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

1 In this question you must show detailed reasoning.
The equation \(x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\). The equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
Find the values of \(p\) and \(q\).

2 In this question you must show detailed reasoning.
Given that \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = - 1 - \mathrm { i }\), find the following, giving each in the form \(a + b \mathrm { i }\).

1. \(z _ { 1 } ^ { * } z _ { 2 }\)
2. \(\frac { z _ { 1 } + 2 z _ { 2 } } { z _ { 2 } }\)

3

1. You are given two matrices, A and B, where $$\mathbf { A } = \left( \begin{array} { l l } 1 & 2
   2 & 1 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } - 1 & 2
   2 & - 1 \end{array} \right)$$ Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
2. You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where $$\mathbf { C } = \left( \begin{array} { r r r } 2 & 1 & 5
   1 & 1 & 3
   - 1 & 2 & 2 \end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r } - 4 & 8 & - 2
   - 5 & 9 & - 1
   3 & - 5 & 1 \end{array} \right)$$ Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
3. The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 }
   3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant. Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).

4 Draw the region of the Argand diagram for which \(| z - 3 - 4 i | \leq 5\) and \(| z | \leq | z - 2 |\).

5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 }
\frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\).

1. The diagram in the Printed Answer Booklet shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
2. Find the equation of the line of invariant points of this transformation.
3. (a) Find the determinant of \(\mathbf { M }\).
   (b) Describe briefly how this value relates to the transformation represented by \(\mathbf { M }\).

6 At the beginning of the year John had a total of \(\pounds 2000\) in three different accounts. He has twice as much money in the current account as in the savings account.

• The current account has an interest rate of \(2.5 \%\) per annum.
• The savings account has an interest rate of \(3.7 \%\) per annum.
• The supersaver account has an interest rate of \(4.9 \%\) per annum.

John has predicted that he will earn a total interest of \(\pounds 92\) by the end of the year.

1. Model this situation as a matrix equation.
2. Find the amount that John had in each account at the beginning of the year.
3. In fact, the interest John will receive is \(\pounds 92\) to the nearest pound. Explain how this affects the calculations.

7 In this question you must show detailed reasoning.
It is given that \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 13 z ^ { 2 } + 65 z - 125\).
The points representing the three roots of the equation \(\mathrm { f } ( z ) = 0\) are plotted on an Argand diagram.
Show that these points lie on the circle \(| z | = k\), where \(k\) is a real number to be determined.

8 Prove that \(n ! > 2 ^ { n }\) for \(n \geq 4\).

9

1. Find the value of \(k\) such that \(\left( \begin{array} { l } 1
   2
   1 \end{array} \right)\) and \(\left( \begin{array} { r } - 2
   3
   k \end{array} \right)\) are perpendicular. Two lines have equations \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 3
   2
   7 \end{array} \right) + \lambda \left( \begin{array} { r } 1
   - 1
   3 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6
   5
   2 \end{array} \right) + \mu \left( \begin{array} { r } 2
   1
   - 1 \end{array} \right)\).
2. Find the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
3. The vector \(\left( \begin{array} { l } 1
   a
   b \end{array} \right)\) is perpendicular to the lines \(l _ { 1 }\) and \(l _ { 2 }\). Find the values of \(a\) and \(b\). \section*{END OF QUESTION PAPER} \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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