OCR MEI Further Pure with Technology (Further Pure with Technology) 2023 June

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. 1. On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.
      • \(a = - 2\)
2. \(a = - 1\)
3. \(a = 0\)
   (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 \(a = 0\).
4. 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 \(a\).
(iii) With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which
• \(y \geqslant 0\) 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.
Find its coordinates and fully justify that it is a cusp.

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

1. Show that \(\mathrm { a } ^ { 2 } = \mathrm { b } + \mathrm { c }\) if and only if \(\mathrm { c } = \mathrm { b } + 1\).
2. 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.
3. Write down the number of Pythagorean triples found by your program in (b).
4. Prove that there is no Pythagorean triple, \(( a , b , c )\), in which \(\mathrm { b } ^ { 2 } = \mathrm { a } + \mathrm { c }\).

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

1. Use Wilson's theorem to show that \(17 ! \equiv 1 ( \bmod 19 )\).
2. 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.

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. 1. Determine the values of \(P\) for which
      • \(\frac { \mathrm { dP } } { \mathrm { dt } } = 0\)
2. \(\frac { \mathrm { dP } } { \mathrm { dt } } > 0\)
3. \(\frac { \mathrm { dP } } { \mathrm { dt } } < 0\)
   (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 \(t\).
4. In this question you will explore the limitations of using the Euler method to approximate solutions to the differential equation
$$\frac { d P } { d t } = 2 P ^ { 1.25 } \left( 1 - \frac { P } { 1000 } \right) ^ { 1.5 } , t \geqslant 0 , P ( t ) \geqslant 0 ( * * )$$ 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}
(i) 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 \(P _ { 0 } = 1\).
• 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).