Edexcel FP2 AS (Further Pure 2 AS) 2022 June

1. Sketch on an Argand diagram the region defined by

$$z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg ( z + 2 ) < \frac { \pi } { 4 } \cap \{ z \in \mathbb { C } : - 1 < \operatorname { Re } ( z ) \leqslant 1 \}$$ On your sketch

• shade the part of the diagram that is included in the region
• use solid lines to show the parts of the boundary that are included in the region
• use dashed lines to show the parts of the boundary that are not included in the region

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

$$\mathbf { M } = \left( \begin{array} { r r } 4 & 2
3 & - 1 \end{array} \right)$$ Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }$$

1. (i) Let \(G\) be a group of order 5291848

Without performing any division, use proof by contradiction to show that \(G\) cannot have a subgroup of order 11
(ii) (a) Complete the following Cayley table for the set \(X = \{ 2,4,8,14,16,22,26,28 \}\) with the operation of multiplication modulo 30 A copy of this table is given on page 11 if you need to rewrite your Cayley table.
(b) Hence determine whether the set \(X\) with the operation of multiplication modulo 30 forms a group.
[0pt] [You may assume multiplication modulo \(n\) is an associative operation.] Only use this grid if you need to rewrite your Cayley table. (Total for Question 3 is 9 marks)

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

1. (a) Use the Euclidean algorithm to find the highest common factor \(h\) of 416 and 72
   (b) Hence determine integers \(a\) and \(b\) such that $$416 a + 72 b = h$$ (c) Determine the value \(c\) in the set \(\{ 0,1,2 \ldots , 415 \}\) such that $$23 \times 72 \equiv c ( \bmod 416 )$$
2. Evaluate \(5 ^ { 10 } ( \bmod 13 )\) giving your answer as the smallest positive integer solution.

1. A person takes a course of a particular vitamin.

Before the course there was none of the vitamin in the person's body.
During the course, vitamin tablets are taken at the same time each day.
Initially two tablets are taken and on each following day only one tablet is taken.
Each tablet contains 10 mg of the vitamin.
Between doses the amount of the vitamin in the person's body decreases naturally by 60\% Let \(u _ { n } \mathrm { mg }\) be the amount of the vitamin in the person's body immediately after a tablet is taken, \(n\) days after the initial two tablets were taken.

1. Explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { 0 } = 20 \quad u _ { n + 1 } = 0.4 u _ { n } + 10$$ The general solution to this recurrence relation has the form \(u _ { n } = a ( 0.4 ) ^ { n } + b\)
2. Determine the value of \(a\) and the value of \(b\). The course is only effective if the amount of the vitamin in the person's body remains above 6 mg at all times throughout the course.
3. Determine whether this course of the vitamin will be effective for this person, giving a reason for your answer.