Higher November 2022 Paper 2 Q25
25 Two congruent parallelograms, PQRV and VRST, are joined.
\(\overrightarrow{QP} = \mathbf{a} \qquad \overrightarrow{PV} = \mathbf{b}\)
X is the midpoint of VT.
\(VW : WR = 1 : 2\)
Prove that Q, W and X lie on a straight line. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any one of \((\overrightarrow{QW} =)\ \mathbf{a} + \mathbf{b} - \dfrac{1}{3}\mathbf{a}\) \((\overrightarrow{WX} =)\ \dfrac{1}{3}\mathbf{a} + \dfrac{1}{2}\mathbf{b}\) \((\overrightarrow{QX} =)\ \mathbf{a} + \mathbf{b} + \dfrac{1}{2}\mathbf{b}\) | M1 | oe eg \((\overrightarrow{QW} =)\ \dfrac{2}{3}\mathbf{a} + \mathbf{b}\) or \((\overrightarrow{WX} =)\ -\dfrac{2}{3}\mathbf{a} + \mathbf{b} + \mathbf{a} - \dfrac{1}{2}\mathbf{b}\) or \((\overrightarrow{QX} =)\ \mathbf{a} + \dfrac{3}{2}\mathbf{b}\) allow use of \(\overrightarrow{WQ}\) and/or \(\overrightarrow{XW}\) and/or \(\overrightarrow{XQ}\) |
| Any two of \((\overrightarrow{QW} =)\ \mathbf{a} + \mathbf{b} - \dfrac{1}{3}\mathbf{a}\) \((\overrightarrow{WX} =)\ \dfrac{1}{3}\mathbf{a} + \dfrac{1}{2}\mathbf{b}\) \((\overrightarrow{QX} =)\ \mathbf{a} + \mathbf{b} + \dfrac{1}{2}\mathbf{b}\) | M1dep | oe allow use of \(\overrightarrow{WQ}\) and/or \(\overrightarrow{XW}\) and/or \(\overrightarrow{XQ}\) |
| Any valid pair of vectors and indication that one vector is a multiple of the other eg \(\overrightarrow{QW} = \dfrac{2}{3}\mathbf{a} + \mathbf{b}\) and \(\overrightarrow{WX} = \dfrac{1}{3}\mathbf{a} + \dfrac{1}{2}\mathbf{b}\) and \(\dfrac{2}{3}\mathbf{a} + \mathbf{b} = 2\left(\dfrac{1}{3}\mathbf{a} + \dfrac{1}{2}\mathbf{b}\right)\) | A1 | eg \(\overrightarrow{QW} = \dfrac{2}{3}\mathbf{a} + \mathbf{b}\) and \(\overrightarrow{XQ} = -\mathbf{a} - \dfrac{3}{2}\mathbf{b}\) and \(3\overrightarrow{QW} = -2\overrightarrow{XQ}\) or \(\overrightarrow{QX} = \mathbf{a} + \dfrac{3}{2}\mathbf{b}\) and \(\overrightarrow{WX} = \dfrac{1}{3}\mathbf{a} + \dfrac{1}{2}\mathbf{b}\) and \(WX\) is \(\dfrac{1}{3}\) of \(QX\) and \(WX\) is parallel to \(QX\) |
Additional guidance
Up to M2 may be awarded for correct work with no answer, or incorrect answer, even if this is seen amongst multiple attempts