2014 WAEC Mathematics Theory (a)  In the Venn diagram, P, Qand R are subsets of the universal set U....

(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.

(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).

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