OCR MEI Further Pure Core AS (Further Pure Core AS) Specimen

1 The complex number \(z _ { 1 }\) is \(1 + \mathrm { i }\) and the complex number \(z _ { 2 }\) has modulus 4 and argument \(\frac { \pi } { 3 }\).

1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { i }\), giving \(a\) and \(b\) in exact form.
2. Express \(\frac { z _ { 2 } } { z _ { 1 } }\) in the form \(c + d i\), giving \(c\) and \(d\) in exact form.
3. Describe fully the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2
   0 & 1 \end{array} \right)\).
4. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2
   0 & 1 \end{array} \right)\). Explaining your reasoning, find the area of the image of the triangle following this transformation.

3

1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7728fdf9-2000-4265-a0cb-f34a6561c2ca-2_917_825_1334_699} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg ( z - 2 \mathrm { i } ) = \frac { \pi } { 4 }\).

4

1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$\begin{aligned} x - 2 y - z & = 6
   3 x + y + 5 z & = - 4
   - 4 x + 2 y - 3 z & = a \end{aligned}$$
2. Determine whether the intersection of the three planes could be on the \(z\)-axis.

5 The cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { 2 } { \alpha }\) and \(\alpha + \frac { 2 } { \alpha }\). Find

• the values of the roots of the equation,
• the value of \(p\).

6

1. Show that, when \(n = 5 , \sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = 330\).
2. Find, in terms of \(n\), a fully factorised expression for \(\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 }\).

7 The plane \(\Pi\) has equation \(3 x - 5 y + z = 9\).

1. Show that \(\Pi\) contains
   • the point (4,1,2)
     and
   • the vector \(\left( \begin{array} { l } 1
     1
     2 \end{array} \right)\).
   • Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through ( \(4,1,2\) ).

8 In this question you must show detailed reasoning.

1. Explain why all cubic equations with real coefficients have at least one real root.
2. Points representing the three roots of the equation \(z ^ { 3 } + 9 z ^ { 2 } + 27 z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices.

9 You are given that matrix \(\mathbf { M } = \left( \begin{array} { l l } - 3 & 8
- 2 & 5 \end{array} \right)\).

1. Prove that, for all positive integers \(n , \mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 - 4 n & 8 n
   - 2 n & 1 + 4 n \end{array} \right)\).
2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf { M }\). It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf { M } ^ { n }\), for any positive integer \(n\).
3. Explain geometrically why this claim is true.
4. Verify algebraically that this claim is true. \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }