[Maximum number: 18]

The binary operator multiplication modulo 14, denoted by *, is defined on the set $S=\{2,4,6,8,10,12\}$ .

(a)

Copy and complete the following operation table.

[ 4 ]

(b)

(i)

Show that $\{S, *\}$ is a group.

(ii)

Find the order of each element of $\{S, *\}$ .

(iii)

Hence show that $\{S, *\}$ is cyclic and find all the generators.

[ 11 ]

(c)

The set T is defined by $\{x * x: x \in S\}$ . Show that $\{T, *\}$ is a subgroup of $\{S, *\}$ .

[ 3 ]

[Maximum number: 4]

1.
One root of the equation $x^{2}+a x+b=0$ is 2+3 i where $a, b \in \mathbb{R}$ . Find the value of a and the value of b.

[Maximum number: 5]

When the polynomial $3 x^{3}+a x+b$ is divided by ( x-2 ), the remainder is 2 , and when divided by ( x+1 ), it is 5 . Find the value of a and the value of b.

(a)

If w=2+2 i, find the modulus and argument of w.

[ 2 ]

(b)

Given $z=\cos \left(\frac{5 \pi}{6}\right)+\mathrm{i} \sin \left(\frac{5 \pi}{6}\right)$ , find in its simplest form $w^{4} z^{6}$ .

[ 4 ]

(a)

Use the Euclidean algorithm to find the greatest common divisor of the numbers 56 and 315.

[ 4 ]

(b)

(i)

Find the general solution to the diophantine equation 56 x+315 y=21.

[ 9 ]

(ii)

Hence or otherwise find the smallest positive solution to the congruence $315 x \equiv 21$ (modulo 56).

[ 9 ]

[Maximum number: 8]

In this question, you will be investigating the family of functions of the form $f(x)=x^{n} \mathrm{e}^{-x}$ .
Consider the family of functions $f_{n}(x)=x^{n} \mathrm{e}^{-x}$ , where $x \geq 0$ and $n \in \mathbb{Z}^{+}$ .
When n=1, the function $f_{1}(x)=x \mathrm{e}^{-x}$ , where $x \geq 0$ .

(a)

Use mathematical induction to prove your conjecture from part (e). You may assume that, for any value of $m, \lim _{x \rightarrow \infty} x^{m} \mathrm{e}^{-x}=0$ .

[ 8 ]

(a)

Consider the following Cayley table for the set $G=\{1,3,5,7,9,11,13,15\}$ under the operation $\times_{16}$ , where $\times_{16}$ denotes multiplication modulo 16 .

[ 7 ]

(i)

Find the values of a, b, c, d, e, f, g, h, i and j.

(ii)

Given that $\times_{16}$ is associative, show that the set G, together with the operation $\times_{16}$ , forms a group.

[ 7 ]

(b)

The Cayley table for the set $H=\left\{e, a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, b_{4}\right\}$ under the operation *, is shown below.

[ 8 ]

(i)

Given that * is associative, show that H together with the operation * forms a group.

(ii)

Find two subgroups of order 4.

[ 8 ]

(c)

Show that $\left\{G, \times_{16}\right\}$ and $\{H, *\}$ are not isomorphic.

[ 2 ]

(d)

Show that $\{H, *\}$ is not cyclic.

[ 3 ]

[Maximum number: 6]

In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

(a)

(i)

Prove by mathematical induction that $f_{q}(x)=\sum_{i=1}^{n} i^{q} x^{i}, q \in \mathbb{Z}^{+}$ .

[ 6 ]

(a)

(i)

One version of Fermat's little theorem states that, under certain conditions,

a^{p-1} \equiv 1(\bmod p)

Show that this result is not valid when a=4, p=9 and state which condition is not satisfied.

(ii)

Given that $5^{64} \equiv n(\bmod 7)$ , where $0 \leq n \leq 6$ , find the value of n.

[ 8 ]

(b)

Find the general solution to the simultaneous congruences

\begin{aligned} x & \equiv 3(\bmod 4) \\ 3 x & \equiv 2(\bmod 5) . \end{aligned}

[ 6 ]

[Maximum number: 19]

The binary operation multiplication modulo 10 , denoted by $\mathrm{x}_{10}$ , is defined on the set $T=\{2,4,6,8\}$ and represented in the following Cayley table.

(a)

Show that $\left\{T, x_{10}\right\}$ is a group. (You may assume associativity.)

[ 4 ]

(b)

By making reference to the Cayley table, explain why T is Abelian.

[ 1 ]

(c)

(i)

Find the order of each element of $\left\{T, \times_{10}\right\}$ .

[ 6 ]

(ii)

Hence show that $\left\{T, \times_{10}\right\}$ is cyclic and write down all its generators.

The binary operation multiplication modulo 10 , denoted by $x_{10}$ , is defined on the set $V=\{1,3,5,7,9\}$ .

[ 6 ]

(d)

Show that $\left\{V, \times_{10}\right\}$ is not a group.

[ 2 ]

Question 1

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