ECZ 2020 · O-Level Paper 2 · Question 2 — Vectors In Two Dimensions
Of the 115 students who attended an end of year party, 74 took fanta, 93 took sprite, 87 took coke, 61 took fanta and sprite, 71 took sprite and coke, 60 took fanta and coke and 50 took all the three drinks. In the diagram, AVC is a straight line, AB = 8p, AC = 4p + 9q, BV = -6p + kq and AM = MB.
(a(i)) Of the 115 students who attended an end of year party, 74 took fanta, 93 took sprite, 87 took coke, 61 took fanta and sprite, 71 took sprite and coke, 60 took fanta and coke and 50 took all the three drinks. Illustrate this information on a Venn diagram.[2]
Verified working — step 1
Centre first: n(all three) = 50.(a(ii)(a)) How many students took none of the drinks?[1]
Verified working — step 1
115 students altogether and 112 took at least one drink, so 115 - 112 = …(a(ii)(b)) How many students took fanta and sprite but not coke?[1]
Verified working — step 1
Fanta and sprite = 61, of whom 50 also took coke, so 61 - 50 = …(a(ii)(c)) How many students took at least two different drinks?[1]
Verified working — step 1
At least two = the three "exactly two" regions plus the centre:\n11 + 21 + 10 + 50 = …(b(i)(a)) In the diagram, AVC is a straight line, AB = 8p, AC = 4p + 9q, BV = -6p + kq and AM = MB. Express AM in terms of p, q andork.[1]
Verified working — step 1
M is the midpoint of AB because AM = …(b(i)(b)) Express AV in terms of p, q andork.[1]
Verified working — step 1
AV = AB + BV = …(b(ii)) Given that AV = hAC, by forming an equation involving p, q, h and k or otherwise, find the numerical values of h and k.[3]
Verified working — step 1
AV = hAC gives 2p + kq = h(4p + 9q) = 4hp + 9hq.The full step-by-step working and final answer are included with Premium — checked against ECZ marking standards.
Original examination question © Examinations Council of Zambia. Worked solution and commentary © G12 Titan — not to be reproduced without permission.
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