Edexcel FP2 AS (Further Pure 2 AS) 2023 June

1. The operation * is defined on the set \(G = \{ 0,1,2,3 \}\) by

$$x ^ { * } y \equiv x + y - 2 x y ( \bmod 4 )$$

1. Complete the Cayley table below.

   *	0	1	2	3
   0
   1
   2
   3

2. Show that \(G\) is a group under the operation *
   (You may assume the associative law is satisfied.)
3. State the order of each element of \(G\).
4. State whether \(G\) is a cyclic group, giving a reason for your answer.

1. A linear transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix

$$\mathbf { M } = \left( \begin{array} { c c } 5 & 1
k & - 3 \end{array} \right)$$ where \(k\) is a constant.
Given that matrix \(\mathbf { M }\) has a repeated eigenvalue,

1. determine
   1. the value of \(k\)
   2. the eigenvalue.
2. Hence determine a Cartesian equation of the invariant line under \(T\).

1. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.

Given that $$\arg \left( \frac { z - 4 - i } { z - 2 - 7 i } \right) = \frac { \pi } { 2 }$$

1. sketch the locus of \(P\) as \(z\) varies,
2. determine the exact maximum possible value of \(| z |\)

1. A student takes out a loan for \(\pounds 1000\) from a bank.

The bank charges \(0.5 \%\) monthly interest on the amount of the loan yet to be repaid.
At the end of each month

• the interest is added to the loan
• the student then repays \(\pounds 50\)

Let \(U _ { n }\) be the amount of money owed \(n\) months after the loan was taken out.
The amount of money owed by the student is modelled by the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ where \(A\) is a constant.

1. 1. State the value of the constant \(A\).
   2. Explain, in the context of the problem, the value 1.005 Using the value of \(A\) found in part (a)(i),
2. solve the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
3. Hence determine, according to the model, the number of months it will take to completely repay the loan.

1. 1. Making your reasoning clear and using modulo arithmetic, show that

$$214 ^ { 6 } \text { is divisible by } 8$$ (ii) The following 7-digit number has four unknown digits $$a 5 \square b \square a b 0$$ Given that the number is divisible by 11

1. determine the value of the digit \(a\). Given that the number is also divisible by 3
2. determine the possible values of the digit \(b\).