OCR MEI Further Extra Pure (Further Extra Pure) 2024 June

1 A surface, \(S\), is defined in 3-D by \(z = f ( x , y )\) where \(f ( x , y ) = 12 x - 30 y + 6 x y\).

1. Determine the coordinates of any stationary points on the surface.
2. The equation \(\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { a } )\), where \(a\) is a constant, defines a section of S . Given that this equation is \(\mathrm { z } = 24 \mathrm { x } + \mathrm { b }\), find the value of \(a\) and the value of \(b\). The diagram shows the contour \(z = 12\) and its associated asymptotes.
   \includegraphics[max width=\textwidth, alt={}, center]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-2_860_1143_742_242}
3. Find the equations of the asymptotes.
4. By forming grad \(g\), where \(g ( x , y , z ) = f ( x , y ) - z\), find the equation of the tangent plane to \(S\) at the point where \(x = 3\) and \(y = 2\). Give your answer in vector form. The point \(( 0,4 , - 120 )\), which lies on S , is denoted by A .
   The plane with equation \(\mathbf { r }\). \(\left( \begin{array} { r } 3
   3
   - 2 \end{array} \right) = 52\) is denoted by \(\Pi\).
5. Show that the normal to S at A intersects \(\Pi\) at the point \(( - 360,304 , - 110 )\).

2

1. Determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 0\).
2. Using your answer to part (a), determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 20 n ^ { 2 } + 60 n\). In the rest of this question the sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation in part (b). You are given that \(u _ { 0 } = - 9\) and \(u _ { 1 } = - 12\).
3. Determine the particular solution for \(\mathrm { u } _ { \mathrm { n } }\). You are given that, as \(n\) increases, once the values of \(u _ { n }\) start to increase, then from that point onwards the sequence is an increasing sequence.
4. Use your answer to part (c) to determine, by direct calculation, the least value taken by terms in the sequence. You should show any values that you rely on in your argument.

3 Fig. 3.1 shows an equilateral triangle, with vertices \(\mathrm { A } , \mathrm { B }\) and C , and the three axes of symmetry of the triangle, \(\mathrm { S } _ { \mathrm { a } } , \mathrm { S } _ { \mathrm { b } }\) and \(\mathrm { S } _ { \mathrm { c } }\). The axes of symmetry are fixed in space and all intersect at the point O . \begin{figure}[h] \end{figure} There are six distinct transformations under which the image of the triangle is indistinguishable from the triangle itself, ignoring labels.
These are denoted by \(\mathrm { I } , \mathrm { M } _ { a ^ { \prime } } \mathrm { M } _ { \mathrm { b } ^ { \prime } } , \mathrm { M } _ { \mathrm { c } ^ { \prime } } , \mathrm { R } _ { 120 }\) and \(\mathrm { R } _ { 240 }\) where

• I is the identity transformation
• \(\mathrm { M } _ { \mathrm { a } }\) is a reflection in the mirror line \(\mathrm { S } _ { \mathrm { a } }\) (and likewise for \(\mathrm { M } _ { \mathrm { b } }\) and \(\mathrm { M } _ { \mathrm { c } }\) )
• \(\mathrm { R } _ { 120 }\) is an anticlockwise rotation by \(120 ^ { \circ }\) about O (and likewise for \(\mathrm { R } _ { 240 }\) ).

Composition of transformations is denoted by ○.
Fig. 3.2 illustrates the composition of \(R _ { 120 }\) followed by \(R _ { 240 }\), denoted by \(R _ { 240 } \circ R _ { 120 }\). This shows that \(\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 }\) is equivalent to the identity transformation, so that \(\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 } = \mathrm { I }\). \begin{figure}[h] \end{figure}

