5 Let $\mathrm { f } ( x , y ) = x ^ { 3 } + y ^ { 3 } - 2 x y + 1$. The surface $S$ has equation $z = \mathrm { f } ( x , y )$.\\
(i) (a) Find $f _ { x }$.\\
(b) Find $\mathrm { f } _ { y }$.\\
(c) Show that $S$ has a stationary point at ( $0,0,1$ ).\\
(d) Find the coordinates of the second stationary point of $S$.\\
(ii) The section $z = \mathrm { f } ( a , y )$, where $a$ is a constant, has exactly one stationary point. Determine the equation of the section.

A customer takes out a loan of $\pounds P$ from a bank at an annual interest rate of $4.9 \%$. Interest is charged monthly at an equivalent monthly interest rate. This interest is added to the outstanding amount of the loan at the end of each month, and then the customer makes a fixed monthly payment of $\pounds M$ in order to reduce the outstanding amount of the loan.

Let $L _ { n }$ denote the outstanding amount of the loan at the end of month $n$ after the fixed payment has been made, with $L _ { 0 } = P$.\\
(i) Explain how the outstanding amount of the loan from one month to the next is modelled by the recurrence relation

$$L _ { n + 1 } = 1.004 L _ { n } - M$$

with $L _ { 0 } = P , n \geq 0$.\\
(ii) Solve, in terms of $n , M$ and $P$, the first order recurrence relation given in part (i).\\
(iii) The loan amount is $\pounds 100000$ and will be fully repaid after 10 years. Find, to the nearest pound, the value of the monthly repayment.\\
(iv) The bank's procedures only allow for calculations using integer amounts of pounds. When each monthly amount of the outstanding $\operatorname { debt } \left( L _ { n } \right)$ is calculated it is always rounded up to the nearest pound before the monthly repayment ( $M$ ) is subtracted.\\
Rewrite (*) to take this into account.\\
(i) Let $N = 10 a + b$ and $M = a - 5 b$ where $a$ and $b$ are integers such that $a \geq 1$ and $0 \leq b \leq 9$. $N$ is to be tested for divisibility by 17 .\\
(a) Prove that $17 \mid N$ if and only if $17 \mid M$.\\
(b) Demonstrate step-by-step how an algorithm based on these forms can be used to show that $17 \mid 4097$.\\
(ii) (a) Show that, for $n \geq 2$, any number of the form $1001 _ { n }$ is composite.\\
(b) Given that $n$ is a positive even number, provide a counter-example to show that the statement "any number of the form $10001 _ { n }$ is prime" is false.

\section*{END OF QUESTION PAPER}
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