Edexcel FP2 AS (Further Pure 2 AS) 2020 June

1. The set \(G = \{ 1,3,7,9,11,13,17,19 \}\) under the binary operation of multiplication modulo 20 forms a group.
   1. Find the inverse of each element of \(G\).
   2. Find the order of each element of \(G\).
   3. Find a subgroup of \(G\) of order 4
   4. Explain how the subgroup you found in part (c) satisfies Lagrange's theorem.

1. The highest common factor of 963 and 657 is \(c\).
   1. Use the Euclidean algorithm to find the value of \(c\).
   2. Hence find integers \(a\) and \(b\) such that
   $$963 a + 657 b = c$$

$$A = \left( \begin{array} { r r } 1 & - 2
1 & 4 \end{array} \right)$$

1. Show that the characteristic equation for \(\mathbf { A }\) is \(\lambda ^ { 2 } - 5 \lambda + 6 = 0\)
2. Use the Cayley-Hamilton theorem to find integers \(p\) and \(q\) such that $$\mathbf { A } ^ { 3 } = p \mathbf { A } + q \mathbf { I }$$ (ii) Given that the \(2 \times 2\) matrix \(\mathbf { M }\) has eigenvalues \(- 1 + \mathrm { i }\) and \(- 1 - \mathrm { i }\), with eigenvectors \(\binom { 1 } { 2 - \mathrm { i } }\) and \(\binom { 1 } { 2 + \mathrm { i } }\) respectively, find the matrix \(\mathbf { M }\).

1. Sam borrows \(\pounds 10000\) from a bank to pay for an extension to his house.

The bank charges \(5 \%\) annual interest on the portion of the loan yet to be repaid. Immediately after the interest has been added at the end of each year and before the start of the next year, Sam pays the bank a fixed amount, \(\pounds F\). Given that \(\pounds A _ { n }\) (where \(A _ { n } \geqslant 0\) ) is the amount owed at the start of year \(n\),

1. write down an expression for \(A _ { n + 1 }\) in terms of \(A _ { n }\) and \(F\),
2. prove, by induction that, for \(n \geqslant 1\) $$A _ { n } = ( 10000 - 20 F ) 1.05 ^ { n - 1 } + 20 F$$
3. Find the smallest value of \(F\) for which Sam can repay all of the loan by the start of year 16 .

5. \begin{figure}[h] \end{figure} Figure 1 shows an Argand diagram.
The set of points, \(A\), that lies within the shaded region, including its boundaries, is defined by $$A = \{ z : p \leqslant \arg ( z ) \leqslant q \} \cap \{ z : | z | \leqslant r \}$$ where \(p , q\) and \(r\) are positive constants.

1. Write down the values of \(p , q\) and \(r\). Given that \(w = - 2 \sqrt { 3 } + 2 \mathrm { i }\) and \(\mathrm { z } \in A\),
2. find the maximum value of \(| w - z | ^ { 2 }\) giving your answer in an exact simplified form.