O-LevelMathematicsTrigonometry4024/22 · May/June 2015 · Q812 Marks

Two ports, A and B, are 15 km apart and B is due south of A. A boat sails from A on a bearing of 141°. [Figure 8.1]

(a)State the bearing of A from the boat.[1]
(b)Calculate the shortest distance between the boat and B.[2]
(c)When the boat is due east of B, calculate its distance from A.[2]
(d)The scale drawing, drawn to a scale of 1 cm to 2km, shows A, B and a third port, C. [Figure 8.2]
(d)(i)When the boat has travelled 24 km, it stops at the point X. Mark and label X on the diagram.[2]
(d)(ii)A second boat is located I less than 12km from A II nearer to BC than to BA. Shade the region in which this second boat must lie.[3]
(d)(iii)The point Y is the position of the second boat when it is as far as possible from X. Mark and label Y on the diagram and hence find the maximum possible distance between the two boats.[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

[1] 321
[2] M1 for sin 39 = y/15 oe; 9.43 to 9.44
[2] B1 for cos 39 = 15/X oe; 19.3 to 19.31
[2] B1 for either a correct distance or bearing; X marked 12cm from A on bearing of 141°
[3] B1 for arc, min length 3 cm, radius 6 cm, centre A; B1 for bisector of ∠ABC, min length 3 cm; B1 for shading; Correct region shaded
[2] M1 for Y established at northern end of shading; 17.6 to 18.4 dependent on an acceptable X and Y

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