ECZ 2022 · GCE Paper 2 · Question 3 — Sets

In a particular locality, 25 people like walking, 22 people like running and 15 people like cycling during their morning routine exercises. 11 people like both walking and running, 9 people like both running and cycling, 7 people like both walking and cycling, 3 people like all three activities while 7 people do not like any of the activities. The position vectors of the points A, B and C are (-1, -2), (3, 6) and (9, 18).

(a(i)) Draw a Venn diagram to illustrate this information.[2]

Verified working — step 1

Let W, R, C represent Walking, Running, Cycling.

(a(ii)(a)) How many people are in this locality?[1]

Verified working — step 1

Total = (W only)+(R only)+(C only)+(W∩R only)+(R∩C only)+(W∩C only)+(all three)+(none)

(a(ii)(b)) How many people like walking only?[1]

Verified working — step 1

From the Venn diagram, Walking only = n(W) - [n(W∩R only)+n(W∩C only)+n(all three)]

(a(ii)(c)) How many people like two different activities only?[1]

Verified working — step 1

People liking exactly two activities = (W∩R only)+(R∩C only)+(W∩C only)

(b) The position vectors of the points A, B and C are (-1, -2), (3, 6) and (9, 18). Show that the points A, B and C are collinear.[5]

Verified working — step 1

A(-1,-2), B(3,6), C(9,18)

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Original examination question © Examinations Council of Zambia. Worked solution and commentary © G12 Titan — not to be reproduced without permission.

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