OCR MEI Further Extra Pure (Further Extra Pure) 2019 June

The matrix A is \(\begin{pmatrix} 0.6 & 0.8 \\ 0.8 & -0.6 \end{pmatrix}\)

1. Given that A represents a reflection, write down the eigenvalues of A. [1]
2. Hence find the eigenvectors of A. [3]
3. Write down the equation of the mirror line of the reflection represented by A. [1]

A surface \(S\) is defined by \(z = 4x^2 + 4y^2 - 4x + 8y + 11\).

1. Show that the point P\((0.5, -1, 6)\) is the only stationary point on \(S\). [2]
2. 1. On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to \(z = 42\). [2]
   2. By using the sketch in part (b)(i), deduce that P must be a minimum point on \(S\). [3]
3. In the section of \(S\) corresponding to \(y = c\), the minimum value of \(z\) occurs at the point where \(x = a\) and \(z = 22\). Find all possible values of \(a\) and \(c\). [4]

The matrix A is \(\begin{pmatrix} -1 & 2 & 4 \\ 0 & -1 & -25 \\ -3 & 5 & -1 \end{pmatrix}\). Use the Cayley-Hamilton theorem to find A\(^{-1}\). [8]

\(T\) is the set \(\{1, 2, 3, 4\}\). A binary operation \(\bullet\) is defined on \(T\) such that \(a \bullet a = 2\) for all \(a \in T\). It is given that \((T, \bullet)\) is a group.

1. Deduce the identity element in \(T\), giving a reason for your answer. [2]
2. Find the value of \(1 \bullet 3\), showing how the result is obtained. [3]
3. 1. Complete a group table for \((T, \bullet)\). [2]
   2. State with a reason whether the group is abelian. [1]

A financial institution models the repayment of a loan to a client in the following way.

• An amount, \(£C\), is loaned to the client at the start of the repayment period.
• The amount owed \(n\) years after the start of the repayment period is \(£L_n\), so that \(L_0 = C\).
• At the end of each year, interest of \(\alpha\%\) (\(\alpha > 0\)) of the amount owed at the start of that year is added to the amount owed.
• Immediately after interest has been added to the amount owed a repayment of \(£R\) is made by the client.
• Once \(L_n\) becomes negative the repayment is finished and the overpayment is refunded to the client.

1. Show that during the repayment period, \(L_{n+1} = aL_n + b\), giving \(a\) and \(b\) in terms of \(\alpha\) and \(R\). [2]
2. Find the solution of the recurrence relation \(L_{n+1} = aL_n + b\) with \(L_0 = C\), giving your solution in terms of \(a\), \(b\), \(C\) and \(n\). [5]
3. Deduce from parts (a) and (b) that, for the repayment scheme to terminate, \(R > \frac{\alpha C}{100}\). [2]

A client takes out a £30000 loan at 8% interest and agrees to repay £3000 at the end of each year.

1. 1. Use an algebraic method to find the number of years it will take for the loan to be repaid. [3]
   2. Taking into account the refund of overpayment, find the total amount that the client repays over the lifetime of the loan. [3]

1. Given that \(\sqrt{7}\) is an irrational number, prove that \(a^2 - 7b^2 \neq 0\) for all \(a, b \in \mathbb{Q}\) where \(a\) and \(b\) are not both 0. [2]
2. A set \(G\) is defined by \(G = \{a + b\sqrt{7} : a, b \in \mathbb{Q}, a\) and \(b\) not both 0\(\}\). Prove that \(G\) is a group under multiplication. (You may assume that multiplication is associative.) [7]
3. A subset \(H\) of \(G\) is defined by \(H = \{1 + c\sqrt{7} : c \in \mathbb{Q}\}\). Determine whether or not \(H\) is a subgroup of \((G, \times)\). [2]
4. Using \((G, \times)\), prove by counter-example that the statement 'An infinite group cannot have a non-trivial subgroup of finite order' is false. [2]