Questions Further Extra Pure

Given that $\mathbf { A }$ represents a reflection, write down the eigenvalues of $\mathbf { A }$.
Hence find the eigenvectors of $\mathbf { A }$.
Write down the equation of the mirror line of the reflection represented by $\mathbf { A }$.

OCR MEI Further Extra Pure 2019 June Q2

2 A surface $S$ is defined by $z = 4 x ^ { 2 } + 4 y ^ { 2 } - 4 x + 8 y + 11$.

Show that the point $\mathrm { P } ( 0.5 , - 1,6 )$ is the only stationary point on $S$.
1. On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to $z = 42$.
2. By using the sketch in part (b)(i), deduce that P must be a minimum point on $S$.
In the section of $S$ corresponding to $y = c$, the minimum value of $z$ occurs at the point where $x = a$ and $z = 22$.
Find all possible values of $a$ and $c$.

OCR MEI Further Extra Pure 2019 June Q3

3 The matrix $\mathbf { A }$ is $\left( \begin{array} { r r r } - 1 & 2 & 4
0 & - 1 & - 25
- 3 & 5 & - 1 \end{array} \right)$. Use the Cayley-Hamilton theorem to find $\mathbf { A } ^ { - 1 }$.
$4 T$ is the set $\{ 1,2,3,4 \}$. A binary operation • is defined on $T$ such that $a \cdot a = 2$ for all $a \in T$. It is given that ( $T , \cdot$ ) is a group.

Deduce the identity element in $T$, giving a reason for your answer.
Find the value of $1 \cdot 3$, showing how the result is obtained.
1. Complete a group table for ( $T , \bullet$ ).
2. State with a reason whether the group is abelian.

OCR MEI Further Extra Pure 2019 June Q5

5 A financial institution models the repayment of a loan to a client in the following way.

An amount, $\pounds C$, is loaned to the client at the start of the repayment period.
The amount owed $n$ years after the start of the repayment period is $\pounds L _ { n }$, so that $L _ { 0 } = C$.
At the end of each year, interest of $\alpha \% ( \alpha > 0 )$ of the amount owed at the start of that year is added to the amount owed.
Immediately after interest has been added to the amount owed a repayment of $\pounds R$ is made by the client.
Once $L _ { n }$ becomes negative the repayment is finished and the overpayment is refunded to the client.
1. Show that during the repayment period, $L _ { n + 1 } = a L _ { n } + b$, giving $a$ and $b$ in terms of $\alpha$ and $R$.
2. Find the solution of the recurrence relation $L _ { n + 1 } = a L _ { n } + b$ with $L _ { 0 } = C$, giving your solution in terms of $a , b , C$ and $n$.
3. Deduce from parts (a) and (b) that, for the repayment scheme to terminate, $R > \frac { \alpha C } { 100 }$.

A client takes out a $\pounds 30000$ loan at $8 \%$ interest and agrees to repay $\pounds 3000$ at the end of each year.

Use an algebraic method to find the number of years it will take for the loan to be repaid.
Taking into account the refund of overpayment, find the total amount that the client repays over the lifetime of the loan.

OCR MEI Further Extra Pure 2019 June Q6

Given that $\sqrt { 7 }$ is an irrational number, prove that $a ^ { 2 } - 7 b ^ { 2 } \neq 0$ for all $a , b \in \mathbb { Q }$ where $a$ and $b$ are not both 0 .
A set $G$ is defined by $G = \{ a + b \sqrt { 7 } : a , b \in \mathbb { Q } , a$ and $b$ not both $0 \}$. Prove that $G$ is a group under multiplication. (You may assume that multiplication is associative.)
A subset $H$ of $G$ is defined by $H = \{ 1 + c \sqrt { 7 } : c \in \mathbb { Q } \}$. Determine whether or not $H$ is a subgroup of ( $G , \times$ ).
Using $( G , \times )$, prove by counter-example that the statement 'An infinite group cannot have a non-trivial subgroup of finite order' is false.

