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

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

1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = \mathrm{f}(2, y)\) giving the coordinates of any turning points and any points of intersection with the axes. [4]
2. Find the stationary points on \(S\). [7]

\(G\) is a group of order 8.

1. Explain why there is no subgroup of \(G\) of order 6. [1]

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.

1. Write down the possible generators of \(H\). [2]

\(M\) 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 \(G\).

1. Specify all possible isomorphisms between \(M\) and \(G\). [4]

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

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

The sequence \(u_0, u_1, u_2, \ldots\) satisfies the recurrence relation \(u_{n+2} - 3u_{n+1} - 10u_n = 24n - 10\).

1. Determine the general solution of the recurrence relation. [6]
2. Hence determine the particular solution of the recurrence relation for which \(u_0 = 6\) and \(u_1 = 10\). [3]
3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u_2\). [1]

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). You are given that \(v_n\) converges to a finite non-zero limit, \(q\), as \(n \to \infty\).

1. Determine \(p\) and \(q\). [4]

A surface \(S\) is defined for \(z \geqslant 0\) by \(x^2 + y^2 + 2z^2 = 126\). \(C\) 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 \(\frac{1}{4}\pi\) 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 $$S = \frac{1}{(q+1)} + \frac{1}{(q+1)(q+2)} + \frac{1}{(q+1)(q+2)(q+3)} + \cdots$$

1. By considering a suitable geometric series show that \(S < \frac{1}{q}\). [3]
2. Deduce that \(S \notin \mathbb{Z}\). [2]

You are also given that \(\mathrm{e} = \sum_{r=0}^{\infty} \frac{1}{r!}\).

1. Assume that \(\mathrm{e} = \frac{p}{q}\), where \(p\) and \(q\) are positive integers. By writing the infinite series for \(\mathrm{e}\) in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that \(\mathrm{e}\) is irrational. [3]