Questions Further Pure Core AS

Find a vector, $\mathbf { b }$, which is perpendicular to both lines.
Show that $\mathbf { b } \cdot \left( \begin{array} { c } - 12
a
- 1 \end{array} \right) = \mathbf { b } \cdot \left( \begin{array} { l } 2
0
5 \end{array} \right)$.
Hence, or otherwise, find the value of $a$.

OCR Further Pure Core AS 2020 November Q8

8 Two loci, $C _ { 1 }$ and $C _ { 2 }$, are defined by $$\begin{aligned} & C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\}
& C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 i ) = \frac { 1 } { 4 } \pi \right\} \end{aligned}$$ where $d$ is a real number.

Find, in terms of $d$, the complex number which is represented on an Argand diagram by the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.
[0pt] [You may assume that $C _ { 1 } \cap C _ { 2 } \neq \varnothing$.]
Explain why the solution found in part (a) is not valid when $d = 3$. \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}

OCR Further Pure Core AS Specimen Q1

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 $q$.

OCR Further Pure Core AS Specimen Q2

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 $a + b \mathrm { i }$.

$z _ { 1 } ^ { * } z _ { 2 }$
$\frac { z _ { 1 } + 2 z _ { 2 } } { z _ { 2 } }$

OCR Further Pure Core AS Specimen Q3

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.
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.
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 $\mathbf { E F } = \binom { - 2 k } { 6 }$.

OCR Further Pure Core AS Specimen Q4

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

OCR Further Pure Core AS Specimen Q5

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

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 $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$.
Find the equation of the line of invariant points of this transformation.
(a) Find the determinant of $\mathbf { M }$.
(b) Describe briefly how this value relates to the transformation represented by $\mathbf { M }$.

OCR Further Pure Core AS Specimen Q6

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.

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

OCR Further Pure Core AS Specimen Q7

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.

OCR Further Pure Core AS Specimen Q8

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

OCR Further Pure Core AS Specimen Q9

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 $l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6
5
2 \end{array} \right) + \mu \left( \begin{array} { r } 2
1
- 1 \end{array} \right)$.
Find the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.
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.
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.

OCR MEI Further Pure Core AS 2018 June Q1

1 The matrices $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$ are defined as follows: $$\mathbf { A } = \left( \begin{array} { l } 1

OCR MEI Further Pure Core AS 2018 June Q3

3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 3
1 & - 1 & 3 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 3 \end{array} \right)$$ Calculate all possible products formed from two of these three matrices. 2 Find, to the nearest degree, the angle between the vectors $\left( \begin{array} { r } 1
0
- 2 \end{array} \right)$ and $\left( \begin{array} { r } - 2
3
- 3 \end{array} \right)$. 3 Find real numbers $a$ and $b$ such that $( a - 3 i ) ( 5 - i ) = b - 17 i$.

OCR MEI Further Pure Core AS 2018 June Q4

4 Find a cubic equation with real coefficients, two of whose roots are $2 - \mathrm { i }$ and 3.

OCR MEI Further Pure Core AS 2018 June Q5

5 A transformation of the $x - y$ plane is represented by the matrix $\left( \begin{array} { r r } \cos \theta & 2 \sin \theta
2 \sin \theta & - \cos \theta \end{array} \right)$, where $\theta$ is a positive acute angle.

Write down the image of the point $( 2,3 )$ under this transformation.
You are given that this image is the point ( $a , 0$ ). Find the value of $a$.

OCR MEI Further Pure Core AS 2018 June Q6

6 Find the invariant line of the transformation of the $x - y$ plane represented by the matrix $\left( \begin{array} { r r } 2 & 0
4 & - 1 \end{array} \right)$.

OCR MEI Further Pure Core AS 2018 June Q7

Express $\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 }$ as a single fraction.
Find how many terms of the series $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 3 \times 5 } + \frac { 2 } { 5 \times 7 } + \ldots + \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) } + \ldots$$ are needed for the sum to exceed 0.999999.

OCR MEI Further Pure Core AS 2018 June Q8

8 Prove by induction that $\left( \begin{array} { l l } 1 & 1
0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 2 ^ { n } - 1
0 & 2 ^ { n } \end{array} \right)$ for all positive integers $n$.

OCR MEI Further Pure Core AS 2018 June Q9

9 Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9ef04b56-c6e5-46ea-a485-fe872932e9d8-3_549_520_397_751} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure}

Find, in modulus-argument form, the complex number represented by the point P .
Find, in the form $a + \mathrm { i } b$, where $a$ and $b$ are exact real numbers, the complex number represented by the point Q .
In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
- $3 + 5 \mathrm { i }$
- $5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )$
  lie within this region.

OCR MEI Further Pure Core AS 2018 June Q10

10 Three planes have equations $$\begin{aligned} - x + 2 y + z & = 0
2 x - y - z & = 0
x + y & = a \end{aligned}$$ where $a$ is a constant.

Investigate the arrangement of the planes:
- when $a = 0$;
- when $a \neq 0$.
- Chris claims that the position vectors $- \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , 2 \mathbf { i } - \mathbf { j } - \mathbf { k }$ and $\mathbf { i } + \mathbf { j }$ lie in a plane. Determine whether or not Chris is correct.

OCR MEI Further Pure Core AS 2019 June Q1

1 In this question you must show detailed reasoning.
Find $\sum _ { r = 1 } ^ { 100 } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)$, giving your answer correct to 4 decimal places.

OCR MEI Further Pure Core AS 2019 June Q2

2 The roots of the equation $3 x ^ { 2 } - x + 2 = 0$ are $\alpha$ and $\beta$.
Find a quadratic equation with integer coefficients whose roots are $2 \alpha - 3$ and $2 \beta - 3$.

OCR MEI Further Pure Core AS 2019 June Q3

3 In this question you must show detailed reasoning.
$\mathbf { A }$ and $\mathbf { B }$ are matrices such that $\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1
- 1 & 1 \end{array} \right)$.

Find $\mathbf { A B }$.
Given that $\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1
0 & 1 \end{array} \right)$, find $\mathbf { B }$.

OCR MEI Further Pure Core AS 2019 June Q4

Find $\mathbf { M } ^ { - 1 }$, where $\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3
- 1 & 1 & 2
- 2 & 1 & 2 \end{array} \right)$.
Hence find, in terms of the constant $k$, the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19
- x + y + 2 z & = 4
- 2 x + y + 2 z & = k \end{aligned}$$
In this question you must show detailed reasoning. Find the acute angle between the planes $x + 2 y + 3 z = 19$ and $- x + y + 2 z = 4$.

OCR MEI Further Pure Core AS 2019 June Q5

5 Prove by induction that, for all positive integers $n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)$.

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