O-LevelAdditional MathematicsTrigonometry4037/22 · Oct/Nov 2020 · Q88 Marks

DO NOT USE A CALCULATOR IN THIS QUESTION. In this question lengths are in centimetres. [Figure 8.1: A diagram showing a triangle ABC with angle A = 30°, side AB = 2√3+1, side BC, and side AC. A box shows trigonometric ratios: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.]

(a)Given that the area of the triangle ABC is 5.5 cm², find the exact length of AC. Write your answer in the form a+b√3, where a and b are integers.[4]
(b)Show that BC² =c+d√3, where c and d are integers to be found.[4]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[M1] (a) 1/2 (2√3 + 1) AC sin 30° = 11/2 Correct use of area of a triangle
[A1] (a) (2√3 + 1) AC = 22 oe
[M1] (a) AC = 22 / (2√3 + 1) * (2√3 - 1) / (2√3 - 1) Multiply by their (2√3 + 1)
[A1] (a) AC = 4√3 - 2
[M1] (b) BC² = (2√3+1)² + (4√3−2)² - 2(2√3+1)(4√3-2)cos 30 Correct use of cosine rule with their AC.
[A2] (b) BC² =[13+4√3]+[52-16√3]+[-22√3] A1 for one correct expanded bracket A1 for the other two correct expanded brackets
[A1] (b) BC² = 65-34√3

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