O-LevelAdditional MathematicsVectors in two dimensions4037/22 · Oct/Nov 2025 · Q68 Marks

In this question lengths are in centimetres and time, t, is in seconds. A particle P is moving in a straight line with a speed of 26 in the direction of the vector ( 5 \ -12 ). When t = 0, P passes through a point A which has position vector ( 3 \ 6 ).

(a)Find the velocity vector of P.[2]
(b)Write down the position vector of P at time t.[2]
(c)At the same time that P passes through A, a particle Q passes through a point B. The position vector of Q at time t is given by ( 8t-5 \ 2-25t ). The distance between P and Q at time t is d. Show that d² = mt² +nt+r, where m, n and r are integers to be found.[3]
(d)Hence show that P and Q do not collide.[1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

[B1] (a) for the magnitude of the direction vector is 13 soi or M1 for √5² +(-12)² oe or M1 for (26cos1.18 / -26sin1.18) oe
[B1] (a) ( 10 / -24 ) oe, isw
[M1 FT for (3/6) + t their (10/-24)] (b) ( 3 / 6 ) + t ( 10 / -24 ) oe, isw
[A1] (b) FT their velocity vector (implied by 2 marks)
[B1] (c) ( (3+10t)-(8t-5) / (6-24t)-(2-25t) ) oe or ( (8t-5)-(3+10t) / (2-25t)-(6-24t) ) oe, soi
[M1 FT] (c) ( (3/6)+t(10/-24) - (8t-5/2-25t) )
[A1] (c) d² = 5t² + 40t +80 mark final answer (implied by 3 marks)
[B1 dep on part (c) being correct] (d) Correct argument including d e.g. When d=0 → t=-4, but t ≥ 0 [so no collision] or When t=-4 → d=0, but t ≥ 0 [so no collision] or d > 0 when t ≥ 0 so d ≠ 0 [so no collision]

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