Edexcel FP2 (Further Pure Mathematics 2) 2023 June

1. $$\mathbf { A } = \left( \begin{array} { r r } - 1 & a
3 & 8 \end{array} \right)$$ where \(a\) is a constant.

1. Determine, in expanded form in terms of \(a\), the characteristic equation for \(\mathbf { A }\).
2. Hence use the Cayley-Hamilton theorem to determine values of \(a\) and \(b\) such that $$\mathbf { A } ^ { 3 } = \mathbf { A } + b \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. A complex number \(z\) is represented by the point \(P\) in the complex plane.

Given that \(z\) satisfies $$| z - 6 | = 2 | z + 3 i |$$

1. show that the locus of \(P\) passes through the origin and the points - 4 and - 8 i
2. Sketch on an Argand diagram the locus of \(P\) as \(z\) varies.
3. On your sketch, shade the region which satisfies both $$| z - 6 | \geqslant 2 | z + 3 i | \text { and } | z | \leqslant 4$$

1. In a model for the number of subscribers to a new social media channel it is assumed that

• each week \(20 \%\) of the subscribers at the start of the week cancel their subscriptions
• between the start and end of week \(n\) the channel gains \(20 n\) new subscribers

Given that at the end of week 1 there were 25 subscribers,

1. explain why the number of subscribers at the end of week \(n , U _ { n }\), is modelled by the recurrence relation $$U _ { 1 } = 25 \quad U _ { n + 1 } = 0.8 U _ { n } + 20 ( n + 1 ) \quad n = 1,2,3 , \ldots$$
2. Prove by induction that for \(n \geqslant 1\) $$U _ { n } = 325 \left( \frac { 4 } { 5 } \right) ^ { n - 1 } + 100 n - 400$$ Given that 6 months after starting the channel there were approximately 1800 subscribers,
3. evaluate the model in the light of this information.

1. (a) Use the Euclidean algorithm to show that the highest common factor of 168 and 66 is 6
   (b) Use back substitution to determine integers \(a\) and \(b\) such that

$$168 a + 66 b = 6$$ (c) Explain why there are no integer solutions to the equation $$168 x + 66 y = 10$$ (d) Solve the congruence equation $$11 v \equiv 8 ( \bmod 28 )$$

1. 1. A security code is made up of 4 numerical digits followed by 3 distinct uppercase letters.

Given that the digits must be from the set \(\{ 1,2,3,4,5 \}\) and the letters from the set \{A, B, C, D\}

1. determine the total number of possible codes using this system. To enable more codes to be generated, the system is adapted so that the 3 letters can appear anywhere in the code but no letter can be next to another letter.
2. Determine the increase in the number of codes using this adapted system.
   (ii) A combination lock code consists of four distinct digits that can be read as a positive integer, \(N = a b c d\), satisfying
   • all the digits are odd
   • \(\quad N\) is divisible by 9
   • the digits appear in either ascending or descending order
   • \(\quad N \equiv e ( \bmod a b )\) where \(a b\) is read as a two-digit number and \(e\) is the odd digit that is not used in the code
   • Use the first two properties to determine the four digits used in the code.
   • Hence determine the code on the lock.

1. Determine a closed form for the recurrence relation

$$\begin{aligned} & u _ { 0 } = 1 \quad u _ { 1 } = 4
& u _ { n + 2 } = 2 u _ { n + 1 } - \frac { 4 } { 3 } u _ { n } + n \quad n \geqslant 0 \end{aligned}$$

1. The set \(\mathrm { G } = \mathbb { R } - \left\{ - \frac { 3 } { 2 } \right\}\) with the operation of \(x \bullet y = 3 ( x + y + 1 ) + 2 x y\) forms a group.
   1. Determine the identity element of this group.
   2. Determine the inverse of a general element \(x\) in this group.
   3. Explain why the value \(- \frac { 3 } { 2 }\) must be excluded from \(G\) in order for this to be a group.

8. $$I _ { n } = \int _ { 0 } ^ { 2 } ( x - 2 ) ^ { n } \mathrm { e } ^ { 4 x } \mathrm {~d} x \quad n \geqslant 0$$

1. Prove that for \(n \geqslant 1\) $$I _ { n } = - a ^ { n - 2 } - \frac { n } { 4 } I _ { n - 1 }$$ where \(a\) is a constant to be determined.
2. Hence determine the exact value of $$\int _ { 0 } ^ { 2 } ( x - 2 ) ^ { 2 } e ^ { 4 x } d x$$

9. \begin{figure}[h] \end{figure} Figure 1 shows a locus in the complex plane.
The locus is an arc of a circle from the point represented by \(z _ { 1 } = 3 + 2 i\) to the point represented by \(z _ { 2 } = a + 4 \mathrm { i }\), where \(a\) is a constant, \(a \neq 1\) Given that

• the point \(z _ { 3 } = 1 + 4 \mathrm { i }\) also lies on the locus
• the centre of the circle has real part equal to - 1
  1. determine the value of \(a\).
  2. Hence determine a complex equation for the locus, giving any angles in the equation as positive values.

10. \begin{figure}[h] \end{figure} A solid playing piece for a board game is modelled by rotating the curve \(C\), shown in Figure 2, through \(2 \pi\) radians about the \(x\)-axis. The curve \(C\) has equation $$y = \sqrt { 1 + \frac { x ^ { 2 } } { 9 } } \quad - 4 \leqslant x \leqslant 4$$ with units as centimetres.

1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the playing piece is given by $$S = p \pi \int _ { - 4 } ^ { 4 } \sqrt { 81 + 10 x ^ { 2 } } \mathrm {~d} x + q \pi$$ where \(p\) and \(q\) are constants to be determined. Using the substitution \(x = \frac { 9 } { \sqrt { 10 } } \sinh u\), or another algebraic integration method, and showing all your working,
2. determine the total surface area of the playing piece, giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\)