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OCR MEI Further Extra Pure 2024 June Q5

4 marks Standard +0.8

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

5 marks Moderate -0.8

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

5 marks Standard +0.8

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

12 marks Challenging +1.8

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

13 marks Challenging +1.8

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

17 marks Challenging +1.2

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}

OCR MEI Further Extra Pure 2021 November Q1

11 marks Challenging +1.8

1 In this question you must show detailed reasoning.
A surface $S$ is defined by $z = f ( x , y )$ where $f ( x , y ) = x ^ { 3 } + x ^ { 2 } y - 2 y ^ { 2 }$.

On the coordinate axes in the Printed Answer Booklet, sketch the section $z = f ( 2 , y )$ giving the coordinates of any turning points and any points of intersection with the axes.
Find the stationary points on $S$. $2 G$ is a group of order 8.
Explain why there is no subgroup of $G$ of order 6 . You are now given that $G$ is a cyclic group with the following features:
- $e$ is the identity element of $G$,
- $g$ is a generator of $G$,
- $H$ is the subgroup of $G$ of order 4.
- Write down the possible generators of $H$. $M$ is the group ( $\{ 0,1,2,3,4,5,6,7 \} , + _ { 8 }$ ) where $+ _ { 8 }$ denotes the binary operation of addition modulo 8. You are given that $M$ is isomorphic to $G$.
- Specify all possible isomorphisms between $M$ and $G$.

OCR MEI Further Extra Pure 2021 November Q3

14 marks Standard +0.3

3 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l l } 3 & 3 & 0 \\ 0 & 2 & 2 \\ 1 & 3 & 4 \end{array} \right)$.

Determine the characteristic equation of $\mathbf { A }$.
Hence verify that the eigenvalues of $\mathbf { A }$ are 1, 2 and 6 .
For each eigenvalue of $\mathbf { A }$ determine an associated eigenvector.
Use the results of parts (b) and (c) to find $\mathbf { A } ^ { n }$ as a single matrix, where $n$ is a positive integer.

OCR MEI Further Extra Pure 2021 November Q4

14 marks Challenging +1.8

4 The sequence $u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots$ satisfies the recurrence relation $u _ { n + 2 } - 3 u _ { n + 1 } - 10 u _ { n } = 24 n - 10$.

Determine the general solution of the recurrence relation.
Hence determine the particular solution of the recurrence relation for which $u _ { 0 } = 6$ and $u _ { 1 } = 10$.
Show, by direct calculation, that your solution in part (b) gives the correct value for $u _ { 2 }$. The sequence $v _ { 0 } , v _ { 1 } , v _ { 2 } , \ldots$ is defined by $v _ { n } = \frac { u _ { n } } { p ^ { n } }$ for some constant $p$, where $u _ { n }$ denotes the
particular solution found in part (b). particular solution found in part (b). You are given that $\mathrm { v } _ { \mathrm { n } }$ converges to a finite non-zero limit, $q$, as $n \rightarrow \infty$.
Determine $p$ and $q$.

OCR MEI Further Extra Pure 2021 November Q5

6 marks Challenging +1.8

5 A surface $S$ is defined for $z \geqslant 0$ by $x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 126$. $C$ is the set of points on $S$ for which the tangent plane to $S$ at that point intersects the $x - y$ plane at an angle of $\frac { 1 } { 3 } \pi$ radians. Show that $C$ lies in a plane, $\Pi$, whose equation should be determined.

OCR MEI Further Extra Pure 2021 November Q6

8 marks Challenging +1.8

6 You are given that $q \in \mathbb { Z }$ with $q \geqslant 1$ and that $\mathrm { S } = \frac { 1 } { ( \mathrm { q } + 1 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) ( \mathrm { q } + 3 ) } + \ldots$.

By considering a suitable geometric series show that $\mathrm { S } < \frac { 1 } { \mathrm { q } }$.
Deduce that $S \notin \mathbb { Z }$. You are also given that $\mathrm { e } = \sum _ { r = 0 } ^ { \infty } \frac { 1 } { r ! }$.
Assume that $\mathrm { e } = \frac { \mathrm { p } } { \mathrm { q } }$, where $p$ and $q$ are positive integers. By writing the infinite series for e in a form using $q$ and $S$ and using the result from part (b), prove by contradiction that e is irrational.

