Theory question 109
Sets(a) In the Venn diagram, P, Q and R are subsets of the universal set U. If n(U) = 125, find: (i) the value of x; (ii) n(P ∪ Q ∩ 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.
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(a)(i) (16 − 2x) + 5x + (6 + x) + 4x + 8x + 7x + (19 − 3x) + 4 = 125 → 45 + 20x = 125 → x = 4.
(ii) (P ∪ Q) ∩ R′ = the parts of P and Q outside R: (16 − 2x) + 5x + (6 + x) = 22 + 4x = 38.
(b) XW is a diameter, so ∠XYW = 90° and ∠XWY = 180° − 90° − 50° = 40°.
(i) WX ∥ YZ, so ∠WYZ = ∠XWY = 40° (alternate).
(ii) ∠WOZ = 2 × ∠WYZ = 80°. In ΔOEW: ∠OEW = 180° − 80° − 40° = 60°. ∠YEZ = ∠OEW = 60° (vertically opposite).
The angle in a semicircle is 90°; the angle at the centre is twice the angle at the circumference.