Edexcel FP2 (Further Pure Mathematics 2) 2020 June

1. A small sports club has 12 adult members and 14 junior members.

The club needs to enter a team of 8 players for a particular competition.
Determine the number of ways in which the team can be selected if

1. there are no restrictions on the team,
2. the team must contain 4 adults and 4 juniors,
3. more than half the team must be adults.

1. Solve the recurrence system

$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4
9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$

3. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & - 2
2 & - 4 & 1
1 & 2 & 3 \end{array} \right)$$ where \(k\) is a constant.

1. Show that, in terms of \(k\), a characteristic equation for \(\mathbf { M }\) is given by $$\lambda ^ { 3 } - ( 2 k + 13 ) \lambda + 5 ( k + 6 ) = 0$$ Given that \(\operatorname { det } \mathbf { M } = 5\)
2. 1. find the value of \(k\)
   2. use the Cayley-Hamilton theorem to find the inverse of \(\mathbf { M }\).

4. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a design for a road speed bump of width 2.35 metres. The speed bump has a uniform cross-section with vertical ends and its length is 30 cm . A side profile of the speed bump is shown in Figure 2. The curve \(C\) shown in Figure 2 is modelled by the polar equation $$r = 30 \left( 1 - \theta ^ { 2 } \right) \quad 0 \leqslant \theta \leqslant 1$$ The units for \(r\) are centimetres and the initial line lies along the road surface, which is assumed to be horizontal. Once the speed bump has been fixed to the road, the visible surfaces of the speed bump are to be painted. Determine, in \(\mathrm { cm } ^ { 2 }\), the area that is to be painted, according to the model.

1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$ The circle with equation \(| z + \mathrm { i } | = 3\) is mapped by \(T\) onto the circle \(C\).

1. Show that the equation for \(C\) can be written as $$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$
2. Hence find
   1. a Cartesian equation for \(C\),
   2. the centre and radius of \(C\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, \(G\).

1. Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6
   3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
2. Write down, in similar two-line notation, the remaining elements of the group \(G\).
3. Explain why each of the following statements is false, making your reasoning clear.
   1. \(G\) has a subgroup of order 4
   2. \(G\) is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
4. Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
   \section*{Diagram 1} \section*{Spare copy of Diagram 1}
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
   Only use this diagram if you need to redraw your answer to part (d).

7. $$I _ { n } = \int \left( 4 - x ^ { 2 } \right) ^ { - n } \mathrm {~d} x \quad n > 0$$

1. Show that, for \(n > 0\) $$I _ { n + 1 } = \frac { x } { 8 n \left( 4 - x ^ { 2 } \right) ^ { n } } + \frac { 2 n - 1 } { 8 n } I _ { n }$$
2. Find \(I _ { 2 }\)

1. The four digit number \(n = a b c d\) satisfies the following properties:
   (1) \(n \equiv 3 ( \bmod 7 )\)
   (2) \(n\) is divisible by 9
   (3) the first two digits have the same sum as the last two digits
   (4) the digit \(b\) is smaller than any other digit
   (5) the digit \(c\) is even
   1. Use property (1) to explain why \(6 a + 2 b + 3 c + d \equiv 3 ( \bmod 7 )\)
   2. Use properties (2), (3) and (4) to show that \(a + b = 9\)
   3. Deduce that \(c \equiv 5 ( a - 1 ) ( \bmod 7 )\)
   4. Hence determine the number \(n\), verifying that it is unique. You must make your reasoning clear.