OCR MEI Further Extra Pure Specimen Q1

10 marks Challenging +1.8

1 The set $G = \{ 1,4,5,6,7,9,11,16,17 \}$ is a group of order 9 under the binary operation of multiplication modulo 19.

Show that $G$ is a cyclic group generated by the element 4 .
Find another generator for $G$. Justify your answer.
Specify two distinct isomorphisms from the group $J = \{ 0,1,2,3,4,5,6,7,8 \}$ under addition modulo 9 to $G$.

OCR MEI Further Extra Pure Specimen Q2

4 marks Challenging +1.2

2 A binary operation * is defined on the set $S = \{ p , q , r , s , t \}$ by the following composition table.

$*$	$p$	$q$	$r$	$s$	$t$
$p$	$p$	$q$	$r$	$s$	$t$
$q$	$q$	$p$	$s$	$t$	$r$
$r$	$r$	$t$	$p$	$q$	$s$
$s$	$s$	$r$	$t$	$p$	$q$
$t$	$t$	$s$	$q$	$r$	$p$

Determine whether ( $S , *$ ) is a group.

Find the general solution of $$u _ { n } = 8 u _ { n - 1 } - 16 u _ { n - 2 } , n \geq 2 .$$ A new sequence $v _ { n }$ is defined by $v _ { n } = \frac { u _ { n } } { u _ { n - 1 } }$ for $n \geq 1$.
(A) Use $( * )$ to show that $v _ { n } = 8 - \frac { 16 } { v _ { n - 1 } }$. for $n \geq 2$.
(B) Deduce that if $v _ { n }$ tends to a limit then it must be 4 .
Use your general solution in part (i) to show that $\lim _ { n \rightarrow \infty } v _ { n } = 4$.
Deduce the value of $\lim _ { n \rightarrow \infty } \left( \frac { u _ { n } } { u _ { n - 2 } } \right)$.

OCR MEI Further Extra Pure Specimen Q4

16 marks Challenging +1.2

4 A surface $S$ has equation $\mathrm { g } ( x , y , z ) = 0$, where $\mathrm { g } ( x , y , z ) = ( y - 2 x ) ( y + z ) ^ { 2 } - 18$.

Show that $\frac { \partial \mathrm { g } } { \partial y } = ( y + z ) ( - 4 x + 3 y + z )$.
Show that $\frac { \partial \mathrm { g } } { \partial x } + 2 \frac { \partial \mathrm {~g} } { \partial y } - 2 \frac { \partial \mathrm {~g} } { \partial \mathrm { z } } = 0$.
Hence identify a vector which lies in the tangent plane of every point on $S$, explaining your reasoning.
Find the cartesian equation of the tangent plane to the surface $S$ at the point $\mathrm { P } ( 1,4 , - 7 )$. The tangent plane to the surface $S$ at the point $\mathrm { Q } ( 0,2,1 )$ has equation $6 x - 7 y - 4 z = - 18$.
Find a vector equation for the line of intersection of the tangent planes at P and Q .

OCR MEI Further Extra Pure Specimen Q5

18 marks Challenging +1.2

5 In this question you must show detailed reasoning. You are given that the matrix $\mathbf { M } = \left( \begin{array} { c c c } \frac { 1 } { 2 } & - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 } \\ \frac { 1 } { \sqrt { 2 } } & 0 & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { 2 } & \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 } \end{array} \right)$ represents a rotation in 3-D space.

