Edexcel CP1 (Core Pure 1) 2019 June

1. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are real constants.
Given that \(- 1 + 2 \mathrm { i }\) and \(3 - \mathrm { i }\) are two roots 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 values of \(a , b , c\) and \(d\).

1. Show that

$$\int _ { 0 } ^ { \infty } \frac { 8 x - 12 } { \left( 2 x ^ { 2 } + 3 \right) ( x + 1 ) } \mathrm { d } x = \ln k$$ where \(k\) is a rational number to be found.

3. \begin{figure}[h] \end{figure} Diagram not to scale Figure 1 shows the design for a table top in the shape of a rectangle \(A B C D\). The length of the table, \(A B\), is 1.2 m . The area inside the closed curve is made of glass and the surrounding area, shown shaded in Figure 1, is made of wood. The perimeter of the glass is modelled by the curve with polar equation $$r = 0.4 + a \cos 2 \theta \quad 0 \leqslant \theta < 2 \pi$$ where \(a\) is a constant.

1. Show that \(a = 0.2\) Hence, given that \(A D = 60 \mathrm {~cm}\),
2. find the area of the wooden part of the table top, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

1. Prove that, for \(n \in \mathbb { Z } , n \geqslant 0\)

$$\sum _ { r = 0 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { ( n + a ) ( n + b ) } { c ( n + 2 ) ( n + 3 ) }$$ where \(a\), \(b\) and \(c\) are integers to be found.

1. A tank at a chemical plant has a capacity of 250 litres. The tank initially contains 100 litres of pure water.

Salt water enters the tank at a rate of 3 litres every minute. Each litre of salt water entering the tank contains 1 gram of salt. It is assumed that the salt water mixes instantly with the contents of the tank upon entry.
At the instant when the salt water begins to enter the tank, a valve is opened at the bottom of the tank and the solution in the tank flows out at a rate of 2 litres per minute. Given that there are \(S\) grams of salt in the tank after \(t\) minutes,

1. show that the situation can be modelled by the differential equation $$\frac { \mathrm { d } S } { \mathrm {~d} t } = 3 - \frac { 2 S } { 100 + t }$$
2. Hence find the number of grams of salt in the tank after 10 minutes. When the concentration of salt in the tank reaches 0.9 grams per litre, the valve at the bottom of the tank must be closed.
3. Find, to the nearest minute, when the valve would need to be closed.
4. Evaluate the model.

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

$$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5
(6)

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

$$\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 4 } { 3 }$$ The line \(l _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where \(t\) is a scalar parameter.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) lie in the same plane.
2. Write down a vector equation for the plane containing \(l _ { 1 }\) and \(l _ { 2 }\)
3. Find, to the nearest degree, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)

1. A scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish.
   At time \(t\) years, the number of white-clawed crayfish, \(w\), and the number of signal crayfish, \(s\), are modelled by the differential equations

$$\begin{aligned} & \frac { \mathrm { d } w } { \mathrm {~d} t } = \frac { 5 } { 2 } ( w - s )
& \frac { \mathrm { d } s } { \mathrm {~d} t } = \frac { 2 } { 5 } w - 90 \mathrm { e } ^ { - t } \end{aligned}$$

1. Show that $$2 \frac { \mathrm {~d} ^ { 2 } w } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} w } { \mathrm {~d} t } + 2 w = 450 \mathrm { e } ^ { - t }$$
2. Find a general solution for the number of white-clawed crayfish at time \(t\) years.
3. Find a general solution for the number of signal crayfish at time \(t\) years. The model predicts that, at time \(T\) years, the population of white-clawed crayfish will have died out. Given that \(w = 65\) and \(s = 85\) when \(t = 0\)
4. find the value of \(T\), giving your answer to 3 decimal places.
5. Suggest a limitation of the model.