Questions - OCR 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 Further Pure Core AS 2022 June Q3

3 In this question you must show detailed reasoning. The roots of the equation $5 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 9 = 0$ are $\alpha , \beta$ and $\gamma$.
Find a cubic equation with integer coefficients whose roots are $\alpha \beta , \beta \gamma$ and $\gamma \alpha$.

OCR Further Pure Core AS 2024 June Q9

9 In this question you must show detailed reasoning. You are given that $a$ is a real root of the equation $x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0$.
You are also given that $a + 2 + 3 \mathrm { i }$ is one root of the equation
$z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0$. Determine all possible values of $z$.

OCR Further Pure Core AS 2020 November Q1

1 In this question you must show detailed reasoning. Use an algebraic method to find the square roots of $- 77 - 36 \mathrm { i }$.
$2 \mathrm { P } , \mathrm { Q }$ and T are three transformations in 2-D.
P is a reflection in the $x$-axis. $\mathbf { A }$ is the matrix that represents P .

Write down the matrix $\mathbf { A }$. Q is a shear in which the $y$-axis is invariant and the point $\binom { 1 } { 0 }$ is transformed to the point $\binom { 1 } { 2 }$. $\mathbf { B }$ is the
matrix that represents Q . matrix that represents Q.
Find the matrix $\mathbf { B }$. T is P followed by Q. C is the matrix that represents T.
Determine the matrix $\mathbf { C }$.
$L$ is the line whose equation is $y = x$.
Explain whether or not $L$ is a line of invariant points under $T$. An object parallelogram, $M$, is transformed under T to an image parallelogram, $N$.
Explain what the value of the determinant of $\mathbf { C }$ means about
- the area of $N$ compared to the area of $M$,
- the orientation of $N$ compared to the orientation of $M$.

OCR Further Pure Core AS 2020 November Q3

3 In this question you must show detailed reasoning. The complex number $7 - 4 \mathrm { i }$ is denoted by $z$.

Giving your answers in the form $a + b \mathrm { i }$, where $a$ and $b$ are rational numbers, find the following.
1. $3 z - 4 z ^ { * }$
2. $( z + 1 - 3 i ) ^ { 2 }$
3. $\frac { z + 1 } { z - 1 }$
Express $z$ in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
The complex number $\omega$ is such that $z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )$. Find the following.
- $| \omega |$
- $\arg ( \omega )$, giving your answer correct to 3 significant figures

OCR Further Pure Core AS 2020 November Q5

5 In this question you must show detailed reasoning. The cubic equation $5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0$ has roots $\alpha , \beta$ and $\gamma$.
Find a cubic equation with integer coefficients whose roots are $\alpha + \beta , \beta + \gamma$ and $\gamma + \alpha$.

OCR Further Pure Core AS 2019 June Q1

1 You are given that $z = 3 - 4 \mathrm { i }$.

Find
- $| z |$,
- $\arg ( z )$,
- $Z ^ { * }$.
On an Argand diagram the complex number $w$ is represented by the point $A$ and $w ^ { * }$ is represented by the point $B$.
Describe the geometrical relationship between the points $A$ and $B$.

OCR Further Pure Core AS 2019 June Q2

2 Matrices $\mathbf { P }$ and $\mathbf { Q }$ are given by $\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0
- 2 & 1 & 3 \end{array} \right)$ and $\mathbf { Q } = ( ( 1 + k ) - 1 )$ where $k$ is a constant.
Exactly one of statements A and B is true.
Statement A: $\quad \mathbf { P }$ and $\mathbf { Q }$ (in that order) are conformable for multiplication.
Statement B: $\quad \mathbf { Q }$ and $\mathbf { P }$ (in that order) are conformable for multiplication.

State, with a reason, which one of A and B is true.
Find either $\mathbf { P Q }$ or $\mathbf { Q P }$ in terms of $k$.

OCR Further Pure Core AS 2019 June Q3

3 The position vector of point $A$ is $\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }$.
The line $l$ passes through $A$ and is perpendicular to $\mathbf { a }$.

