Edexcel FP2 (Further Pure Mathematics 2) 2021 June

1. In this question you must show detailed reasoning.

Without performing any division, explain why \(n = 20210520\) is divisible by 66

1. A binary operation ★ on the set of non-negative integers, \(\mathbb { Z } _ { 0 } ^ { + }\), is defined by

$$m \star n = | m - n | \quad m , n \in \mathbb { Z } _ { 0 } ^ { + }$$

1. Explain why \(\mathbb { Z } _ { 0 } ^ { + }\)is closed under the operation
2. Show that 0 is an identity for \(\left( \mathbb { Z } _ { 0 } ^ { + } , \star \right)\)
3. Show that all elements of \(\mathbb { Z } _ { 0 } ^ { + }\)have an inverse under ★
4. Determine if \(\mathbb { Z } _ { 0 } ^ { + }\)forms a group under ★, giving clear justification for your answer.

1. (a) Use the Euclidean Algorithm to find integers \(a\) and \(b\) such that

$$125 a + 87 b = 1$$ (b) Hence write down a multiplicative inverse of 87 modulo 125
(c) Solve the linear congruence $$87 x \equiv 16 ( \bmod 125 )$$

1. Let \(G\) be a group of order \(46 ^ { 46 } + 47 ^ { 47 }\)

Using Fermat's Little Theorem and explaining your reasoning, determine which of the following are possible orders for a subgroup of \(G\)

1. 11
2. 21

1. The point \(P\) in the complex plane represents a complex number \(z\) such that

$$| z + 9 | = 4 | z - 12 i |$$ Given that, as \(z\) varies, the locus of \(P\) is a circle,

1. determine the centre and radius of this circle.
2. Shade on an Argand diagram the region defined by the set $$\{ z \in \mathbb { C } : | z + 9 | < 4 | z - 12 i | \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg \left( z - \frac { 3 + 44 i } { 5 } \right) < \frac { \pi } { 4 } \right\}$$

1. A recurrence system is defined by

$$\begin{aligned} u _ { n + 2 } & = 9 ( n + 1 ) ^ { 2 } u _ { n } - 3 u _ { n + 1 } \quad n \geqslant 1
u _ { 1 } & = - 3 , u _ { 2 } = 18 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { N }\), $$u _ { n } = ( - 3 ) ^ { n } n !$$

1. In this question you must show all stages of your working.

You must not use the integration facility on your calculator. $$I _ { n } = \int t ^ { n } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t \quad n \geqslant 0$$

1. Show that, for \(n > 1\) $$I _ { n } = \frac { t ^ { n - 1 } } { 5 ( n + 2 ) } \left( 4 + 5 t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - \frac { 4 ( n - 1 ) } { 5 ( n + 2 ) } I _ { n - 2 }$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1241b133-4161-4c04-9b50-067904cc25c2-20_385_394_829_833} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The curve shown in Figure 1 is defined by the parametric equations $$x = \frac { 1 } { \sqrt { 5 } } t ^ { 5 } \quad y = \frac { 1 } { 2 } t ^ { 4 } \quad 0 \leqslant t \leqslant 1$$ This curve is rotated through \(2 \pi\) radians about the \(x\)-axis to form a hollow open shell.
2. Show that the external surface area of the shell is given by $$\pi \int _ { 0 } ^ { 1 } t ^ { 7 } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t$$ Using the results in parts (a) and (b) and making each step of your working clear,
3. determine the value of the external surface area of the shell, giving your answer to 3 significant figures.

8. $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 2 & 5
0 & 3 & p
- 6 & 6 & - 4 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2
1
- 2 \end{array} \right)\) is an eigenvector for \(\mathbf { A }\)

1. 1. determine the eigenvalue corresponding to this eigenvector
   2. hence show that \(p = 2\)
   3. determine the remaining eigenvalues and corresponding eigenvectors of \(\mathbf { A }\)
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\)
3. 1. Solve the differential equation \(\dot { u } = k u\), where \(k\) is a constant. With respect to a fixed origin \(O\), the velocity of a particle moving through space is modelled by $$\left( \begin{array} { c } \dot { x }
      \dot { y }
      \dot { z } \end{array} \right) = \mathbf { A } \left( \begin{array} { l } x
      y
      z \end{array} \right)$$ By considering \(\left( \begin{array} { c } u
      v
      w \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } x
      y
      z \end{array} \right)\) so that \(\left( \begin{array} { c } \dot { u }
      \dot { v }
      \dot { w } \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } \dot { x }
      \dot { y }
      \dot { z } \end{array} \right)\)
   2. determine a general solution for the displacement of the particle.