OCR MEI Further Extra Pure 2022 June Q1

1 Three sequences, $\mathrm { a } _ { \mathrm { n } } , \mathrm { b } _ { \mathrm { n } }$ and $\mathrm { c } _ { \mathrm { n } }$, are defined for $n \geqslant 1$ by the following recurrence relations. $$\begin{aligned} & \left( a _ { n + 1 } - 2 \right) \left( 2 - a _ { n } \right) = 3 \text { with } a _ { 1 } = 3
& b _ { n + 1 } = - \frac { 1 } { 2 } b _ { n } + 3 \text { with } b _ { 1 } = 1.5
& c _ { n + 1 } - \frac { c _ { n } ^ { 2 } } { n } = 1 \text { with } c _ { 1 } = 2.5 \end{aligned}$$ The output from a spreadsheet which presents the first 10 terms of $a _ { n } , b _ { n }$ and $c _ { n }$, is shown below.

	A	B	C	D
1	$n$	$a _ { n }$	$b _ { n }$	$c _ { n }$
2	1	3	1.5	2.5
3	2	-1	2.25	7.25
4	3	3	1.875	27.28125
5	4	-1	2.0625	249.0889
6	5	3	1.96875	15512.32
7	6	-1	2.01563	48126390
8	7	3	1.99219	3.86E+14
9	8	-1	2.00391	$2.13 \mathrm { E } + 28$
10	9	3	1.99805	5.66E+55
11	10	-1	2.00098	3.6E+110

Without attempting to solve any recurrence relations, describe the apparent behaviour, including as $n \rightarrow \infty$, of

$a _ { n }$
$\mathrm { b } _ { \mathrm { n } }$
$\mathrm { C } _ { \mathrm { n } }$

OCR MEI Further Extra Pure 2022 June Q2

2 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r r } 10 & 12 & - 8
- 1 & 2 & 4
3 & 6 & 2 \end{array} \right)$.

In this question you must show detailed reasoning. Show that the characteristic equation of $\mathbf { A }$ is $- \lambda ^ { 3 } + 14 \lambda ^ { 2 } - 56 \lambda + 64 = 0$.
Use the Cayley-Hamilton theorem to determine $\mathbf { A } ^ { - 1 }$. A matrix $\mathbf { E }$ and a diagonal matrix $\mathbf { D }$ are such that $\mathbf { A } = \mathbf { E D E } ^ { - 1 }$. The elements in the diagonal of $\mathbf { D }$ increase from top left to bottom right.
Determine the matrix $\mathbf { D }$.

OCR MEI Further Extra Pure 2022 June Q3

3 A sequence is defined by the recurrence relation $5 t _ { n + 1 } - 4 t _ { n } = 3 n ^ { 2 } + 28 n + 6$, for $n \geqslant 0$, with $t _ { 0 } = 7$.

Find an expression for $t _ { n }$ in terms of $n$. Another sequence is defined by $\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { n } ^ { \mathrm { m } } }$, for $n \geqslant 1$, where $m$ is a constant.
In each of the following cases determine $\lim _ { n \rightarrow \infty } \mathrm {~V} _ { n }$, if it exists, or show that the sequence is divergent.
1. $m = 3$
2. $m = 2$
3. $m = 1$

OCR MEI Further Extra Pure 2022 June Q4

4 A binary operation, ○, is defined on a set of numbers, $A$, in the following way.
$a \circ b = \mathrm { k } _ { 1 } \mathrm { a } - \mathrm { k } _ { 2 } \mathrm {~b} + \mathrm { k } _ { 3 }$, for $a , b \in A$,
where $k _ { 1 } , k _ { 2 }$ and $k _ { 3 }$ are constants (which are not necessarily in $A$ ) and the operations addition, subtraction and multiplication of numbers have their usual notation and meaning. You are initially given the following information about ○ and $A$.

