Edexcel FP2 AS (Further Pure 2 AS) 2018 June

1. (i) Using a suitable algorithm and without performing any division, determine whether 23738 is divisible by 11
   (ii) Use the Euclidean algorithm to find the highest common factor of 2322 and 654

2. \begin{figure}[h] \end{figure} Figure 1 shows an equilateral triangle \(A B C\). The lines \(x , y\) and \(z\) and their point of intersection, \(O\), are fixed in the plane. The triangle \(A B C\) is transformed about these fixed lines and the fixed point \(O\). The lines \(x , y\) and \(z\) each pass through a vertex of the triangle and the midpoint of the opposite side. The transformations \(I , X , Y , Z , R _ { 1 }\) and \(R _ { 2 }\) of the plane containing triangle \(A B C\) are defined as follows:

• I: Do nothing
• \(X\) : Reflect in the line \(x\)
• \(Y\) : Reflect in the line \(y\)
• \(Z\) : Reflect in the line \(z\)
• \(R _ { 1 }\) : Rotate \(120 ^ { \circ }\) anticlockwise about \(O\)
• \(R _ { 2 }\) : Rotate \(240 ^ { \circ }\) anticlockwise about \(O\)

The operation * is defined as 'followed by' on the set \(T = \left\{ I , X , Y , Z , R _ { 1 } , R _ { 2 } \right\}\).
For example, \(X { } ^ { * } Y\) means a reflection in the line \(x\) followed by a reflection in the line \(y\).

1. 1. Complete the Cayley table on page 5 Given that the associative law is satisfied,
   2. show that \(T\) is a group under the operation *
2. Show that the element \(R _ { 2 }\) has order 3
3. Explain why \(T\) is not a cyclic group.
4. Write down the elements of a subgroup of \(T\) that has order 3

   \multirow{2}{*}{}		Second transformation
   	*	I	\(X\)	\(Y\)	\(Z\)	\(R _ { 1 }\)	\(R _ { 2 }\)
   \multirow{6}{*}{First Transformation}	I
   	\(X\)		I			\(Z\)
   	\(Y\)
   	\(Z\)
   	\(R _ { 1 }\)		\(Y\)
   	\(R _ { 2 }\)

   \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Only use this grid if you need to re-write your Cayley table}

   \multirow{2}{*}{}		Second transformation
   	*	I	\(X\)	\(Y\)	Z	\(R _ { 1 }\)	\(R _ { 2 }\)
   \multirow{6}{*}{First Transformation}	I
   	X		I			Z
   	Y
   	Z
   	\(R _ { 1 }\)		\(Y\)
   	\(R _ { 2 }\)

   \end{table}

3 A tree at the bottom of a garden needs to be reduced in height. The tree is known to increase in height by 15 centimetres each year. On the first day of every year, the height is measured and the tree is immediately trimmed by \(3 \%\) of this height. When the tree is measured, before trimming on the first day of year 1 , the height is 6 metres.
Let \(H _ { n }\) be the height of the tree immediately before trimming on the first day of year \(n\).

1. Explain, in the context of the problem, why the height of the tree may be modelled by the recurrence relation $$H _ { n + 1 } = 0.97 H _ { n } + 0.15 , \quad H _ { 1 } = 6 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(H _ { n } = 0.97 ^ { n - 1 } + 5 , \quad n \geqslant 1\)
3. Explain what will happen to the height of the tree immediately before trimming in the long term.
4. By what fixed percentage should the tree be trimmed each year if the height of the tree immediately before trimming is to be 4 metres in the long term?

4. $$\mathbf { A } = \left( \begin{array} { r r } 1 & 1 \\ - 2 & 4 \end{array} \right)$$ Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { D } = \mathbf { P } ^ { - 1 } \mathbf { A P }\)

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

Given that \(\arg \left( \frac { z - 6 i } { z - 3 i } \right) = \frac { \pi } { 3 }\)

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