Edexcel FP2 AS (Further Pure 2 AS) 2024 June

1. 1. The table below is a Cayley table for the group \(G\) with operation ∘

1. State which element is the identity of the group.
2. Determine the inverse of the element ( \(b \circ c\) )
3. Give a reason why the set \(\{ a , b , e , f \}\) cannot be a subgroup of \(G\). You must justify your answer.
4. Show that the set \(\{ b , d , f \}\) is a subgroup of \(G\).
   (ii) Given that \(H\) is a group with an element \(x\) of order 3 and an element \(y\) of order 6 satisfying $$y x = x y ^ { 5 }$$ show that \(y ^ { 3 } x y ^ { 3 } x ^ { 2 }\) is the identity element.
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1. Tiles are sold in boxes with 21 tiles in each box.

The tiles are laid out in \(x\) rows of 5 tiles and \(y\) rows of 6 tiles.
All the tiles from a box are used before the next box is opened.
When all the rows of tiles have been laid, there are \(n\) tiles left in the last opened box.

1. Write down a congruence expression for \(n\) in the form $$a x + b y ( \bmod c )$$ where \(a\), \(b\) and \(c\) are integers. Given that
   • exactly 43 rows of tiles are laid
   • there are no tiles left in the last opened box
   • use your congruence expression to determine the minimum number of rows of 6 tiles laid.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$\mathbf { A } = \left( \begin{array} { r r } 3 & k
- 5 & 2 \end{array} \right)$$ where \(k\) is a constant.
Given that there exists a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix where $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { r r } 8 & 0
0 & - 3 \end{array} \right)$$

1. show that \(k = - 6\)
2. determine a suitable matrix \(\mathbf { P }\)

1. A circle \(C\) in the complex plane has equation

$$| z - ( - 3 + 3 i ) | = \alpha | z - ( 1 + 3 i ) |$$ where \(\alpha\) is a real constant with \(\alpha > 1\)
Given that the imaginary axis is a tangent to \(C\)

1. sketch, on an Argand diagram, the circle \(C\)
2. explain why the value of \(\alpha\) is 3 The circle \(C\) is contained in the region $$R = \left\{ z \in \mathbb { C } : \beta \leqslant \arg z \leqslant \frac { \pi } { 2 } \right\}$$
3. Determine the maximum value of \(\beta\) Give your answer in radians to 3 significant figures.

5. \begin{figure}[h] \end{figure} Figure 1 shows the first three stages of a pattern that is created by a recursive process.
The process starts with a square and proceeds as follows

• each square is replaced by 5 smaller squares each \(\frac { 1 } { 9 }\) th the size of the square being replaced
• the 5 smaller squares are the ones in each corner and the one in the centre
• once each of the squares has been replaced, the square immediately to the right and above the centre square of the pattern is then removed

Let \(u _ { n }\) be the number of squares in the pattern in stage \(n\), where stage 1 is the original square.

1. Explain why \(u _ { n }\) satisfies the recurrence system $$u _ { 1 } = 1 \quad u _ { n + 1 } = 5 u _ { n } - 1 \quad ( n = 1,2,3 , \ldots )$$
2. Solve this recurrence system. Given that the initial square has area 25
3. determine the total area of all the squares in stage 8 of the pattern, giving your answer to 2 significant figures.