Edexcel CP1 (Core Pure 1) 2020 June

1. $$f ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
2. find the value of \(p\) and the value of \(q\).

1. (a) Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x\) is an improper integral.
   (b) Prove that

$$\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations $$\begin{array} { l l } C _ { 1 } : r = ( 1 + \sin \theta ) & 0 \leqslant \theta < 2 \pi
C _ { 2 } : r = 3 ( 1 - \sin \theta ) & 0 \leqslant \theta < 2 \pi \end{array}$$ The region \(R\) lies inside \(C _ { 1 }\) and outside \(C _ { 2 }\) and is shown shaded in Figure 1.
Show that the area of \(R\) is $$p \sqrt { 3 } - q \pi$$ where \(p\) and \(q\) are integers to be determined.

1. The plane \(\Pi _ { 1 }\) has equation

$$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find a Cartesian equation for \(\Pi _ { 1 }\) The line \(l\) has equation $$\frac { x - 1 } { 5 } = \frac { y - 3 } { - 3 } = \frac { z + 2 } { 4 }$$
2. Find the coordinates of the point of intersection of \(l\) with \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r . } ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 5$$
3. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)

1. Two compounds, \(X\) and \(Y\), are involved in a chemical reaction. The amounts in grams of these compounds, \(t\) minutes after the reaction starts, are \(x\) and \(y\) respectively and are modelled by the differential equations

$$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = - 5 x + 10 y - 30
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = - 2 x + 3 y - 4 \end{aligned}$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 50$$
2. Find, according to the model, a general solution for the amount in grams of compound \(X\) present at time \(t\) minutes.
3. Find, according to the model, a general solution for the amount in grams of compound \(Y\) present at time \(t\) minutes. Given that \(x = 2\) and \(y = 5\) when \(t = 0\)
4. find
   1. the particular solution for \(x\),
   2. the particular solution for \(y\). A scientist thinks that the chemical reaction will have stopped after 8 minutes.
5. Explain whether this is supported by the model.

1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r + 2 ) = n ( n + 2 ) ( n + 3 )$$ (ii) Prove by induction that for all positive odd integers \(n\) $$f ( n ) = 4 ^ { n } + 5 ^ { n } + 6 ^ { n }$$ is divisible by 15

1. A sample of bacteria in a sealed container is being studied.

The number of bacteria, \(P\), in thousands, is modelled by the differential equation $$( 1 + t ) \frac { \mathrm { d } P } { \mathrm {~d} t } + P = t ^ { \frac { 1 } { 2 } } ( 1 + t )$$ where \(t\) is the time in hours after the start of the study.
Initially, there are exactly 5000 bacteria in the container.

1. Determine, according to the model, the number of bacteria in the container 8 hours after the start of the study.
2. Find, according to the model, the rate of change of the number of bacteria in the container 4 hours after the start of the study.
3. State a limitation of the model.