Paper 2 · Vector Geometry
(a) If g = (-5; 2) and h = (3; 4), express g + 2h in the form (x; y). [2] (b) In the diagram, OABC is a parallelogram in which OA = 4p and OC = 5q. X is a point on AC such that AX : XC = 2 : 3. (i) Express, in terms of p and/or q 1. AC, 2. OX in its simplest terms. (ii) Y is a point on AB such that AY/AB = k, where k is a constant. Express OY in terms of p, q and k. (iii) Given that OY = h OX, where h is a constant, write down another expression for OY in terms of p, q and h. (iv) Using results in (ii) and (iii), find the value of h and the value of k. (v) Express (the area of triangle OAY)/(the area of parallelogram OABC), as a fraction in its simplest form. [10]
Model answer
(a) (1; 10); (b)(i) 1. AC = 5q - 4p 2. OX = (12/5)p + 2q; (ii) OY = 4p + 5kq; (iii) OY = (12h/5)p + 2h q; (iv) h = 5/3, k = 2/3; (v) 1/3
Also accepted: (1; 10); 5q - 4p; 2.4p + 2q; 4p + 5kq; h = 5/3 and k = 2/3; 1/3, (1, 10); h = 5/3, k = 2/3; 1/3, h = 5/3, k = 2/3, 1/3, 0.333, 0,333, (1; 10); 5q - 4p; 2,4p + 2q; 4p + 5kq; 2,4hp + 2hq; h = 5/3, k = 2/3; 1/3
ZIMSEC Mathematics 4008/2, November 2013, Q8, via the Topical Collection