1MA0 Higher June 2016 Paper 1 Q23

EdexcelHigherOld spec6 marksVectors

Diagram NOT accurately drawn

\(OPTR\) is a trapezium.

\(\overrightarrow{OP} = \mathbf{a}\)
\(\overrightarrow{PT} = \mathbf{b}\)
\(\overrightarrow{OR} = 3\mathbf{b}\)

(a)

(i) Find \(\overrightarrow{OT}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\)

(ii) Find \(\overrightarrow{PR}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\)
Give your answer in its simplest form. (2)

\(S\) is the point on \(PR\) such that \(PS : SR = 1 : 3\)

(b) Find \(\overrightarrow{OS}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\)
Give your answer in its simplest form. (2)

*(c) What does your answer to part (b) tell you about the position of point \(S\)? (2)

Answer	Mark	Notes
(i) \(\mathbf{a} + \mathbf{b}\)	2	B1 for \(\mathbf{a} + \mathbf{b}\) oe
(ii) \(-\mathbf{a} + 3\mathbf{b}\)		B1 for \(-\mathbf{a} + 3\mathbf{b}\) oe

Mark scheme (b)

Answer	Mark	Notes
\(\tfrac{3}{4}\mathbf{a} + \tfrac{3}{4}\mathbf{b}\)	2	M1 for \(\overrightarrow{OP} + \tfrac{1}{4}\overrightarrow{PR}\) or \(\overrightarrow{OR} + \tfrac{3}{4}\overrightarrow{RP}\) (may be in terms of \(\mathbf{a}\) and \(\mathbf{b}\)) A1 for \(\tfrac{3}{4}\mathbf{a} + \tfrac{3}{4}\mathbf{b}\) or \(\tfrac{3}{4}(\mathbf{a} + \mathbf{b})\)

Mark scheme (c)

Answer	Mark	Notes
\(OS = \tfrac{3}{4}OT\)	2	C2 (dep A1) for \(S\) divides \(OT\) in the ratio 3:1 oe or \(OS = \tfrac{3}{4}OT\) oe (C1 (dep A1) for \(S\) lies on \(OT\) or that \(OT\) and \(PR\) intersect at \(S\) oe)