O-LevelAdditional MathematicsVectors in two dimensions4037/22 · Oct/Nov 2011 · Q1110 Marks

Answer only one of the following two alternatives. EITHER [Figure 11.1 (EITHER)] In the diagram above OA = a, OB = b, OS = (3/5)OA and OT = (7/5)OB. OR [Figure 11.2 (OR)] In the diagram above OC = c and OD = d. The points P and Q lie on OC and OD produced respectively, so that OC : CP = 1 : 2 and OD : DQ = 2 : 1. The line CD is extended to R so that CD = DR.

(EITHER)(i)Given that AX = µAB, where µ is a constant, express OX in terms of µ, a and b.[2]
(EITHER)(ii)Given that SX = λST, where λ is a constant, express OX in terms of λ, a and b.[4]
(EITHER)(iii)Hence evaluate µ and λ.[4]
(OR)(i)Find, in terms of c and/or d, the vectors OP, OQ and OR.[5]
(OR)(ii)Show that the points P, Q and R are collinear and find the ratio PQ : QR.[5]

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 (Oct/Nov 2011). The mark or mark type is shown in brackets.

[M1A1] (EITHER)(i) OX = a + µ(b-a) or (1 − µ)a + µb
[B1] (EITHER)(ii) OS = 3/5 a
[B1] (EITHER)(ii) OT = 7/5 b
[M1A1] (EITHER)(ii) OX = 3/5 a + λ(7/5 b - 3/5 a) or OX = (1-λ)3/5 a + λ7/5 b
[M1] (EITHER)(iii) Equate components or arrange to (α) a = (β) b and put α = β = 0
[DM1] (EITHER)(iii) Solve simultaneous equations
[A1] (EITHER)(iii) λ = 1/2
[A1] (EITHER)(iii) µ = 7/10
[B1] (OR)(i) OP = 3c
[B1] (OR)(i) OQ = 3/2 d
[B1] (OR)(i) DR = CD = d-c
[M1A1] (OR)(i) OR = OD + DR = 2d - c
[M1] (OR)(ii) Finds two of PQ, QR, PR
[A1+A1] (OR)(ii) Two of PQ = 3/2 d - 3c, QR = 1/2 d - c, PR = 2d - 4c
[M1] (OR)(ii) Express one vector as multiple of another
[A1] (OR)(ii) 3:1

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.