$A = \mathbb { R }$
$0 \circ 0 = 2$
An identity element, $e$, exists for ∘ in $A$
1. Show that $a \circ b = a + b + 2$.
2. State the value of $e$.
3. Explain whether ○ is commutative over $A$.
4. Determine whether or not ( $A , \circ$ ) is a group.
5. Explain whether your answer to part (d) would change in each of the following cases, giving details of any change.
  1. $A = \mathbb { Z }$
  2. $A = \{ 2 m : m \in \mathbb { Z } \}$
  3. $\quad A = \{ n : n \in \mathbb { Z } , n \geqslant - 2 \}$

OCR MEI Further Extra Pure 2022 June Q5

5 A surface $S$ is defined by $z = f ( x , y )$, where $f ( x , y ) = y e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }$.

1. Find $\frac { \partial f } { \partial x }$.
2. Show that $\frac { \partial f } { \partial y } = - \left( x ^ { 2 } y + 2 x y + 2 y - 1 \right) e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }$.
3. Determine the coordinates of any stationary points on $S$. Fig. 5.1 shows the graph of $z = e ^ { - x ^ { 2 } }$ and Fig. 5.2 shows the contour of $S$ defined by $z = 0.25$. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_686_822_244} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
  \end{figure} \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_437_822_1105} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
  \end{figure}
Specify a sequence of transformations which transforms the graph of $\mathrm { z } = \mathrm { e } ^ { - \mathrm { x } ^ { 2 } }$ onto the graph of the section defined by $z = f ( x , 1 )$.
Hence, or otherwise, sketch the section defined by $z = f ( x , 1 )$.
Using Fig. 5.2 and your answer to part (c), classify any stationary points on $S$, justifying your answer. You are given that $P$ is a point on $S$ where $z = 0$.
Find, in vector form, the equation of the tangent plane to $S$ at $P$. The tangent plane found in part (e) intersects $S$ in a straight line, $L$.
Write down, in vector form, the equation of $L$.

OCR MEI Further Extra Pure 2023 June Q1

1 A surface is defined in 3-D by $z = 3 x ^ { 3 } + 6 x y + y ^ { 2 }$.
Determine the coordinates of any stationary points on the surface.

OCR MEI Further Extra Pure 2023 June Q2

2 A sequence is defined by the recurrence relation $4 \mathrm { t } _ { \mathrm { n } + 1 } - \mathrm { t } _ { \mathrm { n } } = 15 \mathrm { n } + 17$ for $\mathrm { n } \geqslant 1$, with $t _ { 1 } = 2$.

Solve the recurrence relation to find the particular solution for $\mathrm { t } _ { \mathrm { n } }$. Another sequence is defined by the recurrence relation $( n + 1 ) u _ { n + 1 } - u _ { n } ^ { 2 } = 2 n - \frac { 1 } { n ^ { 2 } }$ for $n \geqslant 1$, with $u _ { 1 } = 2$.
1. Explain why the recurrence relation for $\mathrm { u } _ { \mathrm { n } }$ cannot be solved using standard techniques for non-homogeneous first order recurrence relations.
2. Verify that the particular solution to this recurrence relation is given by $u _ { n } = a n + \frac { b } { n }$ where $a$ and $b$ are constants whose values are to be determined. A third sequence is defined by $\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { u } _ { \mathrm { n } } }$ for $n \geqslant 1$.
Determine $\lim _ { n \rightarrow \infty } \mathrm { v } _ { \mathrm { n } }$.

OCR MEI Further Extra Pure 2023 June Q3

3 A surface, $S$, is defined by $g ( x , y , z ) = 0$ where $g ( x , y , z ) = 2 x ^ { 3 } - x ^ { 2 } y + 2 x y ^ { 2 } + 27 z$. The normal to $S$ at the point $\left( 1,1 , - \frac { 1 } { 9 } \right)$ and the tangent plane to $S$ at the point $( 3,3 , - 3 )$ intersect at $P$. Determine the position vector of P .

OCR MEI Further Extra Pure 2023 June Q4

4 The set $G$ is given by $G = \{ \mathbf { M } : \mathbf { M }$ is a real $2 \times 2$ matrix and det $\mathbf { M } = 1 \}$.