Explain why it follows that $\mathbf { M }$ has 1 as an eigenvalue.
Find a vector equation for the axis of the rotation.
Show that the characteristic equation of $\mathbf { M }$ can be written as $$\lambda ^ { 3 } - \lambda ^ { 2 } + \lambda - 1 = 0 .$$
Find the smallest positive integer $n$ such that $\mathbf { M } ^ { n } = \mathbf { I }$.
Find the magnitude of the angle of the rotation which $\mathbf { M }$ represents. Give your reasoning. {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.
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OCR MEI Further Pure with Technology 2019 June Q1

20 marks Challenging +1.8

1 A family of curves is given by the parametric equations $x ( t ) = \cos ( t ) - \frac { \cos ( ( m + 1 ) t ) } { m + 1 }$ and $y ( t ) = \sin ( t ) - \frac { \sin ( ( m + 1 ) t ) } { m + 1 }$ where $0 \leqslant t < 2 \pi$ and $m$ is a positive integer.

1. Sketch the curves in the cases $m = 3 , m = 4$ and $m = 5$ on separate axes in the Printed Answer Booklet.
2. State one common feature of these three curves.
3. State a feature for the case $m = 4$ which is absent in the cases $m = 3$ and $m = 5$.
1. Determine, in terms of $m$, the values of $t$ for which $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$ but $\frac { \mathrm { d } y } { \mathrm {~d} t } \neq 0$.
2. Describe the tangent to the curve at the points corresponding to such values of $t$.
1. Show that the curve lies between the circle centred at the origin with radius $$1 - \frac { 1 } { m + 1 }$$ and the circle centred at the origin with radius $$1 + \frac { 1 } { m + 1 }$$
2. Hence, or otherwise, show that the area $A$ bounded by the curve satisfies $$\frac { m ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } } < A < \frac { ( m + 2 ) ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } }$$
3. Find the limit of the area bounded by the curve as $m$ tends to infinity.
The arc length of a curve defined by parametric equations $x ( t )$ and $y ( t )$ between points corresponding to $t = c$ and $t = d$, where $c < d$, is $$\int _ { c } ^ { d } \sqrt { \left( \frac { \mathrm {~d} x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } } \mathrm {~d} t$$ Use this to show that the length of the curve is independent of $m$.

OCR MEI Further Pure with Technology 2019 June Q2

20 marks Challenging +1.8

Prove that if $x$ and $y$ are integers which satisfy $x ^ { 2 } - 2 y ^ { 2 } = 1$, then $x$ is odd and $y$ is even.
Create a program to find, for a fixed positive integer $s$, all the positive integer solutions $( x , y )$ to the equation $x ^ { 2 } - 2 y ^ { 2 } = 1$ where $x \leqslant s$ and $y \leqslant s$. Write out your program in the Printed Answer Booklet.
Use your program to find all the positive integer solutions $( x , y )$ to the equation $x ^ { 2 } - 2 y ^ { 2 } = 1$ where $x \leqslant 600$ and $y \leqslant 600$. Give the solutions in ascending order of the value of $x$.
By writing the equation $x ^ { 2 } - 2 y ^ { 2 } = 1$ in the form $( x + \sqrt { 2 } y ) ( x - \sqrt { 2 } y ) = 1$ show how the first solution (the one with the lowest value of $x$ ) in your answer to part (c) can be used to generate the other solutions you found in part (c).
What can you deduce about the number of positive integer solutions $( x , y )$ to the equation $x ^ { 2 } - 2 y ^ { 2 } = 1$ ? In the remainder of this question $T _ { m }$ is the $m ^ { \text {th } }$ triangular number, the sum of the first $m$ positive integers, so that $T _ { m } = \frac { m ( m + 1 ) } { 2 }$.
Create a program to find, for a fixed positive integer $t$, all pairs of positive integers $m$ and $n$ which satisfy $T _ { m } = n ^ { 2 }$ where $m \leqslant t$ and $n \leqslant t$. Write out your program in the Printed Answer Booklet.
Use your program to find all pairs of positive integers $m$ and $n$ which satisfy $T _ { m } = n ^ { 2 }$ where $m \leqslant 300$ and $n \leqslant 300$. Give the pairs in ascending order of the value of $m$.
By comparing your answers to part (c) and part (g), or otherwise, prove that there are infinitely many triangular numbers which are perfect squares.

OCR MEI Further Pure with Technology 2019 June Q3

20 marks Challenging +1.2

3 This question concerns the family of differential equations $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - x ^ { a } y \left( { } ^ { * } \right)$ where $a$ is $- 1,0$ or 1 .

