) lies between 2 and 2.5.
3. Use an iterative formula based on the equation in (a) to calculate this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8
\includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-12_437_686_274_719} The diagram shows the curve \(y = x ^ { 3 } \ln x\), for \(x > 0\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { 1 } { 2 }\). [5]

9 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 3 y } \sin ^ { 2 } 2 x$$ It is given that \(y = 0\) when \(x = 0\).
Solve the differential equation and find the value of \(y\) when \(x = \frac { 1 } { 2 }\).

10 With respect to the origin \(O\), the points \(A , B , C\) and \(D\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
- 1
2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 1
2
- 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 1
- 2
5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 5
- 6

11 \end{array} \right)$$

1. Find the obtuse angle between the vectors \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\).
   The line \(l\) passes through the points \(A\) and \(B\).
2. Find a vector equation for the line \(l\).
3. Find the position vector of the point of intersection of the line \(l\) and the line passing through \(C\) and \(D\).
   11 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + x + 11 } { \left( 4 + x ^ { 2 } \right) ( 1 + x ) }\).
4. Express \(\mathrm { f } ( x )\) in partial fractions.
5. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln 54 - \frac { 1 } { 8 } \pi\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.