OCR MEI Further Pure Core AS (Further Pure Core AS) 2023 June

1 The transformation R of the plane is reflection in the line \(x = 0\).

1. Write down the matrix \(\mathbf { M }\) associated with R .
2. Find \(\mathbf { M } ^ { 2 }\).
3. Interpret the result of part (b) in terms of the transformation \(R\).

2 In this question you must show detailed reasoning.
The equation \(\mathrm { x } ^ { 2 } - \mathrm { kx } + 2 \mathrm { k } = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\).
Find \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha }\) in terms of \(k\), simplifying your answer.

3 In this question you must show detailed reasoning.
The function \(\mathrm { f } ( \mathrm { z } )\) is given by \(\mathrm { f } ( \mathrm { z } ) = 2 \mathrm { z } ^ { 3 } - 7 \mathrm { z } ^ { 2 } + 16 \mathrm { z } - 15\).
By first evaluating \(\mathrm { f } \left( \frac { 3 } { 2 } \right)\), find the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\).

4 You are given that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { ar } + \mathrm { b } ) = \mathrm { n } ^ { 2 }\) for all \(n\), where \(a\) and \(b\) are constants.
By finding \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { ar } + \mathrm { b } )\) in terms of \(a , b\) and \(n\), determine the values of \(a\) and \(b\).

5 The Argand diagram below shows the points representing 1 and \(z\), where \(| z | = 2\).
\includegraphics[max width=\textwidth, alt={}, center]{26cec6f9-78a7-4f0b-969a-13ad02510c25-3_577_595_312_242} Mark the points representing the following complex numbers on the copy of the diagram in the Printed Answer Booklet, labelling them clearly.

• \(\mathrm { Z } ^ { * }\)
• \(\frac { 1 } { z }\)
• \(1 + z\)
• iz

6 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 2 & 1
- 1 & 0 \end{array} \right)\).

1. Calculate \(\mathbf { M } ^ { 2 } , \mathbf { M } ^ { 3 }\) and \(\mathbf { M } ^ { 4 }\).
2. Hence make a conjecture about the matrix \(\mathbf { M } ^ { n }\).
3. Prove your conjecture.

1. By expanding \(( \sqrt { 3 } + \mathrm { i } ) ^ { 5 }\), express \(z ^ { 5 }\) in the form \(\mathrm { a } +\) bi where \(a\) and \(b\) are real and exact.
2. 1. Express \(z\) in modulus-argument form.
   2. Hence find \(z ^ { 5 }\) in modulus-argument form.
   3. Use this result to verify your answers to part (a).

8 The equations of three planes are $$\begin{array} { r } 2 x + y + 3 z = 3
3 x - y - 2 z = 2
- 4 x + 3 y + 7 z = k \end{array}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the planes do not meet at a single point.
2. Given that the planes form a sheaf, determine the value of \(k\).

9 A transformation T of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { c c } k + 1 & - 1
1 & k \end{array} \right)\), where \(k\) is a
constant. constant. Show that, for all values of \(k , \mathrm {~T}\) has no invariant lines through the origin.

10 The plane P has normal vector \(2 \mathbf { i } + a \mathbf { j } - \mathbf { k }\), where \(a\) is a positive constant, and the point ( \(3 , - 1,1\) ) lies in P . The plane \(\mathrm { x } - \mathrm { z } = 3\) makes an angle of \(45 ^ { \circ }\) with P . Find the cartesian equation of P . \section*{END OF QUESTION PAPER}