O-LevelMathematicsVectors and transformations4024/22 · May/June 2014 · Q812 Marks

In this question you may use the grid below to help you. The point P has position vector (4, 2) and the point Q has position vector (8, -3). [Grid] [Diagram of O, A, B, C, D and vectors a, b]

(a)(i)Find PQ.[1]
(a)(ii)Find |PQ|.[1]
(a)(iii)Find the equation of the line PQ.[2]
(a)(iv)Given that Q is the midpoint of the line PR, find the coordinates of R.[2]
(b)(i)(a)In the diagram triangles OAB and OCD are similar. OA = a, OB = b and BC = 4a – b. Express, as simply as possible, in terms of a and/or b AB,[1]
(b)(i)(b)AC,[1]
(b)(i)(c)CD.[2]
(b)(ii)(a)Find, in its simplest form, the ratio perimeter of triangle OAB : perimeter of triangle OCD,[1]
(b)(ii)(b)area of triangle OAB : area of trapezium ABDC.[1]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.

[1] (a)(i) (4, -5)
[1] (a)(ii) 6.4(0) to 6.41 or √41 cao
[2] (a)(iii) y = -1.25x + 7 oe
[2] (a)(iii) B1 for gradient = –1.25 or y-intercept = +7 soi in a final equation
[2] (a)(iv) (12,-8)
[2] (a)(iv) B1 for one value correct
[1] (b)(i)(a) b-a
[1] (b)(i)(b) 3a cao
[2] (b)(i)(c) 4(b-a)
[2] (b)(i)(c) B1 for correct unsimplified CD or for 3(b - a)
[1] (b)(ii)(a) 1:4
[1] (b)(ii)(b) 1:15

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