O-LevelAdditional MathematicsVectors in two dimensions4037/22 · Oct/Nov 2019 · Q89 Marks

A particle A travels with a speed of 6.5 ms⁻¹ in the direction -5i – 12j. Find the velocity, vA, of A.

(i)A particle A travels with a speed of 6.5 ms⁻¹ in the direction -5i – 12j. Find the velocity, vA, of A.[2]
(ii)A particle B travels with velocity vB = 12i−9j. Find the speed, in ms⁻¹, of B.[2]
(iii)Find rA, the position vector of A after t seconds, and rB, the position vector of B after t seconds.[2]
(iv)Find the time when the particles collide and the position vector of the point of collision.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

M1 √5² +12² =13
A1 vA = -2.5i - 6j or 1/2(-5i - 12j)
M1 |vB| = √12² +(-9)² Use Pythagoras
A1 15 Do not allow ± 15. Mark final answer.
B1 FT rA = (20) + t(-2.5) or rA = (20-2.5t)i+(-7-6t)j FT on their vA only if of the form k(-5i - 12j) where k ≠ 1 or 0.
B1 rB = (-67) + t(12) or rB = (-67+12t)i+(11-9t)j
M1 20-2.5t = -67 + 12t or -7-6t = 11 - 9t Equate x or y coordinates. Must have two terms in both coordinates.
A1 t = 6 nfww Ignore other value of t.
A1 r = (5) only or r = 5i - 43j A0 if further value of r found.

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