Do not use a calculator in this question. All lengths in this question are in centimetres. [Figure showing a triangle ABC with angle B = 60 degrees, side AB = 4√3-5, side BC = 4√3+5.] The diagram shows the triangle ABC, where AB = 4√3 – 5, BC = 4√3 + 5 and angle ABC = 60°. It is known that sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. M1 For attempt to use the cosine rule AC² = (4√3-5)² + (4√3+5)² - 2(4√3-5)(4√3+5)cos 60°
2. A1 For all correct unsimplified
3. M1dep for attempt to evaluate without use of calculator AC² = 123
4. A1 AC = √123
5. M1 For attempt at the sine rule or trigonometry involving right-angled triangles AC/sin 60° = (4√3-5)/sin ACB or sin ACB = AD/AC
6. M1dep on first M mark cosecACB = 2√123 / (√3(4√3-5)) or 2√41 / (√3(4√3-5)) oe
7. M1dep on previous M mark for a statement involving rationalisation using a√3+b cosec ACB = 2√41/(4√3-5) * (4√3+5)/(4√3+5)
8. A1 For rationalisation using (4√3+5)/(4√3+5) oe and simplification = 2√41/23 (4√3+5)