# iGCSE Mathematics (0580) :E6.4 Solve problems using the sine and cosine rules.iGCSE Style Questions Paper 4

### Question

The diagram shows a field, ABCD, on horizontal ground.
BC = 192m, CD = 287.9m, BD = 168m and AD = 205.8m.
(a) (i) Calculate angle CBD and show that it rounds to 106.0°, correct to 1 decimal place.
(ii) The bearing of D from B is 038°.
Find the bearing of C from B.
………………………………………….
(iii) A is due east of B.
Calculate the bearing of D from A.
………………………………………….
(b) (i) Calculate the area of triangle BCD.
……………………………………..$$m^{2}$$
(ii) Tomas buys the triangular part of the field, BCD.
The cost is $$35 750$$ per hectare.
Calculate the amount he pays.
Give your answer correct to the nearest $$100.$$
[1 hectare = $$10000m^{2}$$]
$$…………………………………………$$

(a)(i) 106.01 to 106.02
(ii) 292.0 or 291.98 to 291.99
(iii) 310.0 or 310.03 to 310.04
(b)(i) 15 500 or 15 501 to 15 503. …
(ii) 55 400

### Question

The diagram shows a triangular field, ABC, on horizontal ground.

(a) Olav runs from A to B at a constant speed of 4m/s and then from B to C at a constant speed of 3m/s.
He then runs at a constant speed from C to A.
His average speed for the whole journey is 3.6m/s.

Calculate his speed when he runs from C to A.

m/s [3]

(b) Use the cosine rule to find angle BAC.

Angle BAC = [4]

(c) The bearing of C from A is 210°.

(i) Find the bearing of B from A.[1]

(ii) Find the bearing of A from B.[2]

(d) D is the point on AC that is nearest to B.
Calculate the distance from D to A.

m [2]

Ans:

5(a) 4.29 or 4.285 to 4.286

5(b) 82.8 or 82.81 to 82.82 using cosine rule

5(c)(i) 127.2 or 127.1 to 127.2 or 127

5(c)(ii) 307.2 or 307.1 to 307.2 or 307

5(d) 15 or 14.99 to 15.04

### Question

The diagram shows two ports, L and P, and a buoy, M.
The bearing of L from P is 201° and LP = 248km.
The bearing of M from P is 127°.
Angle PML = 42°.
(a) Use the sine rule to calculate LM.
LM = ………………………………….. km
(b) A ship sails directly from L to P.
(i) Calculate the shortest distance from M to LP.
…………………………………… km
(ii) The ship leaves L at 2045 and travels at a speed of 40km/h.
Calculate the time the next day that the ship arrives at P.
………………………………………….