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OCR MEI Further Extra Pure 2023 June Q5

15 marks Challenging +1.2

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

17 marks Standard +0.3

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

12 marks Challenging +1.8

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

12 marks Challenging +1.2

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.

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

15 marks Standard +0.8

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

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 $n \rightarrow \infty$.

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 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
  - $a = 0$
  - $a = 0.5$
  - $a = 2$
  - 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 $a = 2$.
  - 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

2 \begin{enumerate}[label=(\alph*)] \item 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 $n$.

Find all the integers $x$, where $0 \leqslant x < 1000$ which satisfy $x ^ { 2 } \equiv 481 ( \bmod 1000 )$.
Explain why 481 is a quadratic residue modulo 1000.
Determine the quadratic residues modulo 11.
Determine the quadratic residues modulo 13.
Show that, for any integer $m , m ^ { 2 } \equiv ( n - m ) ^ { 2 } ( \bmod n )$.
Prove that if $p$ is prime, where $p > 2$, then the number of quadratic residues modulo $p$ is $\frac { p - 1 } { 2 }$.

\item Fermat's little theorem states that if $p$ is prime and $a$ is an integer which is co-prime to $p$, then $a ^ { p - 1 } \equiv 1 ( \bmod p )$.

Use Fermat's little theorem to show that 91 is not prime.
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

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 $\frac { \mathrm { dy } } { \mathrm { dx } } < 0$.
  - For $b > 0$, find, in terms of $b$, the solution to $( * )$ which passes through the point $( 0 , b )$.
  - 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 $0 \leqslant x \leqslant 0.5$.
  \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_648_1777_301} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
  \end{figure} \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_652_1777_1117} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
  \end{figure}
  1. For the case in Fig. 3.1 suggest a possible value of $a$.
  2. For the case in Fig. 3.2 suggest a possible value of $a$.
  3. In each case, continue the sketch of the solution curves for $0.5 \leqslant x \leqslant 5$ in the Printed Answer Booklet.
  4. State a feature which is present in one of the curves in part (iii) for $0.5 \leqslant x \leqslant 5$ but not in the other.
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 $y _ { 0 } = 2$.
  Use your spreadsheet with $h = 0.1$ to approximate the value of $y$ when $x = 3$ for the solution to (*) in which $y = 2$ when $x = 0$.
2. In this part of the question $a = - 0.2$. Use your spreadsheet to approximate, to $\mathbf { 1 }$ decimal place, the $x$-coordinate of the local maximum for the solution to (*) in which $y = 2$ when $x = 0$.

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 $y = f ( x )$. \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve $y = f ( x )$ in each of the following cases.

$a = - 2$
$a = - 1$
$a = 0$
State a feature which is common to the curve in all three cases, $a = - 2$, $a = - 1$ and $a = 0$.
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 ( $b , \mathrm { f } ( b )$ ).

Show that the $y$-coordinate of the point at which the chord to the curve $y = f ( x )$ between $A$ and $B$ meets the $y$-axis is independent of $a$.

With $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$, determine the range of values of $a$ for which

Find its coordinates and fully justify that it is 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
  $$\frac { d P } { d t } = 2 P ^ { 1.25 } \left( 1 - \frac { P } { 1000 } \right) ^ { 1.5 } , t \geqslant 0 , P ( t ) \geqslant 0 ( * * )$$ The diagram shows the tangent field to (**), and a solution in which $P = 1$ when $t = 0$, produced using a much more accurate numerical method. \includegraphics[max width=\textwidth, alt={}, center]{4715d0f0-a860-4189-802f-1d2d019e1115-4_899_1552_1763_319}
  1. The Euler method for the solution of the differential equation $f ( t , P ) = \frac { d P } { d t }$ is as follows $$P _ { n + 1 } = P _ { n } + h f \left( t _ { n } , P _ { n } \right)$$ It is given that $t _ { 0 } = 0$ and $P _ { 0 } = 1$.

OCR MEI Further Pure with Technology 2024 June Q1

17 marks Standard +0.8

1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter $a$ is a real number.

1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
  - $a = - 0.5$
  - $a = - 0.1$
  - $a = 0.5$
  - State one feature of the curve for the cases $a = - 0.5$ and $a = - 0.1$ that is not a feature of the curve in the case $a = 0.5$.
  - By using a slider for $a$, or otherwise, write down the non-zero value of $a$ for which the points on the curve (\textit{) all lie on a straight line.
  - Write down the equation of the vertical asymptote of the curve (}).
  The equation of the curve (*) can be written in the form $y = x + A + \frac { a ^ { 2 } - a } { x - 1 }$, where $A$ is a constant.
2. Show that $A = 0$.
3. Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$
4. Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
5. In this part of the question the value of the parameter $a$ satisfies $0 < a < 1$. For values of $a$ in this range the curve intersects the $x$-axis at points X and Y . The point Z has coordinates $( 0 , - 1 )$. These three points form a triangle XYZ.
  1. Determine, in terms of $a$, the area of the triangle XYZ.
  2. Find the maximum area of the triangle XYZ.

