1MA0 Higher June 2017 Paper 1 Q24
24
Diagram NOT accurately drawn
\(\overrightarrow{OA} = 6\mathbf{a}\) and \(\overrightarrow{OB} = 6\mathbf{b}\)
\(M\) is the midpoint of \(AB\).
(a) Write \(\overrightarrow{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
Give your answer in its simplest form. (2)
Give your answer in its simplest form. (2)
\(N\) is the midpoint of \(OB\).
\(G\) is the point on \(OM\) such that \(OG : GM = 2 : 1\)
*(b) Show that \(AGN\) is a straight line. (4)
| Answer | Mark | Notes |
|---|---|---|
| \(3\mathbf{a} + 3\mathbf{b}\) | 2 | M1 for \(\overrightarrow{AB} = 6\mathbf{b} - 6\mathbf{a}\) oe or \(\overrightarrow{BA} = 6\mathbf{a} - 6\mathbf{b}\) oe or \((\overrightarrow{OM}) = \dfrac{1}{2}(6\mathbf{a} + 6\mathbf{b})\) oe A1 cao |
| Working | Answer | Mark | Notes |
|---|---|---|---|
| Method 1 Show that \(\overrightarrow{AG}\) is parallel to \(\overrightarrow{AN}\) | Shown | 4 | M1 for a method to find \(\overrightarrow{OG}\) or \(\overrightarrow{GM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) eg \(\overrightarrow{OG} = \dfrac{2}{3}\text{``}(3\mathbf{a} + 3\mathbf{b})\text{''}\) (\(= 2\mathbf{a} + 2\mathbf{b}\)) eg \(\overrightarrow{GM} = \dfrac{1}{3}\text{``}(3\mathbf{a} + 3\mathbf{b})\text{''}\) (\(= \mathbf{a} + \mathbf{b}\)) M1 for a method to find \(\overrightarrow{AN}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) eg \(-6\mathbf{a} + 6\mathbf{b} \div 2\) (\(= 3\mathbf{b} - 6\mathbf{a}\)) M1 for a method to find \(\overrightarrow{AG}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) eg \(-6\mathbf{a} + \text{``}2\mathbf{a} + 2\mathbf{b}\text{''}\) (\(= 2\mathbf{b} - 4\mathbf{a}\)) C1 for correct simplified expressions in terms of \(\mathbf{a}\) and \(\mathbf{b}\) for \(\overrightarrow{AG}\) and \(\overrightarrow{AN}\) followed by conclusion |
| Method 2 Show that \(\overrightarrow{AG}\) is parallel to \(\overrightarrow{GN}\) | OR M1 for a method to find \(\overrightarrow{OG}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) eg \(\dfrac{2}{3}\text{``}(3\mathbf{a} + 3\mathbf{b})\text{''}\) (\(= 2\mathbf{a} + 2\mathbf{b}\)) M1 for a method to find \(\overrightarrow{GN}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) eg \(-\text{``}(2\mathbf{a} + 2\mathbf{b})\text{''} + 6\mathbf{b} \div 2\) (\(= \mathbf{b} - 2\mathbf{a}\)) M1 for a method to find \(\overrightarrow{AG}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) eg \(\dfrac{1}{2}(6\mathbf{b} - 6\mathbf{a}) - \dfrac{1}{3}\text{``}(3\mathbf{a} + 3\mathbf{b})\text{''}\) (\(= 2\mathbf{b} - 4\mathbf{a}\)) C1 for correct simplified expressions in terms of \(\mathbf{a}\) and \(\mathbf{b}\) for \(\overrightarrow{AG}\) and \(\overrightarrow{GN}\) followed by conclusion |