Higher November 2019 Paper 3 Q24
24 \(OXYZ\) is a parallelogram.
\(\overrightarrow{OX} = \mathbf{a}\)
\(\overrightarrow{OY} = \mathbf{b}\)
\(P\) is the point on \(OX\) such that \(OP : PX = 1 : 2\)
\(R\) is the point on \(OY\) such that \(OR : RY = 1 : 3\)
Work out, in its simplest form, the ratio \(ZP : ZR\)
You must show all your working. (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| 4 : 3 | P1 | Process to find a missing vector using the given ratios as fractions, eg \(\dfrac{1}{3}\) of \(\overrightarrow{OX}\ \left(= \dfrac{1}{3}\mathbf{a}\right)\) or \(\dfrac{1}{4}\) of \(\overrightarrow{OY}\ \left(= \dfrac{1}{4}\mathbf{b}\right)\) |
| P1 | for a process to use \(\overrightarrow{ZO} = \overrightarrow{YX} = \mathbf{a} - \mathbf{b}\) oe | |
| P1 | for a process to find either \(\overrightarrow{ZP}\) or \(\overrightarrow{ZR}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\), eg either \(\overrightarrow{ZP} = \mathbf{a} - \mathbf{b} + \dfrac{1}{3}\mathbf{a}\) or \(\overrightarrow{ZR} = \mathbf{a} - \mathbf{b} + \dfrac{1}{4}\mathbf{b}\) | |
| P1 | for a process to write \(\overrightarrow{ZP}\) and \(\overrightarrow{ZR}\) as multiples of the same vector, eg multiplying both by 12 to get the ratio, \(\dfrac{4}{3}(\mathbf{a} - 0.75\mathbf{b})\) and \(\mathbf{a} - 0.75\mathbf{b}\) respectively | |
| A1 | oe |
Additional guidance
Might be embedded in their answer for \(ZP\)
The award of this mark implies the first two process marks.