OCR Further Pure Core 1 (Further Pure Core 1) 2021 June

2 In this question you must show detailed reasoning.
You are given that \(z = \sqrt { 3 } + \mathrm { i }\). \(n\) is the smallest positive whole number such that \(z ^ { n }\) is a positive whole number.

1. Determine the value of \(n\).
2. Find the value of \(z ^ { n }\).

3 Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4.

4

1. Determine an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) giving your answer in the form \(\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )\).
2. Find the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

5 In a predator-prey environment the population, at time \(t\) years, of predators is \(x\) and prey is \(y\). The populations of predators and prey are measured in hundreds. The populations are modelled by the following simultaneous differential equations. \(\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x\)

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x\).
2. 1. Find the general solution for \(x\).
   2. Find the equivalent general solution for \(y\). Initially there are 100 predators and 300 prey.
3. Find the particular solutions for \(x\) and \(y\).
4. Determine whether the model predicts that the predators will die out before the prey.