Determine and describe geometrically the isoclines of (\textit{) when
1. $a = - 1$,
2. $a = 0$,
3. $a = 1$.
In this part of the question $a = 0$.
1. Write down the solution to $( * )$ which passes through the point $( 0 , b )$ where $b \neq 1$.
2. Write down the equation of the asymptote to this solution.
In this part of the question $a = - 1$.
1. Write down the solution to $( * )$ which passes through the point $( c , d )$ where $c \neq 0$.
2. Describe the relationship between $c$ and $d$ when the solution in part (i) has a stationary point.
In this part of the question $a = 1$.
1. The standard Runge-Kutta method of order 4 for the solution of the differential equation $\mathrm { f } ( x , y ) = \frac { \mathrm { d } y } { \mathrm {~d} x }$ is as follows. $k _ { 1 } = h \mathrm { f } \left( x _ { n } , y _ { n } \right)$ $k _ { 2 } = h \mathrm { f } \left( x _ { n } + \frac { h } { 2 } , y _ { n } + \frac { k _ { 1 } } { 2 } \right)$ $k _ { 3 } = h \mathrm { f } \left( x _ { n } + \frac { h } { 2 } , y _ { n } + \frac { k _ { 2 } } { 2 } \right)$ $k _ { 4 } = h \mathrm { f } \left( x _ { n } + h , y _ { n } + k _ { 3 } \right)$ $y _ { n + 1 } = y _ { n } + \frac { 1 } { 6 } \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right)$.
  Construct a spreadsheet to solve (}) in the case $x _ { 0 } = 0$ and $y _ { 0 } = 1.5$. State the formulae you have used in your spreadsheet.
2. Use your spreadsheet with $h = 0.05$ to find an approximation to the value of $y$ when $x = 1$.
3. The solution to $( * )$ in which $x _ { 0 } = 0$ and $y _ { 0 } = 1.5$ has a maximum point ( $r , s$ ) with $0 < r < 1$. Use your spreadsheet with suitable values of $h$ to estimate $r$ to two decimal places. Justify your answer.

OCR MEI Further Pure with Technology 2022 June Q1

20 marks Challenging +1.8

A family of curves is given by the equation $$x ^ { 2 } + y ^ { 2 } + 2 a x y = 1 ( * )$$ where the parameter $a$ is a real number.
You may find it helpful to use a slider (for $a$ ) to investigate this family of curves.
1. On the axes in the Printed Answer Booklet, sketch the curve in each of the cases
  - $a = 0$
$a = 0.5$
$a = 2$ (ii) State a feature of the curve for the cases $a = 0 , a = 0.5$ that is not a feature of the curve in the case $a = 2$.
(iii) In the case $a = 1$, the curve consists of two straight lines. Determine the equations of these lines.
1. Find an equation of the curve (*) in polar form.
2. Hence, or otherwise, find, in exact form, the area bounded by the curve, the positive part of the $x$-axis and the positive part of the $y$-axis, in the case $a = 2$.
In this part of the question $m$ is any real number.

OCR MEI Further Pure with Technology 2022 June Q2

20 marks Challenging +1.8

In this part of the question $n$ is an integer greater than 1 .
An integer $q$, where $0 < q < n$ is said to be a quadratic residue modulo $n$ if there exists an integer $x$ such that $\mathrm { x } ^ { 2 } \equiv \mathrm { q } ( \bmod n )$. Note that under this definition 0 is not considered to be a quadratic residue modulo $n$.
1. Find all the integers $x$, where $0 \leqslant x < 1000$ which satisfy $x ^ { 2 } \equiv 481 ( \bmod 1000 )$.
2. Explain why 481 is a quadratic residue modulo 1000.
3. Determine the quadratic residues modulo 11.
4. Determine the quadratic residues modulo 13.
5. Show that, for any integer $m , m ^ { 2 } \equiv ( n - m ) ^ { 2 } ( \bmod n )$.
6. Prove that if $p$ is prime, where $p > 2$, then the number of quadratic residues modulo $p$ is $\frac { p - 1 } { 2 }$.
Fermat's little theorem states that if $p$ is prime and $a$ is an integer which is co-prime to $p$, then $a ^ { p - 1 } \equiv 1 ( \bmod p )$.
1. Use Fermat's little theorem to show that 91 is not prime.
2. Create a program to find an integer $n$ between 500 and 600 which is not prime and such that $\mathrm { a } ^ { \mathrm { n } - 1 } \equiv 1 ( \bmod n )$ for all integers $a$ which are co-prime to $n$.
  In the Printed Answer Booklet
  - Write down your program in full.
State the integer found by your program.