OCR MEI Further Pure with Technology 2024 June Q2

23 marks Challenging +1.8

2 Wilson's theorem states that a positive integer $n > 1$ is prime if and only if $( n - 1 ) ! \equiv n - 1 ( \bmod n )$.

1. Calculate 136! (mod 137).
2. Hence, determine if the integer 137 is prime.
1. Create a program that uses Wilson's theorem to find all prime numbers less than or equal to $n$.
  Write down your program in the Printed Answer Booklet.
2. Using part (b)(i), write down all prime numbers $x$, where $260 \leqslant x \leqslant 300$.
1. Explain why there is exactly one prime number congruent to $2 ( \bmod 4 )$.
2. Explain why no prime number is congruent to $0 ( \bmod 4 )$.
3. Using part (b)(ii), write down the three prime numbers $y$, where $260 \leqslant y \leqslant 300$, that are congruent to $3 ( \bmod 4 )$. Label the three prime numbers in part (c)(iii) $c _ { 1 } , c _ { 2 }$ and $c _ { 3 }$. Define the integer $N$ by $N = 4 c _ { 1 } c _ { 2 } c _ { 3 } + 3$.
4. Explain why $N$ is not divisible by $c _ { 1 } , c _ { 2 }$ or $c _ { 3 }$.
5. Write down the value of $N ( \bmod 4 )$.
The fundamental theorem of arithmetic states that every positive integer can be written uniquely as the product of powers of prime numbers. Suppose there are finitely many prime numbers congruent to $3 ( \bmod 4 )$. Label these prime numbers $p _ { 1 } , \ldots , \mathrm { p } _ { \mathrm { n } }$, where $p _ { 1 } = 3$. Using the $n - 1$ integers $p _ { 2 } , \ldots , \mathrm { p } _ { \mathrm { n } }$, define the integer $M$ by $\mathrm { M } = 4 \mathrm { p } _ { 2 } \mathrm { p } _ { 3 } \ldots \mathrm { p } _ { \mathrm { n } } + 3$.
1. Write down the value of $M ( \bmod 4 )$.
2. Explain why $\mathrm { M } = \mathrm { q } _ { 1 } \mathrm { q } _ { 2 } \ldots \mathrm { q } _ { \mathrm { r } }$, where the integers $q _ { 1 } , \ldots , \mathrm { q } _ { \mathrm { r } }$ are all prime.
3. Prove that there is at least one integer $i$, where $1 \leqslant i \leqslant n$, such that $q _ { i } \equiv 3 ( \bmod 4 )$.
4. Hence, deduce that there are infinitely many prime numbers congruent to $3 ( \bmod 4 )$.

OCR MEI Further Pure with Technology 2024 June Q3

20 marks Standard +0.8

3 This question concerns the family of differential equations $$\frac { d y } { d x } = x ^ { 2 } - y + \operatorname { acos } ( x ) \cos ( y ) \quad ( * * )$$ where $a$ is a constant, $x \geqslant 0$ and $y > 0$.

