**Source: Questions from Complete International Mathematics For Cambridge IGCSE - David Rayner, Jim Fenson**

1. A ladder of length 8 m rests against a wall so that the angle

between the ladder and the wall is 31°.

How far is the base of the ladder from the wall?

Let the base of ladder be x m from the wallThe ladder, the wall and the floor form a triangle |

The base of the ladder is

**4.1 m**from the wall.2. A ship sails 200 km on a bearing of 243.7°.

a) How far south has it traveled?

b) How far west has it traveled?

Let OA be the line traveled by the ship. N, E, S and W represent the North, East, south and West directions respectively Bearing of ship = 243.7°( The line of travel of ship makes an angle of 243.7° measured clockwise from North) North-South is a straight line = 180° |

The Ship has traveled 88.6 km to the south and 179.3 km to the west.

Find the distance and bearing of the finishing point from the starting point.

Let the starting point be O.The aircarft flies 500 km on bearing of 100°. It reaches point AThen 600 km on a bearing of 160°. Let the finishing point be X | Bearing is 100° i.e the total angle measured clockwise from North is 100°. The horizontal at O represents West. Therefore the angle between the line of travel of ship and West = 100°-90°=10° The bearing of aircraft is now 160°. Hence the angle it makes with the horizontal at point A is 160°-90°=70° The total distance traveled along horizontal axis is a+b The total distance traveled along vertical axis is c+d |

The Aircraft is at a distance of

**953.8 km**from the starting point.

The bearing of the finishing point is 180 - 47 =

**133°**

4. For the points P(2,5), Q(5,1) and R(0,-3), Calculate the angle PQR

5. From the top of a tower 75 m, a man sees two goats, both due west of him.

If the angle of depression of the two goats are 10° and 17° respectively,

Calculate the distance between them.

Angle of depression of goat G1 = 17° Angle of depression of goat G2 = 10° |

The distance between the goats is

**180 m**Find the radius of the circle.

**9.5 cm**

7. A rocket flies 10 km vertically, then 20 km at an angle of 15° to the vertical and finally 60 km at an angle of 26° to the vertical. Calculate the vertical height of the rocket at the end of the third stage.

**83.2 km**

9. Diagram A shows a kitchen floor tiled with regular octagons and squares.

Diagram B shows and enlarged view of one octagon.

The smaller green squares have sides of length 12 cm.

Find the width of the octagons.

Interior Angle of a regular Octagon = 135° Hence angle in triangle = 45° Width of Octagon = (8.5 X 2) + 12 = 29 cm |

10. A farm gate is 1.3 m high.

The diagonal crossbar has a width of 15 cm and makes an angle of 24° with the horizontal.

Find the length of x and y in the diagram.

Hence find the width of the gate.

POINT A |

**113.6 cm**

Width of gate =

**255 cm**

What is the height of the highest point of the stone above ground?

Point B:Point A: |

**2.7 + 0.45 = 3.13 m**

12. The diagram shows the cross section of a rectangular fish tank.

When AB is inclined at 40°, the water just comes up to A.

The tank is then lowered so that BC is horizontal.

What is now the depth of the water in the tank?

Figure 1:Figure 2: | The area in cross section of Figure 1 and area in cross section of figure 2 will be equal as the volume of water is the same. |

**27.12 cm**