OCR MEI Further Pure with Technology (Further Pure with Technology) 2022 June

1

1. 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 \(a = 2\).
2. In this part of the question \(m\) is any real number.
Describing all possible cases, determine the pairs of values \(a\) and \(m\) for which the curve with equation (*) intersects the straight line given by \(y = m x\).

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 \(n\).

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 \(\frac { p - 1 } { 2 }\).

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

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

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.

1. 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.
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 \(y _ { 0 } = 2\).
      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 \(x = 0\).