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WAEC 2022 · Paper 2 · Q5✱

  1. (a)

    Given that m=tan⁡30∘m = \tan30^\circ and n=tan⁡45∘n = \tan45^\circ, simplify, without using a calculator, m−nmn\dfrac{m - n}{mn}, leaving the answer in the form p+qp + \sqrt q.

  2. (b)

    There are 20 women in a bus. 15 of them wear glasses and 10 wear wrist watches, and each of them wears at least one of the two. If a woman is chosen at random from the bus, find the probability that she wears both glasses and a wrist watch.

Worked solution (try it first)

(a)

  1. m=tan⁡30∘=13m = \tan 30^\circ = \frac{1}{\sqrt3} and n=tan⁡45∘=1n = \tan 45^\circ = 1.
  2. So m−nmn=13−113\frac{m - n}{mn} = \frac{\frac{1}{\sqrt3} - 1}{\frac{1}{\sqrt3}}.
  3. Multiply the top and bottom by 3\sqrt3: 1−31=1−3\frac{1 - \sqrt3}{1} = 1 - \sqrt3.
  4. This is in the form p+qp + \sqrt q with p=1p = 1 and the surd term −3-\sqrt3.

(b)

  1. Let xx women wear both.
  2. Every woman wears at least one (glasses or watch), so 15+10−x=2015 + 10 - x = 20 and x=5x = 5.
  3. Probability =520=14= \frac{5}{20} = \frac14.

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