O-LevelAdditional MathematicsVectors in two dimensions4037/11 · May/June 2017 · Q511 Marks

(a) [Figure OABC diagram] The diagram shows a figure OABC, where OA = a, OB = b and OC = c. The lines AC and OB intersect at the point M where M is the midpoint of the line AC. (b) Vectors i and j are unit vectors parallel to the x-axis and y-axis respectively. The vector p has a magnitude of 39 units and has the same direction as −10i + 24j.

(a)(i)Find, in terms of a and c, the vector OM.[2]
(a)(ii)Given that OM :MB = 2:3, find b in terms of a and c.[2]
(b)(i)Find p in terms of i and j.[2]
(b)(ii)Find the vector q such that 2p + q is parallel to the positive y-axis and has a magnitude of 12 units.[3]
(b)(iii)Hence show that |q| = k√5, where k is an integer to be found.[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

[M1] OM = OA + 1/2(OC - OA) oe or OM = OC + 1/2(OA - OC) oe may be implied by correct answer.
[A1] 1/2 (a+c)
[M1] b = 5/2 OM oe, 5/2 (their (i)) dealing with ratio correctly to relate b or OB to OM
[A1] 5/4 (a+c)
[M1] magnitude of -10i+24j and use with 39
[A1] p = -15i+36j
[M1] realising i component is zero
[DM1] use of 12
[A1] q = 30i-60j
[M1] attempt at magnitude of their q
[A1] |q| = 30√5 Answer Given: must have full and correct working

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