OCR Further Pure Core 1 (Further Pure Core 1) 2018 December

1 Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.

1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).

2 The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8315a796-0e7d-464f-8604-9fe3ab7af359-2_470_657_913_319}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
   2. Find the value of \(r\) at the point \(P\).
   3. Mark the point \(P\) on the copy of the graph in the Printed Answer Booklet.
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve.

3 You are given that \(\mathrm { f } ( x ) = \ln ( 2 + x )\).

1. Determine the exact value of \(\mathrm { f } ^ { \prime } ( 0 )\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }\).
3. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).

4 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 \(z ^ { n }\).

5 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1 \\ - 8 & 4 & 0 \\ 19 & - 8 & - 1 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Hence write down \(\mathbf { A } ^ { - 1 }\).
3. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0 \\ 2 x + 5 y + 2 z = 1 \\ 3 x - 2 y - z = 4 \end{array}$$
   1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
   2. Find this unique solution.

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

7

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 \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

8

1. Given that \(u = \tanh x\), use the definition of \(\tanh x\) in terms of exponentials to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
2. Solve the equation \(4 \tanh ^ { 2 } x + \tanh x - 3 = 0\), giving the solution in the form \(a \ln b\) where \(a\) and \(b\) are rational numbers to be determined.
3. Explain why the equation in part (b) has only one root.

9 In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 11 } \frac { 1 } { \sqrt { x ^ { 2 } + 6 x + 13 } } \mathrm {~d} x\) giving your answer in the form \(\ln ( p + q \sqrt { 2 } )\) where \(p\) and \(q\) are integers to be determined.

10 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.