Edexcel FP2 AS (Further Pure 2 AS) Specimen

1. Given that

$$A = \left( \begin{array} { l l } 3 & 1
6 & 4 \end{array} \right)$$

1. find the characteristic equation of the matrix \(\mathbf { A }\).
2. Hence show that \(\mathbf { A } ^ { 3 } = 43 \mathbf { A } - 42 \mathbf { I }\).

1. (i) Without performing any division, explain why 8184 is divisible by 6
   (ii) Use the Euclidean algorithm to find integers \(a\) and \(b\) such that

$$27 a + 31 b = 1$$

1. A curve \(C\) is described by the equation

$$| z - 9 + 12 i | = 2 | z |$$

1. Show that \(C\) is a circle, and find its centre and radius.
2. Sketch \(C\) on an Argand diagram. Given that \(w\) lies on \(C\),
3. find the largest value of \(a\) and the smallest value of \(b\) that must satisfy $$a \leqslant \operatorname { Re } ( w ) \leqslant b$$

1. The operation * is defined on the set \(S = \{ 0,2,3,4,5,6 \}\) by \(x ^ { * } y = x + y = x y ( \bmod 7 )\)

1. 1. Complete the Cayley table shown above
   2. Show that \(S\) is a group under the operation *
      (You may assume the associative law is satisfied.)
2. Show that the element 4 has order 3
3. Find an element which generates the group and express each of the elements in terms of this generator.

1. A population of deer on a large estate is assumed to increase by \(10 \%\) during each year due to natural causes.

The population is controlled by removing a constant number, \(Q\), of the deer from the estate at the end of each year. At the start of the first year there are 5000 deer on the estate.
Let \(P _ { n }\) be the population of deer at the end of year \(n\).

1. Explain, in the context of the problem, the reason that the deer population is modelled by the recurrence relation $$P _ { n } = 1.1 P _ { n - 1 } - Q , \quad P _ { 0 } = 5000 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(P _ { n } = ( 1.1 ) ^ { n } ( 5000 - 10 Q ) + 10 Q , \quad n \geqslant 0\)
3. Explain how the long term behaviour of this population varies for different values of \(Q\).