OCR MEI Further Extra Pure (Further Extra Pure) 2020 November

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

• the eigenvalues of \(\mathbf { A }\),
• an eigenvector associated with each eigenvalue.

2 A sequence is defined by the recurrence relation \(t _ { n + 1 } = \frac { t _ { n } } { n + 3 }\) for \(n \geqslant 1\), with \(t _ { 1 } = 8\).
Verify that the particular solution to the recurrence relation is given by \(t _ { n } = \frac { a } { ( n + b ) ! }\) where \(a\) and \(b\) are constants whose values are to be determined.

3 A sequence is defined by the recurrence relation \(u _ { n + 2 } = 4 u _ { n + 1 } - 5 u _ { n }\) for \(n \geqslant 0\), with \(u _ { 0 } = 0\) and \(u _ { 1 } = 1\).

1. Find an exact real expression for \(u _ { n }\) in terms of \(n\) and \(\theta\), where \(\tan \theta = \frac { 1 } { 2 }\). A sequence is defined by \(v _ { n } = a ^ { \frac { 1 } { 2 } n } u _ { n }\) for \(n \geqslant 0\), where \(a\) is a positive constant.
2. In each of the following cases, describe the behaviour of \(v _ { n }\) as \(n \rightarrow \infty\).
   • \(a = 0.1\)
   • \(a = 0.2\)
   • \(a = 1\)

4

1. In each of the following cases, a set \(G\) and a binary operation ∘ are given. The operation ∘ may be assumed to be associative on \(G\). Determine which, if any, of the other three group axioms are satisfied by ( \(G , \circ\) ) and which, if any, are not satisfied.
   1. \(G = \{ 2 n + 1 : n \in \mathbb { Z } \}\) and \(\circ\) is addition.
   2. \(G = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \}\) and ∘ is multiplication.
   3. \(G\) is the set of all real numbers and ∘ is multiplication.
2. A group \(M\) consists of eight \(2 \times 2\) matrices under the operation of matrix multiplication. Five of the eight elements of \(M\) are as follows. $$\frac { 1 } { \sqrt { 2 } } \left( \begin{array} { l l } 1 & \mathrm { i }
   \mathrm { i } & 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } - 1 & \mathrm { i }
   \mathrm { i } & - 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } 1 & - \mathrm { i }
   - \mathrm { i } & 1 \end{array} \right) \quad \left( \begin{array} { l l } 0 & \mathrm { i }
   \mathrm { i } & 0 \end{array} \right) \quad \left( \begin{array} { l l } 1 & 0
   0 & 1 \end{array} \right)$$
   1. Find the other three elements of \(M\).
      \(( N , * )\) is another group of order 8, with identity element \(e\). You are given that \(N = \langle a , b , c \rangle\) where \(a * a = b * b = c * c = e\).
   2. State whether \(M\) and \(N\) are isomorphic to each other, giving a reason for your answer.

1. Show that \(\mathbf { f }\) is also an eigenvector of \(\mathbf { A }\).
2. State the eigenvalue associated with \(\mathbf { f }\). You are now given that \(\mathbf { A }\) represents a reflection in 3-D space.
3. Explain the significance of \(\mathbf { e }\) and \(\mathbf { f }\) in relation to the transformation that \(\mathbf { A }\) represents.
4. State the cartesian equation of the plane of reflection of the transformation represented by \(\mathbf { A }\).

6 A surface \(S\) is defined by \(z = \mathrm { f } ( x , y ) = 4 x ^ { 4 } + 4 y ^ { 4 } - 17 x ^ { 2 } y ^ { 2 }\).

1. 1. Show that there is only one stationary point on \(S\). The value of \(z\) at the stationary point is denoted by \(s\).
   2. State the value of \(s\).
   3. By factorising \(\mathrm { f } ( x , y )\), sketch the contour lines of the surface for \(z = s\).
   4. Hence explain whether the stationary point is a maximum point, a minimum point or a saddle point. C is a point on \(S\) with coordinates ( \(a , a , \mathrm { f } ( a , a )\) ) where \(a\) is a constant and \(a \neq 0\). \(\Pi\) is the tangent plane to \(S\) at C .
2. 1. Find the equation of \(\Pi\) in the form r.n \(= p\).
   2. The shortest distance from the origin to \(\Pi\) is denoted by \(d\). Show that \(\frac { d } { a } \rightarrow \frac { 3 \sqrt { 2 } } { 4 }\) as \(a \rightarrow \infty\).
   3. Explain whether the origin lies above or below \(\Pi\). \section*{END OF QUESTION PAPER}