OCR MEI Further Extra Pure (Further Extra Pure) Specimen

The set \(G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}\) is a group of order 9 under the binary operation of multiplication modulo 19.

1. Show that \(G\) is a cyclic group generated by the element 4. [3]
2. Find another generator for \(G\). Justify your answer. [2]
3. Specify two distinct isomorphisms from the group \(J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\) under addition modulo 9 to \(G\). [5]

A binary operation \(*\) is defined on the set \(S = \{p, q, r, s, t\}\) by the following composition table. Determine whether \((S, *)\) is a group. [4]

1. Find the general solution of $$u_n = 8u_{n-1} - 16u_{n-2}, \quad n \geq 2. \quad (*)$$ [4]

A new sequence \(v_n\) is defined by \(v_n = \frac{u_n}{u_{n-1}}\) for \(n \geq 1\).

1. (A) Use (*) to show that \(v_n = 8 - \frac{16}{v_{n-1}}\) for \(n \geq 2\). [2] (B) Deduce that if \(v_n\) tends to a limit then it must be 4. [2]
2. Use your general solution in part (i) to show that \(\lim_{n \to \infty} v_n = 4\). [3]
3. Deduce the value of \(\lim_{n \to \infty} \left(\frac{u_n}{u_{n-2}}\right)\). [1]

A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).

1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]

The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).

1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]

In this question you must show detailed reasoning. You are given that the matrix $\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix}$ represents a rotation in 3-D space.

1. Explain why it follows that \(\mathbf{M}\) has 1 as an eigenvalue. [2]
2. Find a vector equation for the axis of the rotation. [4]
3. Show that the characteristic equation of \(\mathbf{M}\) can be written as $$\lambda^3 - \lambda^2 + \lambda - 1 = 0.$$ [5]
4. Find the smallest positive integer \(n\) such that \(\mathbf{M}^n = \mathbf{I}\). [6]
5. Find the magnitude of the angle of the rotation which \(\mathbf{M}\) represents. Give your reasoning. [1]