WAEC 2014 · Paper 2 · Q2

(a)
Simplify $3\sqrt{75} - \sqrt{12} + \sqrt{108}$ , leaving the answer in surd form (radicals).
(b)
If $124_n = 232_{\text{five}}$ , find $n$ .

Worked solution (try it first)

(a)

Simplify each surd using its largest square factor: $\sqrt{75} = 5\sqrt3$ , $\sqrt{12} = 2\sqrt3$ and $\sqrt{108} = 6\sqrt3$ .
Then $3 \times 5\sqrt3 - 2\sqrt3 + 6\sqrt3 = 15\sqrt3 - 2\sqrt3 + 6\sqrt3$
$= 19\sqrt3$ .

(b)

Change both sides to base ten: $124_n = n^2 + 2n + 4$ and $232_{\text{five}} = 2 \times 25 + 3 \times 5 + 2$
$= 67$ .
So $n^2 + 2n + 4 = 67$ , $n^2 + 2n - 63 = 0$ and $(n + 9)(n - 7) = 0$ .
A base is a positive whole number, larger than every digit used, so $n = 7$ .

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