Higher June 2019 Paper 2 Q20
20 \(CDEF\) is a quadrilateral.
\(\overrightarrow{CD} = \mathbf{a}\), \(\overrightarrow{DE} = \mathbf{b}\) and \(\overrightarrow{FC} = \mathbf{a} - \mathbf{b}\)
(a) Express \(\overrightarrow{FE}\) in terms of \(\mathbf{a}\) and/or \(\mathbf{b}\).
Give your answer in its simplest form. (2)
Give your answer in its simplest form. (2)
\(M\) is the midpoint of \(DE\).
\(X\) is the point on \(FM\) such that \(FX : XM = n : 1\)
\(CXE\) is a straight line.
(b) Work out the value of \(n\). (4)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(2\mathbf{a}\) | M1 | for \(\mathbf{a} - \mathbf{b} + \mathbf{a} + \mathbf{b}\ (= 2\mathbf{a})\) |
| A1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 4 | P1 | for a process to find \(\overrightarrow{MF} = -0.5\mathbf{b} - \mathbf{a} - (\mathbf{a} - \mathbf{b})\ (= 0.5\mathbf{b} - 2\mathbf{a})\) or \(\overrightarrow{CE} = \mathbf{a} + \mathbf{b}\) or \(\overrightarrow{FM} = \mathbf{a} - \mathbf{b} + \mathbf{a} + 0.5\mathbf{b}\ (= 2\mathbf{a} - 0.5\mathbf{b})\) |
| P1 | for finding a suitable vector expression for two of (\(\overrightarrow{CE}\) or \(\overrightarrow{EC}\)), (\(\overrightarrow{CX}\) or \(\overrightarrow{XC}\)) or (\(\overrightarrow{EX}\) or \(\overrightarrow{XE}\)) eg \(\overrightarrow{CX} = \mathbf{a} + 0.5\mathbf{b} + \dfrac{1}{n+1}(0.5\mathbf{b} - 2\mathbf{a})\) or \(\overrightarrow{CX} = -\mathbf{a} + \mathbf{b} + \dfrac{n}{n+1}(2\mathbf{a} - 0.5\mathbf{b})\) \(\overrightarrow{XE} = \dfrac{1}{n+1}(2\mathbf{a} - 0.5\mathbf{b}) + 0.5\mathbf{b}\) or \(\overrightarrow{XE} = \dfrac{n}{n+1}(0.5\mathbf{b} - 2\mathbf{a}) + 2\mathbf{a}\) \(\overrightarrow{XC} = \dfrac{n}{n+1}(0.5\mathbf{b} - 2\mathbf{a}) + \mathbf{a} - \mathbf{b}\) or \(\overrightarrow{XC} = \dfrac{1}{n+1}(2\mathbf{a} - 0.5\mathbf{b}) - 0.5\mathbf{b} - \mathbf{a}\) \(\overrightarrow{EX} = -0.5\mathbf{b} + \dfrac{1}{n+1}(0.5\mathbf{b} - 2\mathbf{a})\) or \(\overrightarrow{EX} = -2\mathbf{a} + \dfrac{n}{n+1}(2\mathbf{a} - 0.5\mathbf{b})\) | |
| P1 | for a complete process to equate the coefficients of \(\mathbf{a}\) and \(\mathbf{b}\), eg \(\dfrac{n - 1}{n + 1} = \dfrac{n + 2}{2(n + 1)}\) | |
| A1 | cao | |
| ALTERNATIVE | ||
| P1 | same as above | |
| P1 | for finding two suitable vector expressions for \(\overrightarrow{FX}\) eg \(\overrightarrow{FX} = \dfrac{n}{n+1}(2\mathbf{a} - 0.5\mathbf{b})\) and \(\overrightarrow{FX} = \mathbf{a} - \mathbf{b} + k\mathbf{a} + k\mathbf{b}\) | |
| P1 | for a complete process to equate the coefficients of \(\mathbf{a}\) and \(\mathbf{b}\), eg \(\dfrac{2n}{n+1} - 1 = 1 - \dfrac{n}{2(n+1)}\) | |
| A1 | cao |
Additional guidance
Accept ft from (a) providing vectors are clearly stated
\(\overrightarrow{CX} = \dfrac{n-1}{n+1}\mathbf{a} + \dfrac{n+2}{2(n+1)}\mathbf{b}\)
\(\overrightarrow{XE} = \dfrac{2}{n+1}\mathbf{a} + \dfrac{n}{2(n+1)}\mathbf{b}\)
\(\overrightarrow{XC} = \dfrac{1-n}{n+1}\mathbf{a} + \dfrac{-n-2}{2(n+1)}\mathbf{b}\)
\(\overrightarrow{EX} = -\dfrac{2}{n+1}\mathbf{a} - \dfrac{n}{2(n+1)}\mathbf{b}\)