O-LevelAdditional MathematicsVectors in two dimensions4037/22 · May/June 2023 · Q108 Marks

The diagram shows a triangle OAB. The point C is the mid-point of OA. The point D lies on CB such that CD : DB = 2 : 3. OC = c CB = b. The point E lies on AB such that OE = λOD and AE = μAB where λ and μ are scalars. Find two expressions for OE, each in terms of b, c and a scalar, and hence find AE : EB.

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Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[B2] OD = c + (2/5)b isw
[B1] for λ(c+kb) where k≠2/5 or 1, k > 0
[B2] OE = 2c + μ(b - nc) or μb + (2− μ)c isw or any of the following with n > 0 2c+µ(b-nc) or nc+µ(b−c) or c+b+(1-µ)(nc-b) or n(c+b)+(1-μ)(c-b)
[M1] Equates components at least once λ = 2-μ or 2/5 = λ = μ soi FT providing of equivalent forms e.g.: λ(sc+tb) and xc+ µ(yb + z c) where s, t, x, y, z are scalars
[A1] λ = 2-μ and 2/5 = λ = μ soi, nfww
[A1] μ = 4/7 [λ = 7/10]
[A1] [AE : EB = ] 4:3 oe must have earned all previous marks

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Topic: Vectors in two dimensions, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, May/June 2023 Paper 2 Variant 2, question 10.

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