In this part of the question $a = 0$.
1. Find the solution to (\textbf{) in which $y = 1$ when $x = 0$.
2. In this part of the question $m$ is a real number. Show that the equation of the isocline $\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { m }$ is a parabola.
3. Using the result given in part (a)(ii), or otherwise, sketch the tangent field for (}) on the axes in the Printed Answer Booklet.
Fig. 3.1 and Fig. 3.2 show the tangent fields for two distinct and unspecified values of $a$. In each case, a sketch of the solution curve $\mathrm { y } = \mathrm { g } ( \mathrm { x } )$ which passes through the point $( 0,2 )$ is shown for $0 \leqslant x \leqslant \frac { 1 } { 2 }$. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 3.1} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_399_666_1324_317}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 3.2} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_397_661_1324_1192}
\end{figure}
1. In each case, continue the sketch of the solution curve for $\frac { 1 } { 2 } \leqslant x \leqslant 3$ on the axes in the Printed Answer Booklet.
2. State one feature which is present in the continued solution curve for Fig. 3.1 that is not a feature of the continued solution curve for Fig. 3.2.
3. Using a slider for $a$, or otherwise, estimate the value of $a$ for the solution curve shown in Fig. 3.2.
The Euler method for the solution of the differential equation $\frac { d y } { d x } = f ( x , y )$ is as follows. $$\begin{aligned} & y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right) \\ & x _ { n + 1 } = x _ { n } + h \end{aligned}$$
1. Construct a spreadsheet to solve (), so that the value of $a$ and the value of $h$ can be varied, in the case $x _ { 0 = 0$ and $y _ { 0 } = 1$. State the formulae you have used in your spreadsheet.
2. In this part of the question $a = 0$. Use your spreadsheet with $h = 0.1$ to approximate the value of $y$ when $x = 0.5$ for the solution to (}) in which $y = 1$ when $x = 0$.
3. Using part (a)(i), state the accuracy of the approximate value of $y$ given in part (c)(ii).
4. State one change to your spreadsheet that could improve the accuracy of the approximate value of $y$ found in part (c)(ii).
The modified Euler method for the solution of the differential equation $\frac { d y } { d x } = f ( x , y )$ is as follows. $k _ { 1 } = h f \left( x _ { n } , y _ { n } \right)$ $k _ { 2 } = h f \left( x _ { n } + h , y _ { n } + k _ { 1 } \right)$ $y _ { n + 1 } = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$ $\mathrm { x } _ { \mathrm { n } + 1 } = \mathrm { x } _ { \mathrm { n } } + \mathrm { h }$
1. Adapt your spreadsheet from part (c)(i) to a spreadsheet to solve (**), so that the value of $a$ and the value of $h$ can be varied, in the case $x _ { 0 } = 0$ and $y _ { 0 } = 1$. State the formulae you have used in your spreadsheet.
2. In this part of the question $a = - 0.5$. Use the spreadsheet from part (d)(i) with $h = 0.1$ to approximate the value of $y$ when $x = 0.5$ for the solution to $( * * )$ in which $y = 1$ when $x = 0$. In this part of the question $a = - 0.5$. The solution to (**) in which $y = 1$ when $x = 0$ has a turning point with coordinates $( c , d )$ where $0 < c < 1$.
3. Use the spreadsheet in part (d)(i) to determine the value of $c$ correct to $\mathbf { 1 }$ decimal place.
4. Use the spreadsheet in part (d)(i) to determine the value of $d$ correct to $\mathbf { 3 }$ decimal places.

OCR MEI Further Pure with Technology Specimen Q1

19 marks Challenging +1.8

1 A family of curves has polar equation $r = \cos n \left( \frac { \theta } { n } \right) , 0 \leq \theta < n \pi$, where $n$ is a positive even integer.

(A) Sketch the curve for the cases $n = 2$ and $n = 4$.
(B) State two points which lie on every curve in the family.
(C) State one other feature common to all the curves.
(A) Write down an integral for the length of the curve for the case $n = 4$.
(B) Evaluate the integral.
(A) Using $t = \theta$ as the parameter, find a parametric form of the equation of the family of curves.
(B) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sin t \sin \left( \frac { t } { n } \right) - \cos t \cos \left( \frac { t } { n } \right) } { \sin t \cos \left( \frac { t } { n } \right) + \cos t \sin \left( \frac { t } { n } \right) }$.
Hence show that there are $n + 1$ points where the tangent to the curve is parallel to the $y$-axis.
By referring to appropriate sketches, show that the result in part (iv) is true in the case $n = 4$.
(A) Create a program to find all the solutions to $x ^ { 2 } \equiv - 1 ( \bmod p )$ where $0 \leq x < p$. Write out your program in full in the Printed Answer Booklet.
(B) Use the program to find the solutions to $x ^ { 2 } \equiv - 1 ( \bmod p )$ for the primes
$$\begin{aligned} ( 4 k ) ! & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( 2 k + 1 ) \times ( 2 k + 2 ) \times \ldots \times ( 4 k - 1 ) \times 4 k ( \bmod p ) \\ & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( - 2 k ) \times ( - ( 2 k - 1 ) ) \times \ldots \times ( - 2 ) \times ( - 1 ) ( \bmod p ) \\ & \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p ) \end{aligned}$$ (A) Explain why ( $2 k + 2$ ) can be written as ( $- ( 2 k - 1 )$ ) in line ( 2 ).
(B) Explain how line (3) has been obtained.
(C) Explain why, if $p$ is a prime of the form $p = 4 k + 1$, then $x ^ { 2 } \equiv - 1 ( \bmod p )$ will have at least one solution.
(D) Hence find a solution of $x ^ { 2 } \equiv - 1 ( \bmod 29 )$.
(A) Create a program that will find all the positive integers $n$, where $n < 1000$, such that $( n - 1 ) ! \equiv - 1 \left( \bmod n ^ { 2 } \right)$. Write out your program in full.
(B) State the values of $n$ obtained.
(C) A Wilson prime is a prime $p$ such that $( p - 1 ) ! \equiv - 1 \left( \bmod p ^ { 2 } \right)$. Write down all the Wilson primes $p$ where $p < 1000$.

◯	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

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