Show that $G$ forms a group under matrix multiplication, × . You may assume that matrix multiplication is associative.
The matrix $\mathbf { A } _ { n }$ is defined by $\mathbf { A } _ { n } = \left( \begin{array} { l l } 1 & 0
n & 1 \end{array} \right)$ for any integer $n$. The set $S$ is defined by $\mathrm { S } = \left\{ \mathrm { A } _ { \mathrm { n } } : \mathrm { n } \in \mathbb { Z } , \mathrm { n } \geqslant 0 \right\}$.
1. Determine whether $S$ is closed under × .
2. Determine whether $S$ is a subgroup of ( $G , \times$ ).
1. Find a subgroup of ( $G , \times$ ) of order 2 .
2. By considering the inverse of the non-identity element in any such subgroup, or otherwise, show that this is the only subgroup of ( $G , \times$ ) of order 2. The set of all real $2 \times 2$ matrices is denoted by $H$.
With the help of an example, explain why ( $H , \times$ ) is not a group.

OCR MEI Further Extra Pure 2023 June Q5

5 The matrix $\mathbf { P }$ is given by $\mathbf { P } = \left( \begin{array} { l l } a & 0
2 & 3 \end{array} \right)$ where $a$ is a constant and $a \neq 3$.

Given that the acute angle between the directions of the eigenvectors of $\mathbf { P }$ is $\frac { 1 } { 4 } \pi$ radians, determine the possible values of $a$.
You are given instead that $\mathbf { P }$ satisfies the matrix equation $\mathbf { I } = \mathbf { P } ^ { 2 } + r \mathbf { P }$ for some rational number $r$.
1. Use the Cayley-Hamilton theorem to determine the value of $a$ and the corresponding value of $r$.
2. Hence show that $\mathbf { P } ^ { 4 } = \mathbf { s } \mathbf { + t } \mathbf { t } \mathbf { P }$ where $s$ and $t$ are rational numbers to be determined. You should not calculate $\mathbf { P } ^ { 4 }$.

OCR MEI Further Extra Pure 2024 June Q1

1 A surface, $S$, is defined in 3-D by $z = f ( x , y )$ where $f ( x , y ) = 12 x - 30 y + 6 x y$.

Determine the coordinates of any stationary points on the surface.
The equation $\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { a } )$, where $a$ is a constant, defines a section of S . Given that this equation is $\mathrm { z } = 24 \mathrm { x } + \mathrm { b }$, find the value of $a$ and the value of $b$. The diagram shows the contour $z = 12$ and its associated asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-2_860_1143_742_242}
Find the equations of the asymptotes.
By forming grad $g$, where $g ( x , y , z ) = f ( x , y ) - z$, find the equation of the tangent plane to $S$ at the point where $x = 3$ and $y = 2$. Give your answer in vector form. The point $( 0,4 , - 120 )$, which lies on S , is denoted by A .
The plane with equation $\mathbf { r }$. $\left( \begin{array} { r } 3
3
- 2 \end{array} \right) = 52$ is denoted by $\Pi$.
Show that the normal to S at A intersects $\Pi$ at the point $( - 360,304 , - 110 )$.

OCR MEI Further Extra Pure 2024 June Q2

Determine the general solution of the recurrence relation $2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 0$.
Using your answer to part (a), determine the general solution of the recurrence relation $2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 20 n ^ { 2 } + 60 n$. In the rest of this question the sequence $u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots$ satisfies the recurrence relation in part (b). You are given that $u _ { 0 } = - 9$ and $u _ { 1 } = - 12$.
Determine the particular solution for $\mathrm { u } _ { \mathrm { n } }$. You are given that, as $n$ increases, once the values of $u _ { n }$ start to increase, then from that point onwards the sequence is an increasing sequence.
Use your answer to part (c) to determine, by direct calculation, the least value taken by terms in the sequence. You should show any values that you rely on in your argument.

