OCR MEI Further Extra Pure (Further Extra Pure) 2023 June

1 A surface is defined in 3-D by \(z = 3 x ^ { 3 } + 6 x y + y ^ { 2 }\).
Determine the coordinates of any stationary points on the surface.

2 A sequence is defined by the recurrence relation \(4 \mathrm { t } _ { \mathrm { n } + 1 } - \mathrm { t } _ { \mathrm { n } } = 15 \mathrm { n } + 17\) for \(\mathrm { n } \geqslant 1\), with \(t _ { 1 } = 2\).

1. Solve the recurrence relation to find the particular solution for \(\mathrm { t } _ { \mathrm { n } }\). Another sequence is defined by the recurrence relation \(( n + 1 ) u _ { n + 1 } - u _ { n } ^ { 2 } = 2 n - \frac { 1 } { n ^ { 2 } }\) for \(n \geqslant 1\), with \(u _ { 1 } = 2\).
2. 1. Explain why the recurrence relation for \(\mathrm { u } _ { \mathrm { n } }\) cannot be solved using standard techniques for non-homogeneous first order recurrence relations.
   2. Verify that the particular solution to this recurrence relation is given by \(u _ { n } = a n + \frac { b } { n }\) where \(a\) and \(b\) are constants whose values are to be determined. A third sequence is defined by \(\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\).
3. Determine \(\lim _ { n \rightarrow \infty } \mathrm { v } _ { \mathrm { n } }\).

3 A surface, \(S\), is defined by \(g ( x , y , z ) = 0\) where \(g ( x , y , z ) = 2 x ^ { 3 } - x ^ { 2 } y + 2 x y ^ { 2 } + 27 z\). The normal to \(S\) at the point \(\left( 1,1 , - \frac { 1 } { 9 } \right)\) and the tangent plane to \(S\) at the point \(( 3,3 , - 3 )\) intersect at \(P\). Determine the position vector of P .

4 The set \(G\) is given by \(G = \{ \mathbf { M } : \mathbf { M }\) is a real \(2 \times 2\) matrix and det \(\mathbf { M } = 1 \}\).

1. Show that \(G\) forms a group under matrix multiplication, × . You may assume that matrix multiplication is associative.
2. The matrix \(\mathbf { A } _ { n }\) is defined by \(\mathbf { A } _ { n } = \left( \begin{array} { l l } 1 & 0
   n & 1 \end{array} \right)\) for any integer \(n\). The set \(S\) is defined by \(\mathrm { S } = \left\{ \mathrm { A } _ { \mathrm { n } } : \mathrm { n } \in \mathbb { Z } , \mathrm { n } \geqslant 0 \right\}\).
   1. Determine whether \(S\) is closed under × .
   2. Determine whether \(S\) is a subgroup of ( \(G , \times\) ).
3. 1. Find a subgroup of ( \(G , \times\) ) of order 2 .
   2. By considering the inverse of the non-identity element in any such subgroup, or otherwise, show that this is the only subgroup of ( \(G , \times\) ) of order 2. The set of all real \(2 \times 2\) matrices is denoted by \(H\).
4. With the help of an example, explain why ( \(H , \times\) ) is not a group.

5 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l } a & 0
2 & 3 \end{array} \right)\) where \(a\) is a constant and \(a \neq 3\).

1. Given that the acute angle between the directions of the eigenvectors of \(\mathbf { P }\) is \(\frac { 1 } { 4 } \pi\) radians, determine the possible values of \(a\).
2. You are given instead that \(\mathbf { P }\) satisfies the matrix equation \(\mathbf { I } = \mathbf { P } ^ { 2 } + r \mathbf { P }\) for some rational number \(r\).
   1. Use the Cayley-Hamilton theorem to determine the value of \(a\) and the corresponding value of \(r\).
   2. Hence show that \(\mathbf { P } ^ { 4 } = \mathbf { s } \mathbf { + t } \mathbf { t } \mathbf { P }\) where \(s\) and \(t\) are rational numbers to be determined. You should not calculate \(\mathbf { P } ^ { 4 }\).