OCR MEI Further Pure Core AS (Further Pure Core AS) 2022 June

1

1. 1. Write the following simultaneous equations as a matrix equation. $$\begin{aligned} x + y + 2 z & = 7
      2 x - 4 y - 3 z & = - 5
      - 5 x + 3 y + 5 z & = 13 \end{aligned}$$
   2. Hence solve the equations.
2. Determine the set of values of the constant \(k\) for which the matrix equation $$\left( \begin{array} { c c } k + 1 & 1
   2 & k \end{array} \right) \binom { x } { y } = \binom { 23 } { - 17 }$$ has a unique solution.

2

1. Show that the vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) is parallel to the plane \(2 \mathrm { x } + \mathrm { y } - 3 \mathrm { z } = 10\).
2. Determine the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x - y - 3 z = 3\).

3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.

4 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.

5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3 i\), and the half line intersects the circle at the origin and at the point representing \(4 + 4 \mathrm { i }\).
\includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242} State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.

6

1. Using standard summation formulae, show that \(\sum _ { r = 1 } ^ { n } r ( r + 2 ) = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 7 )\).
2. Use induction to prove the result in part (a).

7 On an Argand diagram, the point A represents the complex number \(z\) with modulus 2 and argument \(\frac { 1 } { 3 } \pi\). The point B represents \(\frac { 1 } { z }\).

1. Sketch an Argand diagram showing the origin O and the points A and B .
2. The point C is such that OACB is a parallelogram. C represents the complex number \(w\). Determine each of the following.
   • The modulus of \(w\), giving your answer in exact form.
   • The argument of \(w\), giving your answer correct to \(\mathbf { 3 }\) significant figures.

8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta
\sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).

1. Show that T leaves areas unchanged for all values of \(\theta\).
2. Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T . The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2
   0 & 1 \end{array} \right)\).
3. 1. Find \(\mathbf { M N } ^ { - 1 }\).
   2. Hence describe fully a sequence of two transformations of the plane that is equivalent to T . \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its 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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