Determine the shortest distance between the origin, $O$, and $l$.
$l$ is also perpendicular to the vector $\mathbf { b }$ where $\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }$.
Find a vector which is perpendicular to both $\mathbf { a }$ and $\mathbf { b }$.
Write down an equation of $l$ in vector form.
$P$ is a point on $l$ such that $P A = 2 O A$.
Find angle $P O A$ giving your answer to 3 significant figures.
$C$ is a point whose position vector, $\mathbf { c }$, is given by $\mathbf { c } = p \mathbf { a }$ for some constant $p$. The line $m$ passes through $C$ and has equation $\mathbf { r } = \mathbf { c } + \mu \mathbf { b }$. The point with position vector $9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }$ lies on $m$.
Find the value of $p$.

OCR Further Pure Core AS 2019 June Q4

4 In this question you must show detailed reasoning. You are given that $\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185$ and that $2 + \mathrm { i }$ is a root of the equation $f ( z ) = 0$.

Express $\mathrm { f } ( \mathrm { z } )$ as the product of two quadratic factors with integer coefficients.
Solve $f ( z ) = 0$. Two loci on an Argand diagram are defined by $C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}$ and $C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}$ where $r _ { 1 } > r _ { 2 }$. You are given that two of the points representing the roots of $\mathrm { f } ( \mathrm { z } ) = 0$ are on $C _ { 1 }$ and two are on $C _ { 2 } . R$ is the region on the Argand diagram between $C _ { 1 }$ and $C _ { 2 }$.
Find the exact area of $R$.
$\omega$ is the sum of all the roots of $\mathrm { f } ( \mathrm { z } ) = 0$. Determine whether or not the point on the Argand diagram which represents $\omega$ lies in $R$.

OCR Further Pure Core AS 2019 June Q5

5 In this question you must show detailed reasoning. You are given that $\alpha , \beta$ and $\gamma$ are the roots of the equation $5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0$.

Find the value of $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$.
Find a cubic equation whose roots are $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$ giving your answer in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d = 0$ where $a , b , c$ and $d$ are integers.

OCR Further Pure Core AS 2019 June Q6

6 A transformation T is represented by the matrix $\mathbf { T }$ where $\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4
3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)$. A quadrilateral $Q$, whose area is 12 units, is transformed by T to $Q ^ { \prime }$. Find the smallest possible value of the area of $Q ^ { \prime }$.

OCR Further Pure Core AS 2019 June Q7

7 A transformation A is represented by the matrix $\mathbf { A }$ where $\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2
7 - x & - 6 & 1
5 & - 5 x & 2 x \end{array} \right)$.
The tetrahedron $H$ has vertices at $O , P , Q$ and $R$. The volume of $H$ is 6 units.
$P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }$ and $H ^ { \prime }$ are the images of $P , Q , R$ and $H$ under A .

In the case where $x = 5$
- find the volume of $H ^ { \prime }$,
- determine whether A preserves the orientation of $H$.
- Find the values of $x$ for which $O , P ^ { \prime } , Q ^ { \prime }$ and $R ^ { \prime }$ are coplanar (i.e. the four points lie in the same plane).

OCR Further Pure Core AS 2019 June Q8

8 In this question you must show detailed reasoning. $\mathbf { M }$ is the matrix $\left( \begin{array} { l l } 1 & 6
0 & 2 \end{array} \right)$.
Prove that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right)
0 & 2 ^ { n } \end{array} \right)$, for any positive integer $n$. \section*{END OF QUESTION PAPER}

OCR Further Pure Core AS 2023 June Q1

1 The roots of the equation $4 x ^ { 4 } - 2 x ^ { 3 } - 3 x + 2 = 0$ are $\alpha , \beta , \gamma$ and $\delta$. By using a suitable substitution, find a quartic equation whose roots are $\alpha + 2 , \beta + 2 , \gamma + 2$ and $\delta + 2$ giving your answer in the form $a t ^ { 4 } + b t ^ { 3 } + c t ^ { 2 } + d t + e = 0$, where $a , b , c , d$, and $e$ are integers.

Questions — OCR Further Pure Core AS (67 questions)

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