OCR MEI Further Pure with Technology (Further Pure with Technology) 2019 June

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. 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 \(m = 5\).
2. 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 \(t\).
3. 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.
4. 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 \(m\).

2

1. Prove that if \(x\) and \(y\) are integers which satisfy \(x ^ { 2 } - 2 y ^ { 2 } = 1\), then \(x\) is odd and \(y\) is even.
2. 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.
3. 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 \(x\).
4. 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).
5. 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 \(T _ { m } = \frac { m ( m + 1 ) } { 2 }\).
6. 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.
7. 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 \(m\).
8. By comparing your answers to part (c) and part (g), or otherwise, prove that there are infinitely many triangular numbers which are perfect squares.

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 .

1. Determine and describe geometrically the isoclines of (\textit{) when
   1. \(a = - 1\),
   2. \(a = 0\),
   3. \(a = 1\).
2. 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.
3. 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.
4. 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.