OCR MEI Further Extra Pure (Further Extra Pure) 2022 June

1 Three sequences, \(\mathrm { a } _ { \mathrm { n } } , \mathrm { b } _ { \mathrm { n } }\) and \(\mathrm { c } _ { \mathrm { n } }\), are defined for \(n \geqslant 1\) by the following recurrence relations. $$\begin{aligned} & \left( a _ { n + 1 } - 2 \right) \left( 2 - a _ { n } \right) = 3 \text { with } a _ { 1 } = 3
& b _ { n + 1 } = - \frac { 1 } { 2 } b _ { n } + 3 \text { with } b _ { 1 } = 1.5
& c _ { n + 1 } - \frac { c _ { n } ^ { 2 } } { n } = 1 \text { with } c _ { 1 } = 2.5 \end{aligned}$$ The output from a spreadsheet which presents the first 10 terms of \(a _ { n } , b _ { n }\) and \(c _ { n }\), is shown below. Without attempting to solve any recurrence relations, describe the apparent behaviour, including as \(n \rightarrow \infty\), of

• \(a _ { n }\)
• \(\mathrm { b } _ { \mathrm { n } }\)
• \(\mathrm { C } _ { \mathrm { n } }\)

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 10 & 12 & - 8
- 1 & 2 & 4
3 & 6 & 2 \end{array} \right)\).

1. In this question you must show detailed reasoning. Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 14 \lambda ^ { 2 } - 56 \lambda + 64 = 0\).
2. Use the Cayley-Hamilton theorem to determine \(\mathbf { A } ^ { - 1 }\). A matrix \(\mathbf { E }\) and a diagonal matrix \(\mathbf { D }\) are such that \(\mathbf { A } = \mathbf { E D E } ^ { - 1 }\). The elements in the diagonal of \(\mathbf { D }\) increase from top left to bottom right.
3. Determine the matrix \(\mathbf { D }\).

3 A sequence is defined by the recurrence relation \(5 t _ { n + 1 } - 4 t _ { n } = 3 n ^ { 2 } + 28 n + 6\), for \(n \geqslant 0\), with \(t _ { 0 } = 7\).

1. Find an expression for \(t _ { n }\) in terms of \(n\). Another sequence is defined by \(\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { n } ^ { \mathrm { m } } }\), for \(n \geqslant 1\), where \(m\) is a constant.
2. In each of the following cases determine \(\lim _ { n \rightarrow \infty } \mathrm {~V} _ { n }\), if it exists, or show that the sequence is divergent.
   1. \(m = 3\)
   2. \(m = 2\)
   3. \(m = 1\)

4 A binary operation, ○, is defined on a set of numbers, \(A\), in the following way.
\(a \circ b = \mathrm { k } _ { 1 } \mathrm { a } - \mathrm { k } _ { 2 } \mathrm {~b} + \mathrm { k } _ { 3 }\), for \(a , b \in A\),
where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are constants (which are not necessarily in \(A\) ) and the operations addition, subtraction and multiplication of numbers have their usual notation and meaning. You are initially given the following information about ○ and \(A\).

• \(A = \mathbb { R }\)
• \(0 \circ 0 = 2\)
• An identity element, \(e\), exists for ∘ in \(A\)
  1. Show that \(a \circ b = a + b + 2\).
  2. State the value of \(e\).
  3. Explain whether ○ is commutative over \(A\).
  4. Determine whether or not ( \(A , \circ\) ) is a group.
  5. Explain whether your answer to part (d) would change in each of the following cases, giving details of any change.
     1. \(A = \mathbb { Z }\)
     2. \(A = \{ 2 m : m \in \mathbb { Z } \}\)
     3. \(\quad A = \{ n : n \in \mathbb { Z } , n \geqslant - 2 \}\)

5 A surface \(S\) is defined by \(z = f ( x , y )\), where \(f ( x , y ) = y e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }\).

1. 1. Find \(\frac { \partial f } { \partial x }\).
   2. Show that \(\frac { \partial f } { \partial y } = - \left( x ^ { 2 } y + 2 x y + 2 y - 1 \right) e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }\).
   3. Determine the coordinates of any stationary points on \(S\). Fig. 5.1 shows the graph of \(z = e ^ { - x ^ { 2 } }\) and Fig. 5.2 shows the contour of \(S\) defined by \(z = 0.25\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_686_822_244} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_437_822_1105} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
      \end{figure}
2. Specify a sequence of transformations which transforms the graph of \(\mathrm { z } = \mathrm { e } ^ { - \mathrm { x } ^ { 2 } }\) onto the graph of the section defined by \(z = f ( x , 1 )\).
3. Hence, or otherwise, sketch the section defined by \(z = f ( x , 1 )\).
4. Using Fig. 5.2 and your answer to part (c), classify any stationary points on \(S\), justifying your answer. You are given that \(P\) is a point on \(S\) where \(z = 0\).
5. Find, in vector form, the equation of the tangent plane to \(S\) at \(P\). The tangent plane found in part (e) intersects \(S\) in a straight line, \(L\).
6. Write down, in vector form, the equation of \(L\).