Higher June 2023 Paper 3 Q22
22 Given that the vector \(a\begin{pmatrix} 2 \\ 6 \end{pmatrix} + b\begin{pmatrix} 8 \\ 2 \end{pmatrix}\) is parallel to the vector \(\begin{pmatrix} 13 \\ 6 \end{pmatrix}\)
find an expression for \(b\) in terms of \(a\). (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(b = 3a\) | P1 | for \(a\begin{pmatrix} 2 \\ 6 \end{pmatrix} + b\begin{pmatrix} 8 \\ 2 \end{pmatrix} = k\begin{pmatrix} 13 \\ 6 \end{pmatrix}\) oe or for setting up a linear equation in \(a\) or \(b\) eg \(2a + 8b = 13k\) or \(6a + 2b = 6k\), \(k \neq 0, 1\) or for \(2a + 8b = 13\) and \(6a + 2b = 6\) or for \(\dfrac{2a + 8b}{6a + 2b} = \dfrac{13}{6}\) oe |
| P1 | for process to solve the simultaneous equations to get \(b = \dfrac{3k}{2}\) and \(a = \dfrac{k}{2}\) or \(b = \dfrac{3}{2}\) and \(a = \dfrac{1}{2}\) or both \(2a + 8b = 13k\) and \(6a + 2b = 6k\) with process to eliminate \(k\) | |
| A1 | for \(b = 3a\) oe |
Additional guidance
Accept any non zero value substituted for \(k\)