Paper 2 · Vector Geometry
In the diagram, PQ is parallel to OR, PM = (1/3)PR, OP = 2a and OR = 3b. (a) Express in terms of a and/or b (i) PR, (ii) PM, (iii) OM. [4] (b)(i) Given that PQ = h OR, write down in terms of h, a and/or b an expression for (a) PQ, (b) OQ. (ii) Given also that OQ = k OM, write down another expression for OQ in terms of a, b and k. [3] (c) Using the two expressions for OQ, form an equation and use it to find the value of k and the value of h. [3] (d) Write down OQ in terms of a and b only. [1] (e) Find the ratio (area of triangle OPQ)/(area of trapezium OPQR). [1]
Model answer
(a)(i) PR = 3b - 2a (ii) PM = b - (2/3)a (iii) OM = (4/3)a + b; (b)(i)(a) PQ = 3h b (b) OQ = 2a + 3h b (ii) OQ = (4k/3)a + k b; (c) k = 3/2, h = 1/2; (d) OQ = 2a + (3/2)b; (e) 1/3
Also accepted: 3b - 2a; b - 2a/3; 4a/3 + b; 3hb; 2a + 3hb; k(4a/3 + b); k = 1.5, h = 0.5; 2a + 1.5b; 1/3, k = 3/2, h = 1/2, OQ = 2a + (3/2)b, ratio 1 : 3, 1/3, 1:3, 0,333
ZIMSEC Mathematics 4028/2, June 2010, Q7, via the Topical Collection