OCR MEI Further Extra Pure 2024 June Q3

3 Fig. 3.1 shows an equilateral triangle, with vertices $\mathrm { A } , \mathrm { B }$ and C , and the three axes of symmetry of the triangle, $\mathrm { S } _ { \mathrm { a } } , \mathrm { S } _ { \mathrm { b } }$ and $\mathrm { S } _ { \mathrm { c } }$. The axes of symmetry are fixed in space and all intersect at the point O . \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Fig. 3.1} \includegraphics[alt={},max width=\textwidth]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-4_440_394_440_248}

\end{figure} There are six distinct transformations under which the image of the triangle is indistinguishable from the triangle itself, ignoring labels.
These are denoted by $\mathrm { I } , \mathrm { M } _ { a ^ { \prime } } \mathrm { M } _ { \mathrm { b } ^ { \prime } } , \mathrm { M } _ { \mathrm { c } ^ { \prime } } , \mathrm { R } _ { 120 }$ and $\mathrm { R } _ { 240 }$ where

I is the identity transformation
$\mathrm { M } _ { \mathrm { a } }$ is a reflection in the mirror line $\mathrm { S } _ { \mathrm { a } }$ (and likewise for $\mathrm { M } _ { \mathrm { b } }$ and $\mathrm { M } _ { \mathrm { c } }$ )
$\mathrm { R } _ { 120 }$ is an anticlockwise rotation by $120 ^ { \circ }$ about O (and likewise for $\mathrm { R } _ { 240 }$ ).

Composition of transformations is denoted by ○.
Fig. 3.2 illustrates the composition of $R _ { 120 }$ followed by $R _ { 240 }$, denoted by $R _ { 240 } \circ R _ { 120 }$. This shows that $\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 }$ is equivalent to the identity transformation, so that $\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 } = \mathrm { I }$. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Fig. 3.2} \includegraphics[alt={},max width=\textwidth]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-4_321_1447_1628_242}

\end{figure}

Using the blank diagrams in the Printed Answer Booklet, find the single transformation which is equivalent to each of the following.

$M _ { a } \circ M _ { a }$
$M _ { b } \circ M _ { a }$
$\mathrm { R } _ { 120 } \circ \mathrm { M } _ { \mathrm { a } }$

The set of the six transformations is denoted by G and you are given that $( \mathrm { G } , \circ )$ is a group. The table below is a mostly empty composition table for $\circ$. The entry given is that for $R _ { 240 } \circ R _ { 120 }$.
First transformation performed is
followed by

◯	I	$\mathrm { M } _ { \mathrm { a } }$	$\mathrm { M } _ { \mathrm { b } }$	$\mathrm { M } _ { \mathrm { c } }$	$\mathrm { R } _ { 120 }$	$\mathrm { R } _ { 240 }$
I
$\mathrm { M } _ { \mathrm { a } }$
$\mathrm { M } _ { \mathrm { b } }$
$\mathrm { M } _ { \mathrm { c } }$
$\mathrm { R } _ { 120 }$
$\mathrm { R } _ { 240 }$					I

Complete the copy of this table in the Printed Answer Booklet. You can use some or all of the spare copies of the diagram in the Printed Answer Booklet to help.
Explain why there can be no subgroup of $( \mathrm { G } , \circ )$ of order 4.
A student makes the following claim.
"If all the proper non-trivial subgroups of a group are abelian then the group itself is abelian."
Explain why the claim is incorrect, justifying your answer fully.
With reference to the order of elements in the groups, explain why ( $\mathrm { G } , \circ$ ) is not isomorphic to $\mathrm { C } _ { 6 }$, the cyclic group of order 6 .

OCR MEI Further Extra Pure 2024 June Q4

4 The matrix $\mathbf { P }$ is given by $\mathbf { P } = \left( \begin{array} { r r r } 1 & 7 & 8
- 6 & 12 & 12
- 2 & 4 & 8 \end{array} \right)$.