1. Using the blank diagrams in the Printed Answer Booklet, find the single transformation which is equivalent to each of the following.
   • \(M _ { a } \circ M _ { a }\)
   • \(M _ { b } \circ M _ { a }\)
   • \(\mathrm { R } _ { 120 } \circ \mathrm { M } _ { \mathrm { a } }\)
   The set of the six transformations is denoted by G and you are given that \(( \mathrm { G } , \circ )\) is a group. The table below is a mostly empty composition table for \(\circ\). The entry given is that for \(R _ { 240 } \circ R _ { 120 }\).
   First transformation performed is
   followed by

   ◯	I	\(\mathrm { M } _ { \mathrm { a } }\)	\(\mathrm { M } _ { \mathrm { b } }\)	\(\mathrm { M } _ { \mathrm { c } }\)	\(\mathrm { R } _ { 120 }\)	\(\mathrm { R } _ { 240 }\)
   I
   \(\mathrm { M } _ { \mathrm { a } }\)
   \(\mathrm { M } _ { \mathrm { b } }\)
   \(\mathrm { M } _ { \mathrm { c } }\)
   \(\mathrm { R } _ { 120 }\)
   \(\mathrm { R } _ { 240 }\)					I

2. Complete the copy of this table in the Printed Answer Booklet. You can use some or all of the spare copies of the diagram in the Printed Answer Booklet to help.
3. Explain why there can be no subgroup of \(( \mathrm { G } , \circ )\) of order 4.
4. A student makes the following claim.
   "If all the proper non-trivial subgroups of a group are abelian then the group itself is abelian."
   Explain why the claim is incorrect, justifying your answer fully.
5. With reference to the order of elements in the groups, explain why ( \(\mathrm { G } , \circ\) ) is not isomorphic to \(\mathrm { C } _ { 6 }\), the cyclic group of order 6 .

4 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { r r r } 1 & 7 & 8
- 6 & 12 & 12
- 2 & 4 & 8 \end{array} \right)\).

1. Show that the characteristic equation of \(\mathbf { P }\) is \(- \lambda ^ { 3 } + 21 \lambda ^ { 2 } - 126 \lambda + 216 = 0\). You are given that the roots of this equation are 3,6 and 12 .
2. 1. Verify that \(\left( \begin{array} { r } 1
      - 2
      2 \end{array} \right)\) is an eigenvector of \(\mathbf { P }\), stating its associated eigenvalue.
   2. The vector \(\left( \begin{array} { l } x
      y
      z \end{array} \right)\) is an eigenvector of \(\mathbf { P }\) with eigenvalue 6. Given that \(z = 5\), find \(x\) and \(y\). You are given that \(\mathbf { P }\) can be expressed in the form \(\mathbf { E D E } ^ { - 1 }\), where \(\mathbf { E } = \left( \begin{array} { r r r } 3 & 2 & 1
      1 & 2 & - 2
      1 & 1 & 2 \end{array} \right)\) and \(\mathbf { D }\) is a diagonal matrix. The characteristic equation of \(\mathbf { E }\) is \(- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 15 \lambda + 9 = 0\).
3. 1. Use the Cayley-Hamilton theorem to express \(\mathbf { E } ^ { - 1 }\) in terms of positive powers of \(\mathbf { E }\).
   2. Hence find \(\mathbf { E } ^ { - 1 }\).
   3. By identifying the matrix \(\mathbf { D }\) and using \(\mathbf { P } = \mathbf { E D E } ^ { - 1 }\), determine \(\mathbf { P } ^ { 4 }\).

5 In this question you may assume that if \(p\) and \(q\) are distinct prime numbers and \(\mathbf { p } ^ { \alpha } = \mathbf { q } ^ { \beta }\) where \(\alpha , \beta \in \mathbb { Z }\), then \(\alpha = 0\) and \(\beta = 0\).

1. Prove that it is not possible to find \(a\) and \(b\) for which \(\mathrm { a } , \mathrm { b } \in \mathbb { Z }\) and \(3 = 2 ^ { \frac { \mathrm { a } } { \mathrm { b } } }\).
2. Deduce that \(\log _ { 2 } 3 \notin \mathbb { Q }\).