Higher June 2023 Paper 1R Q19
19 The diagram shows a triangle \(ABC\) where \(A\), \(B\) and \(C\) represent the positions of three towns.
Diagram NOT accurately drawn
\(\overrightarrow{AB} = \begin{pmatrix} 7 \\ -2 \end{pmatrix} \qquad \overrightarrow{BC} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\)
Pru travels directly from \(A\) to \(B\) and then directly from \(B\) to \(C\)
Yang travels directly from \(A\) to \(C\)
Given that the values for \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are in kilometres,
work out how much further Pru travels than Yang travels.
Give your answer in km, correct to one decimal place.
(5)
| Scheme | Marks |
|---|---|
| eg \(\begin{pmatrix} 7 \\ -2 \end{pmatrix} + \begin{pmatrix} -3 \\ 5 \end{pmatrix}\) or \(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\) or \(-\begin{pmatrix} 7 \\ -2 \end{pmatrix} - \begin{pmatrix} -3 \\ 5 \end{pmatrix}\) or \(\begin{pmatrix} -4 \\ -3 \end{pmatrix}\) | M1 |
eg \(\left(\overrightarrow{AC} =\right)\sqrt{\text{``}4\text{''}^2 + \text{``}3\text{''}^2}\;\left(= \sqrt{25} = 5\right)\) | M1 |
eg \(\left(\overrightarrow{AB} =\right)\sqrt{7^2 + (\pm 2)^2}\;\left(= \sqrt{53} = 7.28(010)\right)\) or \(\left(\overrightarrow{BC} =\right)\sqrt{(\pm 3)^2 + 5^2}\;\left(= \sqrt{34} = 5.83(095)\right)\) | M1 |
"\(\sqrt{7^2 + (\pm 2)^2}\)" + "\(\sqrt{(\pm 3)^2 + 5^2}\)" or "\(\sqrt{53}\)" + "\(\sqrt{34}\)" (= 13.1(110)) or “7.28” + “5.83” (= 13.1(110)) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 8.1 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for a method for finding \(\overrightarrow{AC}\) or \(\overrightarrow{CA}\)
or
for sight of \(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\) or \(\begin{pmatrix} -4 \\ -3 \end{pmatrix}\)
M1: (dep on previous M1) for a method to find the magnitude of \(\overrightarrow{AC}\) or \(\overrightarrow{CA}\)
M1: (indep) for a method to find the magnitude of either \(\overrightarrow{AB}\) or \(\overrightarrow{BC}\)
M1: (dep on previous M1) for a complete method to find Pru’s distance travelled
A1: accept 8.1 – 8.2, to award full marks \(\overrightarrow{AC}\) must be correct