Study Pure Mathematics notes on Group Theory for A-Level. Useful for revision, homework, and exam preparation in Zimbabwe.
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This is the A-Level Pure Mathematics notes from September 2020. It is a free notes for revision.
Upper $6{ }^{\text {th }}$
LOCKDOWN PURE MATHS TEST 49
Tuesday 22 September 2020
ANSWERS
30 MARKS
NB: If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to $\mathbf{2}$ significant figures.
1. Given that $a * b=\left\{\begin{array}{ll}a+b | \text { if } a+b<6 a+b-6 | \text { if } a+b \geq 6\end{array}\right.$, and it is defined on the set $\{0,1,2,3,4,5\}$.
(i) Construct the operation table for *.
[3]
$*$ | 0 | 1 | 2 | 3 | 4 | 5
0 | 0 | 1 | 2 | 3 | 4 | 5
1 | 1 | 2 | 3 | 4 | 5 | 0
2 | 2 | 3 | 4 | 5 | 0 | 1
3 | 3 | 4 | 5 | 0 | 1 | 2
4 | 4 | 5 | 0 | 1 | 2 | 3
5 | 5 | 0 | 1 | 2 | 3 | 4
(ii) Determine whether $*$ forms a group or not.
[4]
(a) Checking for closure:
There are no new elements in the Cayley table ∴ the set is closed under *.
(b) Checking for Associativity:
$x *(y * z) \equiv(x * y) * z$
Let $x=0, y=2$ and $z=5$
LHS
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