1 A family of curves has polar equation \(r = \cos n \left( \frac { \theta } { n } \right) , 0 \leq \theta < n \pi\), where \(n\) is a positive even integer.
- (A) Sketch the curve for the cases \(n = 2\) and \(n = 4\).
(B) State two points which lie on every curve in the family.
(C) State one other feature common to all the curves. - (A) Write down an integral for the length of the curve for the case \(n = 4\).
(B) Evaluate the integral. - (A) Using \(t = \theta\) as the parameter, find a parametric form of the equation of the family of curves.
(B) Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sin t \sin \left( \frac { t } { n } \right) - \cos t \cos \left( \frac { t } { n } \right) } { \sin t \cos \left( \frac { t } { n } \right) + \cos t \sin \left( \frac { t } { n } \right) }\). - Hence show that there are \(n + 1\) points where the tangent to the curve is parallel to the \(y\)-axis.
- By referring to appropriate sketches, show that the result in part (iv) is true in the case \(n = 4\).
- (A) Create a program to find all the solutions to \(x ^ { 2 } \equiv - 1 ( \bmod p )\) where \(0 \leq x < p\). Write out your program in full in the Printed Answer Booklet.
(B) Use the program to find the solutions to \(x ^ { 2 } \equiv - 1 ( \bmod p )\) for the primes
- \(p = 809\),
- \(p = 811\) and
- \(p = 444001\).
- State Wilson's Theorem.
- The following argument shows that \(( 4 k ) ! \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p )\) for the case \(p = 4 k + 1\).
$$\begin{aligned}
( 4 k ) ! & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( 2 k + 1 ) \times ( 2 k + 2 ) \times \ldots \times ( 4 k - 1 ) \times 4 k ( \bmod p )
& \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( - 2 k ) \times ( - ( 2 k - 1 ) ) \times \ldots \times ( - 2 ) \times ( - 1 ) ( \bmod p )
& \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p )
\end{aligned}$$
(A) Explain why ( \(2 k + 2\) ) can be written as ( \(- ( 2 k - 1 )\) ) in line ( 2 ).
(B) Explain how line (3) has been obtained.
(C) Explain why, if \(p\) is a prime of the form \(p = 4 k + 1\), then \(x ^ { 2 } \equiv - 1 ( \bmod p )\) will have at least one solution.
(D) Hence find a solution of \(x ^ { 2 } \equiv - 1 ( \bmod 29 )\). - (A) Create a program that will find all the positive integers \(n\), where \(n < 1000\), such that \(( n - 1 ) ! \equiv - 1 \left( \bmod n ^ { 2 } \right)\). Write out your program in full.
(B) State the values of \(n\) obtained.
(C) A Wilson prime is a prime \(p\) such that \(( p - 1 ) ! \equiv - 1 \left( \bmod p ^ { 2 } \right)\). Write down all the Wilson primes \(p\) where \(p < 1000\).