Question

(a) Given that m=tan30 degrees and n=tan45 degrees, simplify, without using a calculator, (m-n)/(m+n), leaving the answer in the form p+sqrt(q).
(b) There are 20 women in a bus. 15 of them wear glasses and 10 wear wrist watches. If a woman is chosen at random from the bus, find the probability that she wears both glasses and a wrist watch.

Accepted Answer

(a) tan30 = 1/sqrt(3), tan45 = 1.
(m-n)/(m+n) = (1/sqrt(3) - 1) / (1/sqrt(3) + 1). Multiply numerator and denominator by sqrt(3): (1-sqrt(3))/(1+sqrt(3)).
Rationalise by multiplying by (1-sqrt(3))/(1-sqrt(3)): (1-sqrt(3))^2 / (1-3) = (1-2sqrt(3)+3)/(-2) = (4-2sqrt(3))/(-2) = -2+sqrt(3).
Answer: -2 + sqrt(3) (i.e. p=-2, q=3).

(b) Let x = number who wear both. Using a Venn diagram: (glasses only) + (both) + (watches only) = total: (15-x) + x + (10-x) = 20, so 25 - x = 20, giving x = 5.
Probability(wears both) = 5/20 = 1/4.

GCE Mathematics 2022 Theory — Question 5

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