Higher June 2022 Paper 1 Q15
15 \(A\), \(B\) and \(C\) are three points such that
\(\overrightarrow{AB} = 3\mathbf{a} + 4\mathbf{b}\)
\(\overrightarrow{AC} = 15\mathbf{a} + 20\mathbf{b}\)
(a) Prove that \(A\), \(B\) and \(C\) lie on a straight line. (2)
\(D\), \(E\) and \(F\) are three points on a straight line such that
\(\overrightarrow{DE} = 3\mathbf{e} + 6\mathbf{f}\)
\(\overrightarrow{EF} = -10.5\mathbf{e} - 21\mathbf{f}\)
(b) Find the ratio
length of \(DF\) : length of \(DE\) (3)
length of \(DF\) : length of \(DE\) (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\overrightarrow{AC} = 5\overrightarrow{AB}\) and reason | M1 | for \(\overrightarrow{AC} = 5(3\mathbf{a} + 4\mathbf{b})\) or \(\overrightarrow{BC} = 4(3\mathbf{a} + 4\mathbf{b})\) or indicates that \(15\mathbf{a} + 20\mathbf{b} = 5 \times (3\mathbf{a} + 4\mathbf{b})\) |
| C1 | for \(\overrightarrow{AC} = 5\overrightarrow{AB}\) or \(\overrightarrow{AC} = 5(3\mathbf{a} + 4\mathbf{b})\) or \(15\mathbf{a} + 20\mathbf{b} = 5(3\mathbf{a} + 4\mathbf{b})\) and a correct reason, eg \(AC\) is a multiple of \(AB\) / multiples of each other / \(AB\) is a factor of \(AC\) or they have the same gradient / are parallel / go in the same direction or they have a point in common / both start at \(A\) or \(AC\) is an enlargement of \(AB\) |
Additional guidance
Do not award this mark if any incorrect working seen
Could use \(AB\) and \(BC\) or \(AC\) and \(BC\)
| Answer | Mark | Mark scheme |
|---|---|---|
| 5 : 2 | P1 | for \((\overrightarrow{DF} =)\ 3\mathbf{e} + 6\mathbf{f} + (-10.5\mathbf{e} - 21\mathbf{f})\ \ (= -7.5\mathbf{e} - 15\mathbf{f})\) or for a multiplicative relationship for \(\overrightarrow{DE}\) and \(\overrightarrow{EF}\) eg \(\overrightarrow{EF} = -3.5\,\overrightarrow{DE}\) or for \((DE : EF =)\ 1 : -3.5\) oe |
| P1 | for a multiplicative relationship for \(\overrightarrow{DF}\) and \(\overrightarrow{DE}\) eg \(\overrightarrow{DF} = -2.5\,\overrightarrow{DE}\) or for \((DF : DE =)\ -5 : 2\) oe eg \(-2.5 : 1\) or \(-7.5 : 3\) or for answer of 2 : 5 oe | |
| A1 | oe eg 2.5 : 1 |