Edexcel FP2 (Further Pure Mathematics 2) 2019 June

1. A complex number \(z = x + \mathrm { i } y\) is represented by the point \(P\) in an Argand diagram.

Given that $$| z - 3 | = 4 | z + 1 |$$

1. show that the locus of \(P\) has equation $$15 x ^ { 2 } + 15 y ^ { 2 } + 38 x + 7 = 0$$
2. Hence find the maximum value of \(| z |\)

1. The matrix \(\mathbf { A }\) is given by

$$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 2 & 2
- 2 & 3 & - 1
2 & - 1 & 3 \end{array} \right)$$

1. Show that 2 is a repeated eigenvalue of \(\mathbf { A }\) and find the other eigenvalue.
2. Hence find three non-parallel eigenvectors of \(\mathbf { A }\).
3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix.

1. The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus \(k\) times the number of visits received in the month before that, where \(k\) is a positive constant.

Given that \(V _ { n }\) is the number of visits to the website in month \(n\),

1. write down a general recurrence relation for \(V _ { n + 2 }\) in terms of \(V _ { n + 1 } , V _ { n }\) and \(k\). For a particular website you are given that
   • \(k = 0.24\)
   • In month 1 , there were 65 visits to the website.
   • In month 2 , there were 71 visits to the website.
   • Show that
   $$V _ { n } = 50 ( 1.2 ) ^ { n } - 25 ( - 0.2 ) ^ { n }$$ This model predicts that the number of visits to this website will exceed one million for the first time in month \(N\).
2. Find the value of \(N\).

1. 1. Use Fermat's Little Theorem to find the least positive residue of \(6 ^ { 542 }\) modulo 13
   2. Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if
      1. there are no restrictions where they sit in the row,
   3. Alan, Enid, Felix and Graham sit together,
   4. Brenda sits at one end of the row and Graham sits at the other end of the row,
   5. Charles and Devindra do not sit together.

5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z }$$

1. Prove that, for \(n \geqslant 2\) $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { \operatorname { cosec } ^ { n - 2 } x \cot x } { n - 1 }$$
2. Hence show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \operatorname { cosec } ^ { 6 } x \mathrm {~d} x = \frac { 56 } { 135 } \sqrt { 3 }$$

1. 1. A binary operation * is defined on positive real numbers by

$$a * b = a + b + a b$$ Prove that the operation * is associative.
(ii) The set \(G = \{ 1,2,3,4,5,6 \}\) forms a group under the operation of multiplication modulo 7

1. Show that \(G\) is cyclic. The set \(H = \{ 1,5,7,11,13,17 \}\) forms a group under the operation of multiplication modulo 18
2. List all the subgroups of \(H\).
3. Describe an isomorphism between \(G\) and \(H\).

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

$$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } \quad z \neq - \mathrm { i }$$

1. Show that the circle \(C\) with equation \(| z + \mathrm { i } | = 1\) in the \(z\)-plane is mapped to a circle \(D\) in the \(w\)-plane, giving a Cartesian equation for \(D\).
2. Sketch \(C\) and \(D\) on Argand diagrams.

8. \begin{figure}[h] \end{figure} Figure 1 shows the vertical cross section of a child's spinning top. The point \(A\) is vertically above the point \(B\) and the height of the spinning top is 5 cm . The line \(C D\) is perpendicular to \(A B\) such that \(C D\) is the maximum width of the spinning top.
The spinning top is modelled as the solid of revolution created when part of the curve with polar equation $$r ^ { 2 } = 25 \cos 2 \theta$$ is rotated through \(2 \pi\) radians about the initial line.

1. Show that, according to the model, the surface area of the spinning top is $$k \pi ( 2 - \sqrt { 2 } ) \mathrm { cm } ^ { 2 }$$ where \(k\) is a constant to be determined.
2. Show that, according to the model, the length \(C D\) is \(\frac { 5 \sqrt { 2 } } { 2 } \mathrm {~cm}\).