OCR MEI Further Pure with Technology 2022 June Q3

20 marks Challenging +1.2

3 In this question you are required to consider the family of differential equations $\frac { d y } { d x } = \frac { y ^ { a } } { x + 1 } - \frac { 1 } { y } ( * )$ and its solutions. The parameter $a$ is a real number. You should assume that $x \geqslant 0$ and $y > 0$ throughout this question.

In this part of the question $a = 1$.
1. On the axes in the Printed Answer Booklet
  - Sketch the isocline defined by $\frac { d y } { d x } = 0$.
Shade and label the region in which $\frac { \mathrm { dy } } { \mathrm { dx } } > 0$.
Shade and label the region in which $\frac { \mathrm { dy } } { \mathrm { dx } } < 0$.
(ii) For $b > 0$, find, in terms of $b$, the solution to $( * )$ which passes through the point $( 0 , b )$.
(iii) Determine
The values of $b > 0$ for which the solution in (ii) has a turning point.
The corresponding maximum value of $y$.
Fig. 3.1 and Fig. 3.2 show tangent fields for two distinct but unspecified values of $a$. In each case a sketch of the solution curve $y = \mathrm { g } ( x )$ which passes through $( 0,2 )$ is shown for $0 \leqslant x \leqslant 0.5$.

\includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_648_1777_301} \captionsetup{labelformat=empty} \caption{Fig. 3.1}

\includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_652_1777_1117} \captionsetup{labelformat=empty} \caption{Fig. 3.2}

1. The Euler method for the solution of the differential equation $\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { f } ( x , y )$ is as follows $$y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right)$$ It is given that $x _ { 0 } = 0$ and $y _ { 0 } = 2$.
  - Construct a spreadsheet to solve (*) using the Euler method so that the value of $a$ and the value of $h$ can be varied, in the case $x _ { 0 } = 0$ and $y _ { 0 } = 2$.
State the formulae you have used in your spreadsheet.
[0pt] [3]
(ii) In this part of the question $a = 0.1$.

OCR MEI Further Pure with Technology 2023 June Q1

21 marks Challenging +1.2

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter $a$ is a real number. You may find it helpful to use a slider (for $a$ ) to investigate the family of curves $y = f ( x )$.

1. On the axes in the Printed Answer Booklet, sketch the curve $y = f ( x )$ in each of the following cases.
  - $a = - 2$
$a = - 1$
$a = 0$ (ii) State a feature which is common to the curve in all three cases, $a = - 2$, $a = - 1$ and $a = 0$.
(iii) State a feature of the curve for the cases $a = - 2 , a = - 1$ that is not a feature of the curve in the case $a = 0$.
1. Determine the equation of the oblique asymptote to the curve $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ in terms of $a$.
2. For $b \neq - 1,0,1$ let $A$ be the point with coordinates ( $- b , \mathrm { f } ( - b )$ ) and let $B$ be the point with coordinates ( $b , \mathrm { f } ( b )$ ).

$y \geqslant 0$ for all $x \geqslant 0$
$y \leqslant 0$ for all $x \geqslant 0$
In the case of $a = 0$, the curve $\mathrm { y } = \sqrt [ 4 ] { \mathrm { f } ( \mathrm { x } ) }$ has a cusp.

OCR MEI Further Pure with Technology 2023 June Q2

11 marks Challenging +1.8

2 Throughout this question ( $a , b , c$ ) is a Pythagorean triple with the positive integers $a , b , c$ ordered such that $a \leqslant b \leqslant c$.