Show that the characteristic equation of $\mathbf { P }$ is $- \lambda ^ { 3 } + 21 \lambda ^ { 2 } - 126 \lambda + 216 = 0$. You are given that the roots of this equation are 3,6 and 12 .
1. Verify that $\left( \begin{array} { r } 1
  - 2
  2 \end{array} \right)$ is an eigenvector of $\mathbf { P }$, stating its associated eigenvalue.
2. The vector $\left( \begin{array} { l } x
  y
  z \end{array} \right)$ is an eigenvector of $\mathbf { P }$ with eigenvalue 6. Given that $z = 5$, find $x$ and $y$. You are given that $\mathbf { P }$ can be expressed in the form $\mathbf { E D E } ^ { - 1 }$, where $\mathbf { E } = \left( \begin{array} { r r r } 3 & 2 & 1
  1 & 2 & - 2
  1 & 1 & 2 \end{array} \right)$ and $\mathbf { D }$ is a diagonal matrix. The characteristic equation of $\mathbf { E }$ is $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 15 \lambda + 9 = 0$.
1. Use the Cayley-Hamilton theorem to express $\mathbf { E } ^ { - 1 }$ in terms of positive powers of $\mathbf { E }$.
2. Hence find $\mathbf { E } ^ { - 1 }$.
3. By identifying the matrix $\mathbf { D }$ and using $\mathbf { P } = \mathbf { E D E } ^ { - 1 }$, determine $\mathbf { P } ^ { 4 }$.

OCR MEI Further Extra Pure 2024 June Q5

5 In this question you may assume that if $p$ and $q$ are distinct prime numbers and $\mathbf { p } ^ { \alpha } = \mathbf { q } ^ { \beta }$ where $\alpha , \beta \in \mathbb { Z }$, then $\alpha = 0$ and $\beta = 0$.

Prove that it is not possible to find $a$ and $b$ for which $\mathrm { a } , \mathrm { b } \in \mathbb { Z }$ and $3 = 2 ^ { \frac { \mathrm { a } } { \mathrm { b } } }$.
Deduce that $\log _ { 2 } 3 \notin \mathbb { Q }$.

OCR MEI Further Extra Pure 2020 November Q1

1 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r } 0 & 2
3 & - 1 \end{array} \right)$.
Find

the eigenvalues of $\mathbf { A }$,
an eigenvector associated with each eigenvalue.

OCR MEI Further Extra Pure 2020 November Q2

2 A sequence is defined by the recurrence relation $t _ { n + 1 } = \frac { t _ { n } } { n + 3 }$ for $n \geqslant 1$, with $t _ { 1 } = 8$.
Verify that the particular solution to the recurrence relation is given by $t _ { n } = \frac { a } { ( n + b ) ! }$ where $a$ and $b$ are constants whose values are to be determined.

OCR MEI Further Extra Pure 2020 November Q3

3 A sequence is defined by the recurrence relation $u _ { n + 2 } = 4 u _ { n + 1 } - 5 u _ { n }$ for $n \geqslant 0$, with $u _ { 0 } = 0$ and $u _ { 1 } = 1$.

Find an exact real expression for $u _ { n }$ in terms of $n$ and $\theta$, where $\tan \theta = \frac { 1 } { 2 }$. A sequence is defined by $v _ { n } = a ^ { \frac { 1 } { 2 } n } u _ { n }$ for $n \geqslant 0$, where $a$ is a positive constant.
In each of the following cases, describe the behaviour of $v _ { n }$ as $n \rightarrow \infty$.
- $a = 0.1$
- $a = 0.2$
- $a = 1$

OCR MEI Further Extra Pure 2020 November Q4

In each of the following cases, a set $G$ and a binary operation ∘ are given. The operation ∘ may be assumed to be associative on $G$. Determine which, if any, of the other three group axioms are satisfied by ( $G , \circ$ ) and which, if any, are not satisfied.
1. $G = \{ 2 n + 1 : n \in \mathbb { Z } \}$ and $\circ$ is addition.
2. $G = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \}$ and ∘ is multiplication.
3. $G$ is the set of all real numbers and ∘ is multiplication.
A group $M$ consists of eight $2 \times 2$ matrices under the operation of matrix multiplication. Five of the eight elements of $M$ are as follows. $$\frac { 1 } { \sqrt { 2 } } \left( \begin{array} { l l } 1 & \mathrm { i }
\mathrm { i } & 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } - 1 & \mathrm { i }
\mathrm { i } & - 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } 1 & - \mathrm { i }
- \mathrm { i } & 1 \end{array} \right) \quad \left( \begin{array} { l l } 0 & \mathrm { i }
\mathrm { i } & 0 \end{array} \right) \quad \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right)$$
1. Find the other three elements of $M$.
  $( N , * )$ is another group of order 8, with identity element $e$. You are given that $N = \langle a , b , c \rangle$ where $a * a = b * b = c * c = e$.
2. State whether $M$ and $N$ are isomorphic to each other, giving a reason for your answer.

