2010 - WAEC Mathematics Past Questions and Answers - page 5

From the diagram, OP bisects < ROS

< POS = (\frac{1}{2}) < ROS = (\frac{1}{2}) x 66^o

3x = 33^o

x = (\frac{33^o}{3})

= 11^o

Join SR

< PRS = 90^o(Angle in a semicircle)

< PRS = 55^o (Angle between a chord and a tangent = Angle in the alternate segment)

< PSR + < PRS + < RSP = 180^o

90v + 55^o + < RSp = 180^o

< RSP = 180^o - 145^o

= 35^o

3x = 2 x 60 = 2y (Angle at centre = 2 x angle at circumference)

3x = 2 x 60

x = (\frac{2 \times 60}{3}) = 40^o

2 x 60 = 2y

y = 60^o

x + y = 40 + 60

= 100^o

Using (\frac{x}{a} + \frac{y}{b}) < 1for the equation of the time

where a = intercept on x-axis and b = intercept on y - axis

(\frac{x}{3} + \frac{y}{3} = 1)

= (\frac{x + y}{3} = 1)

= x + y < 3

F = 54^o (Alternate triangle angles)

< GHE = F = 54^o(Angle between a chord and a tangent = angles in the alternate segment)

Now, < GHE + < EHF + < IHF = 180^o(Angles on a straight line)

54^o + < EHF + 54^o = 180^o

< EHF = 180^o - 108^o

= 72^o

Total surface area = sum of the area of the (\bigtriangleup) S + Area of the rectangle

= 2 x Area of (\bigtriangleup) PTQ + 2 x Area of (\bigtriangleup) QUR + Area of rectangle PQRS

2 x (\frac{1}{2})(8 x 4) + 2 x (\frac{1}{2})(5 x 4) + 8 x 5

= 32 + 20 + 40

= 92cm²

Area of remaining = Area of rectangle = Area of semi-circle

22 x 8 - (\frac{1}{2})xar²

ehere r - (\frac{14}{2})cm = 7cm

Area of remaining = 176 - (\frac{1}{2}) x (\frac{22}{4}) x 7 x 7

= 176 - 77

= 99cm²

POQ in a straight line

Hence, < POQ + < QOR = 180^o

56^o + < QOR = 180^o

< QOR = 180^o - 56^o

= 124^o

Now, in (\bigtriangleup) QOR OR = OQ = Radius

< ORQ = < OQR = 2x (Base angles of an Isosceles (\bigtriangleup))

2x + 124 + 2x = 180^o

4x + 124 = 180

4x = 180 - 124

4x = 56

x = (\frac{56}{4})

x = 14^o

The bearing of x from y = 270^o + (\theta)

where (\theta) + 50^o = y

in (\bigtriangleup) OXY

O + X + Y = 180^o

Where O = 40^o + 30^o = 70^o

70^o + 30^o + y = 180^o

y + 100^o = 180^o

y = 180^o - 100^o = 30^o

(\theta) + 50^o = 80^o

80^o - 50^o = 30^o

The bearing of x from y = 270^o + 30^o = 300^o