Show that $\mathrm { a } ^ { 2 } = \mathrm { b } + \mathrm { c }$ if and only if $\mathrm { c } = \mathrm { b } + 1$.
Create a program to find all the Pythagorean triples ( $a , b , c$ ) such that $\mathrm { a } ^ { 2 } = \mathrm { b } + \mathrm { c }$ and $c \leqslant 1000$. Write out your program in full in the Printed Answer Booklet.
Write down the number of Pythagorean triples found by your program in (b).
Prove that there is no Pythagorean triple, $( a , b , c )$, in which $\mathrm { b } ^ { 2 } = \mathrm { a } + \mathrm { c }$.

OCR MEI Further Pure with Technology 2023 June Q3

10 marks Challenging +1.2

3 Wilson's theorem states that an integer $p > 1$ is prime if and only if $( p - 1 ) ! \equiv - 1 ( \bmod p )$.

Use Wilson's theorem to show that $17 ! \equiv 1 ( \bmod 19 )$.
A prime number $p$ is called a Wilson prime if $( p - 1 ) ! \equiv - 1 \left( \bmod p ^ { 2 } \right)$. For example, 5 is a Wilson prime because $( 5 - 1 ) ! \equiv 24 \equiv - 1 ( \bmod 25 )$. At the time of writing all known Wilson primes are less than 1000.
1. Create a program to find all the known Wilson primes. Write out your program in full in the Printed Answer Booklet.
2. Use your program to find and write down all the known Wilson primes.
3. Prove that if there is an integer solution $m$ to the equation $( p - 1 ) ! + 1 = m ^ { 2 }$ where $p$ is prime, then $p$ is a Wilson prime.

OCR MEI Further Pure with Technology 2023 June Q4

18 marks Standard +0.8

4 In this question you are required to consider the family of differential equations $$\frac { d P } { d t } = r P \left( 1 - \frac { P } { K } \right) , \quad t \geqslant 0 , \quad P ( t ) \geqslant 0 \left( ^ { * } \right)$$ where $r$ and $K$ are positive constants. This differential equation can be used as a model for the size of a population $P$ as a function of time $t$.

1. Determine the values of $P$ for which
  - $\frac { \mathrm { dP } } { \mathrm { dt } } = 0$
$\frac { \mathrm { dP } } { \mathrm { dt } } > 0$
$\frac { \mathrm { dP } } { \mathrm { dt } } < 0$ (ii) Solve the equation (*) subject to the initial condition that $P = P _ { 0 }$ when $t = 0$.
(iii) Find a property common to your solution in (ii) in the cases $\mathrm { P } _ { 0 } > \mathrm { K }$ and $\mathrm { P } _ { 0 } < \mathrm { K }$.
(iv) State a feature of your solution in (iii) for the case $\mathrm { P } _ { 0 } > \mathrm { K }$ which is different to the case $P _ { 0 } < K$.
(v) Interpret the value $K$ when $P ( t )$ is the size of a population at time $t$.
In this question you will explore the limitations of using the Euler method to approximate solutions to the differential equation

Construct a spreadsheet to solve (**) using the Euler method so that the value of $h$ can be varied.
State the formulae you have used in your spreadsheet.
(ii) Use your spreadsheet with $h = 0.1$ to approximate
$P ( 1 )$
$P ( 2 )$
$P ( 3 )$ (iii) Use your spreadsheet with $h = 0.05$ to approximate
$P ( 1 )$
$P ( 2 )$
$P ( 3 )$ (iv) State, with reasons, whether the estimates to $P ( t )$ given in your spreadsheet are likely to be overestimates or underestimates to the exact values.
(v) With reference to the diagram, explain any noticeable feature identified when comparing the approximations given to $P ( 2 )$ in (ii) and (iii).

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\(*\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(p\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(q\)	\(q\)	\(p\)	\(s\)	\(t\)	\(r\)
\(r\)	\(r\)	\(t\)	\(p\)	\(q\)	\(s\)
\(s\)	\(s\)	\(r\)	\(t\)	\(p\)	\(q\)
\(t\)	\(t\)	\(s\)	\(q\)	\(r\)	\(p\)

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