OCR MEI Further Extra Pure 2020 November Q6

6 A surface $S$ is defined by $z = \mathrm { f } ( x , y ) = 4 x ^ { 4 } + 4 y ^ { 4 } - 17 x ^ { 2 } y ^ { 2 }$.

1. Show that there is only one stationary point on $S$. The value of $z$ at the stationary point is denoted by $s$.
2. State the value of $s$.
3. By factorising $\mathrm { f } ( x , y )$, sketch the contour lines of the surface for $z = s$.
4. Hence explain whether the stationary point is a maximum point, a minimum point or a saddle point. C is a point on $S$ with coordinates ( $a , a , \mathrm { f } ( a , a )$ ) where $a$ is a constant and $a \neq 0$. $\Pi$ is the tangent plane to $S$ at C .
1. Find the equation of $\Pi$ in the form r.n $= p$.
2. The shortest distance from the origin to $\Pi$ is denoted by $d$. Show that $\frac { d } { a } \rightarrow \frac { 3 \sqrt { 2 } } { 4 }$ as $a \rightarrow \infty$.
3. Explain whether the origin lies above or below $\Pi$. \section*{END OF QUESTION PAPER}

◯	I	\(\mathrm { M } _ { \mathrm { a } }\)	\(\mathrm { M } _ { \mathrm { b } }\)	\(\mathrm { M } _ { \mathrm { c } }\)	\(\mathrm { R } _ { 120 }\)	\(\mathrm { R } _ { 240 }\)
I
\(\mathrm { M } _ { \mathrm { a } }\)
\(\mathrm { M } _ { \mathrm { b } }\)
\(\mathrm { M } _ { \mathrm { c } }\)
\(\mathrm { R } _ { 120 }\)
\(\mathrm { R } _ { 240 }\)					I

	A	B	C	D
1	\(n\)	\(a _ { n }\)	\(b _ { n }\)	\(c _ { n }\)
2	1	3	1.5	2.5
3	2	-1	2.25	7.25
4	3	3	1.875	27.28125
5	4	-1	2.0625	249.0889
6	5	3	1.96875	15512.32
7	6	-1	2.01563	48126390
8	7	3	1.99219	3.86E+14
9	8	-1	2.00391	\(2.13 \mathrm { E } + 28\)
10	9	3	1.99805	5.66E+55
11	10	-1	2.00098	3.6E+110

Questions Further Extra Pure (35 questions)

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OCR MEI Further Extra Pure 2019 June Q1

OCR MEI Further Extra Pure 2019 June Q2

OCR MEI Further Extra Pure 2019 June Q3

OCR MEI Further Extra Pure 2019 June Q5

OCR MEI Further Extra Pure 2019 June Q6

OCR MEI Further Extra Pure 2022 June Q1

OCR MEI Further Extra Pure 2022 June Q2

OCR MEI Further Extra Pure 2022 June Q3

OCR MEI Further Extra Pure 2022 June Q4

OCR MEI Further Extra Pure 2022 June Q5

OCR MEI Further Extra Pure 2023 June Q1

OCR MEI Further Extra Pure 2023 June Q2

OCR MEI Further Extra Pure 2023 June Q3

OCR MEI Further Extra Pure 2023 June Q4

OCR MEI Further Extra Pure 2023 June Q5

OCR MEI Further Extra Pure 2024 June Q1

OCR MEI Further Extra Pure 2024 June Q2

OCR MEI Further Extra Pure 2024 June Q3

OCR MEI Further Extra Pure 2024 June Q4

OCR MEI Further Extra Pure 2024 June Q5

OCR MEI Further Extra Pure 2020 November Q1

OCR MEI Further Extra Pure 2020 November Q2

OCR MEI Further Extra Pure 2020 November Q3

OCR MEI Further Extra Pure 2020 November Q4

OCR MEI Further Extra Pure 2020 November Q6