(a)
In the Venn diagram, P, Qand R are subsets of the universal set U. If n(U) = 125, find : (i) the value of x ; (ii) n((P cup Q cap R’)).
(b) In the diagram, O is the centre of the circle. If WX is parallel to YZ and < WXY = 50°, find the value of (i) , WYZ
(ii) < YEZ.
Explanation
(a)
(i) ((16 – 2x) + (6 + x) + (19 – 3x) + 5x + 7x + 8x + 4x + 4 = 125)
(45 + 20x = 125 implies 20x = 125 – 45 = 80)
(x = frac{80}{20} = 4)
(ii) (n(P cup Q cap R’) = (16 – 2x) + 5x + (6 + x))
= (22 + 4x)
= (22 + 4(4) = 22 + 16 = 38)
(b) In the diagram above, < XYW = 90° (angle in a semi-circle; XW is a diameter)
< XWY + 90° + 50° = 180° (sum of angles in a triangle)
< XWY = 180° – 140° = 40°
(i) < WYZ = 40° (alternate angles; (overline{XW} || overline{YZ}))
(ii) < WOZ = (2 times 40° ) (angle at centre = twice angle at circumference)
= 80°
(therefore < WEO = 180° – (40° + 80°))
= (180° – 120° )
= (60°)
(therefore < YEZ = < WEO = 60°) (vertically opposite angles).