(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^\circ, find the value of (i) , WYZ
(ii) < YEZ.
Verified Correct Answer: .
Step-by-Step Mathematical Solution
(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^\circ (angle in a semi-circle; XW is a diameter)
< XWY + 90^\circ + 50^\circ = 180^\circ (sum of angles in a triangle)
< XWY = 180^\circ - 140^\circ = 40^\circ
(i) < WYZ = 40^\circ (alternate angles; \(\overline{XW} || \overline{YZ}\))
(ii) < WOZ = \(2 \times 40^\circ \) (angle at centre = twice angle at circumference)
= 80^\circ
\(\therefore < WEO = 180^\circ - (40^\circ + 80^\circ)\)
= \(180^\circ - 120^\circ \)
= \(60^\circ\)
\(\therefore < YEZ = < WEO = 60^\circ\) (vertically opposite angles).
Source: Myschool Past Questions Archive (Record #52627)