O-LevelMathematicsVectors and transformations4024/12 · May/June 2017 · Q214 Marks

OACB is a parallelogram. \vec{OA} = \mathbf{a} and \vec{OB} = \mathbf{b}. P and Q are points on OC such that OP = PQ = QC. [Figure: Parallelogram OACB. Vector OA is 'a', OB is 'b'. Points P and Q divide diagonal OC into three equal segments: OP=PQ=QC.]

(a)(i)Express, as simply as possible, in terms of \mathbf{a} and \mathbf{b}, \vec{OP},[1]
(a)(ii)\vec{BP}.[1]
(b)Show that triangles OAQ and CBP are congruent.[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.

[1] (a)(i) 1/3(a + b) or 1/3a + 1/3b or (a + b)/3 final answer
[1] (a)(ii) 1/3a - 2/3b or 1/3(a - 2b) or (a - 2b)/3 final answer
[2] (b) Any two pairs of vectors from \vec{OA} = \vec{BC} oe ; \vec{OQ} = \vec{PC} oe ; \vec{QA} = \vec{BP} oe
B1 for any one pair of vectors stated
Alternative method:
B1 for two of these pairs of sides stated or one of these pairs of sides and this pair of angles stated
\vec{OA} = \vec{BC}
\vec{OQ} = \vec{PC}
\angle AOQ = \angle BCP

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