(a) In the Venn diagram, P, Qand R are subsets of... | WAEC SSCE - General Mathematics - 2014 (Essay)

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(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.

Answer

(a)(i) Adding all the regions of the Venn diagram and equating to \( n(U) = 125 \):

\[ (16 - 2x) + (6 + x) + (19 - 3x) + 5x + 7x + 8x + 4x + 4 = 125 \]

\[ 45 + 20x = 125 \implies 20x = 80 \implies x = 4 \]

(a)(ii) \( n(P \cup Q \cap R') = (16 - 2x) + 5x + (6 + x) = 22 + 4x \)

\[ = 22 + 4(4) = 22 + 16 = \mathbf{38} \]

(b) XW is a diameter, so \( \angle XYW = 90^\circ \) (angle in a semicircle).

In triangle XYW: \( \angle XWY + 90^\circ + 50^\circ = 180^\circ \), so \( \angle XWY = 40^\circ \).

(i) Since \( WX \parallel YZ \), \( \angle WYZ = \angle XWY = \mathbf{40^\circ} \) (alternate angles).

(ii) \( \angle WOZ = 2 \times \angle WYZ = 2 \times 40^\circ = 80^\circ \) (angle at centre is twice angle at circumference). In triangle WEO, \( \angle WEO = 180^\circ - (40^\circ + 80^\circ) = 60^\circ \), therefore \( \angle YEZ = \mathbf{60^\circ} \) (vertically opposite angles).

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Answer Details

(a)(i) Adding all the regions of the Venn diagram and equating to \( n(U) = 125 \):

\[ (16 - 2x) + (6 + x) + (19 - 3x) + 5x + 7x + 8x + 4x + 4 = 125 \]

\[ 45 + 20x = 125 \implies 20x = 80 \implies x = 4 \]

(a)(ii) \( n(P \cup Q \cap R') = (16 - 2x) + 5x + (6 + x) = 22 + 4x \)

\[ = 22 + 4(4) = 22 + 16 = \mathbf{38} \]

(b) XW is a diameter, so \( \angle XYW = 90^\circ \) (angle in a semicircle).

In triangle XYW: \( \angle XWY + 90^\circ + 50^\circ = 180^\circ \), so \( \angle XWY = 40^\circ \).

(i) Since \( WX \parallel YZ \), \( \angle WYZ = \angle XWY = \mathbf{40^\circ} \) (alternate angles).

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(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 d...

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(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 d...

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