Questions - OCR MEI Further Pure with Technology

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

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

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

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

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

3 marks

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

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

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

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

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).

OCR MEI Further Pure with Technology 2024 June Q1

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.
  - $a = - 0.5$
$a = - 0.1$
$a = 0.5$
(ii) 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 $a = 0.5$.
(iii) 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.
(iv) Write down the equation of the vertical asymptote of the curve (}).

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

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

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

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
- $p = 809$,
- $p = 811$ and
- $p = 444001$.
- State Wilson's Theorem.
- The following argument shows that $( 4 k ) ! \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p )$ for the case $p = 4 k + 1$.
$$\begin{aligned} ( 4 k ) ! & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( 2 k + 1 ) \times ( 2 k + 2 ) \times \ldots \times ( 4 k - 1 ) \times 4 k ( \bmod p )
& \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( - 2 k ) \times ( - ( 2 k - 1 ) ) \times \ldots \times ( - 2 ) \times ( - 1 ) ( \bmod p )
& \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p ) \end{aligned}$$ (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 $p < 1000$.

OCR MEI Further Pure with Technology Specimen Q3

3 This question explores the family of differential equations $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + a x + 2 y }$ for various values of the parameter $a$. Fig. 3 shows the tangent field in the case $a = 1$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{141c85ec-5749-4f24-9f6d-fe7a01567511-4_691_696_452_696} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

(A) Sketch the tangent field in the case $a = - 2$.
(B) Explain why the tangent field is not defined for the whole coordinate plane.
(C) Give an inequality which describes the region in which the tangent field is defined.
(D) Find a value of $a$ such that the region for which the tangent field is defined includes the entire $x$-axis.
(A) For the case $a = 1$, with $y = 1$ when $x = 0$, construct a spreadsheet for the Runge-Kutta method of order 2 with formulae as follows, where $\mathrm { f } ( x , y ) = \frac { \mathrm { d } y } { \mathrm {~d} x }$. $$\begin{aligned} k _ { 1 } & = h \mathrm { f } \left( x _ { n } , y _ { n } \right)
k _ { 2 } & = h \mathrm { f } \left( x _ { n } + h , y _ { n } + k _ { 1 } \right)
y _ { n + 1 } & = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right) \end{aligned}$$ State the formulae you have used in your spreadsheet.
(B) Use your spreadsheet to obtain the value of $y$ correct to 4 decimal places when $x = 1$ for
- $h = 0.1$
  and
- $h = 0.05$.
(A) For the case $a = 0$ find the analytical solution that passes through the point ( 0,1 ).
(B) Verify that the solution in part (iii) (A) is a solution to the differential equation.
(C) Use the solution in part (iii) (A) to find the value of $y$ correct to 4 decimal places when $x = 1$.
(A) Verify that $y = - \frac { a } { 2 } x + \frac { a ^ { 2 } } { 8 } - \frac { 1 } { 2 }$ is a solution for all cases when $a \leq 0$.
(B) Show that this is the only straight line solution in these cases. \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.
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Questions — OCR MEI Further Pure with Technology (15 questions)

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