Edexcel CP1 (Core Pure 1) 2024 June

1. $$\mathrm { f } ( z ) = z ^ { 4 } - 6 z ^ { 3 } + a z ^ { 2 } + b z + 145$$ where \(a\) and \(b\) are real constants.
Given that \(2 + 5 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
2. Show all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. The roots of the equation

$$2 x ^ { 3 } - 3 x ^ { 2 } + 12 x + 7 = 0$$ are \(\alpha , \beta\) and \(\gamma\)
Without solving the equation,

1. write down the value of each of $$\alpha + \beta + \gamma \quad \alpha \beta + \alpha \gamma + \beta \gamma \quad \alpha \beta \gamma$$
2. Use the answers to part (a) to determine the value of
   1. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma }\)
   2. \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)
   3. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} Figure 1 shows the design for a bathing pool.
The pool, \(P\), shown unshaded in Figure 1, is surrounded by a tiled area, \(T\), shown shaded in Figure 1. The tiled area is bounded by the edge of the pool and by a circle, \(C\), with radius 6 m .
The centre of the pool and the centre of the circle are the same point.
The edge of the pool is modelled by the curve with polar equation $$r = 4 - a \sin 3 \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ where \(a\) is a positive constant.
Given that the shortest distance between the edge of the pool and the circle \(C\) is 0.5 m ,

1. determine the value of \(a\).
2. Hence, using algebraic integration, determine, according to the model, the exact area of \(T\).

1. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.
   1. Show that
   $$z ^ { n } + \frac { 1 } { z ^ { n } } \equiv 2 \cos n \theta$$ where \(n\) is a positive integer.
2. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
3. Hence, making your reasoning clear, determine all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)

1. A raindrop falls from rest from a cloud. The velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, of the raindrop, \(t\) seconds after the raindrop starts to fall, is modelled by the differential equation

$$( t + 2 ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 v = k ( t + 2 ) - 3 \quad t \geqslant 0$$ where \(k\) is a positive constant.

1. Solve the differential equation to show that $$v = \frac { k } { 4 } ( t + 2 ) - 1 + \frac { 4 ( 2 - k ) } { ( t + 2 ) ^ { 3 } }$$ Given that \(v = 4\) when \(t = 2\)
2. determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.
3. Comment on the validity of the model for very large values of \(t\)

1. Prove by induction that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

1. The line \(l _ { 1 }\) has equation

$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
The plane \(\Pi\) contains \(l _ { 1 }\) and \(l _ { 2 }\)

1. Show that the vector \(3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }\) is perpendicular to \(\Pi\)
2. Hence determine a Cartesian equation of \(\Pi\)
3. Hence determine the value of \(p\) Given that
   • the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\)
   • the point \(B\) has coordinates \(( 12 , - 11,6 )\)
   • determine, to the nearest degree, the acute angle between \(A B\) and \(\Pi\)

1. A scientist is studying the effect of introducing a population of type \(A\) bacteria into a population of type \(B\) bacteria.

At time \(t\) days, the number of type \(A\) bacteria, \(x\), and the number of type \(B\) bacteria, \(y\), are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x \end{aligned}$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
2. Determine a general solution for the number of type \(A\) bacteria at time \(t\) days.
3. Determine a general solution for the number of type \(B\) bacteria at time \(t\) days. The model predicts that, at time \(T\) hours, the number of bacteria in the two populations will be equal. Given that \(x = 100\) and \(y = 275\) when \(t = 0\)
4. determine the value of \(T\), giving your answer to 2 decimal places.
5. Suggest a limitation of the model.