OCR MEI Further Extra Pure (Further Extra Pure) 2021 November

1 In this question you must show detailed reasoning.
A surface \(S\) is defined by \(z = f ( x , y )\) where \(f ( x , y ) = x ^ { 3 } + x ^ { 2 } y - 2 y ^ { 2 }\).

1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = f ( 2 , y )\) giving the coordinates of any turning points and any points of intersection with the axes.
2. Find the stationary points on \(S\).
   \(2 G\) is a group of order 8.
3. Explain why there is no subgroup of \(G\) of order 6 . You are now given that \(G\) is a cyclic group with the following features:
   • \(e\) is the identity element of \(G\),
   • \(g\) is a generator of \(G\),
   • \(H\) is the subgroup of \(G\) of order 4.
   • Write down the possible generators of \(H\).
     \(M\) is the group ( \(\{ 0,1,2,3,4,5,6,7 \} , + _ { 8 }\) ) where \(+ _ { 8 }\) denotes the binary operation of addition modulo 8. You are given that \(M\) is isomorphic to \(G\).
   • Specify all possible isomorphisms between \(M\) and \(G\).

3 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 3 & 0
0 & 2 & 2
1 & 3 & 4 \end{array} \right)\).

1. Determine the characteristic equation of \(\mathbf { A }\).
2. Hence verify that the eigenvalues of \(\mathbf { A }\) are 1, 2 and 6 .
3. For each eigenvalue of \(\mathbf { A }\) determine an associated eigenvector.
4. Use the results of parts (b) and (c) to find \(\mathbf { A } ^ { n }\) as a single matrix, where \(n\) is a positive integer.

4 The sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation \(u _ { n + 2 } - 3 u _ { n + 1 } - 10 u _ { n } = 24 n - 10\).

1. Determine the general solution of the recurrence relation.
2. Hence determine the particular solution of the recurrence relation for which \(u _ { 0 } = 6\) and \(u _ { 1 } = 10\).
3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u _ { 2 }\). The sequence \(v _ { 0 } , v _ { 1 } , v _ { 2 } , \ldots\) is defined by \(v _ { n } = \frac { u _ { n } } { p ^ { n } }\) for some constant \(p\), where \(u _ { n }\) denotes the
   particular solution found in part (b). particular solution found in part (b). You are given that \(\mathrm { v } _ { \mathrm { n } }\) converges to a finite non-zero limit, \(q\), as \(n \rightarrow \infty\).
4. Determine \(p\) and \(q\).

5 A surface \(S\) is defined for \(z \geqslant 0\) by \(x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 126\). \(C\) is the set of points on \(S\) for which the tangent plane to \(S\) at that point intersects the \(x - y\) plane at an angle of \(\frac { 1 } { 3 } \pi\) radians. Show that \(C\) lies in a plane, \(\Pi\), whose equation should be determined.

6 You are given that \(q \in \mathbb { Z }\) with \(q \geqslant 1\) and that
\(\mathrm { S } = \frac { 1 } { ( \mathrm { q } + 1 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) ( \mathrm { q } + 3 ) } + \ldots\).

1. By considering a suitable geometric series show that \(\mathrm { S } < \frac { 1 } { \mathrm { q } }\).
2. Deduce that \(S \notin \mathbb { Z }\). You are also given that \(\mathrm { e } = \sum _ { r = 0 } ^ { \infty } \frac { 1 } { r ! }\).
3. Assume that \(\mathrm { e } = \frac { \mathrm { p } } { \mathrm { q } }\), where \(p\) and \(q\) are positive integers. By writing the infinite series for e in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that e is irrational.