Higher June 2021 Paper 1 Q26
26 \(OACB\) is a trapezium.
Diagram NOT accurately drawn
\(\overrightarrow{OA} = 2\mathbf{a} \qquad \overrightarrow{OB} = 5\mathbf{b} \qquad \overrightarrow{AC} = 3\mathbf{b}\)
The diagonals, \(OC\) and \(AB\), of the trapezium intersect at the point \(P\).
Find and simplify an expression, in terms of a and b, for \(\overrightarrow{OP}\)
Show your working clearly.
(5)
| Scheme | Marks |
|---|---|
| eg \(\overrightarrow{OP} = n(2\mathbf{a} + 3\mathbf{b})\) or \(\overrightarrow{OP} = 2\mathbf{a} + m(5\mathbf{b} - 2\mathbf{a})\) or \(\overrightarrow{OP} = 5\mathbf{b} + x(2\mathbf{a} - 5\mathbf{b})\) | M1 |
| eg \(\overrightarrow{OP} = n(2\mathbf{a} + 3\mathbf{b})\) and \(\overrightarrow{OP} = 2\mathbf{a} + m(5\mathbf{b} - 2\mathbf{a})\) or eg \(\overrightarrow{OP} = n(2\mathbf{a} + 3\mathbf{b})\) and \(\overrightarrow{OP} = 5\mathbf{b} + x(2\mathbf{a} - 5\mathbf{b})\) oe | M1 |
eg \(5m = 3n\) or \(m = \dfrac{3}{5}n\) or \(2n = 2 - 2m\) or \(n = 1 - m\) oe and \(2 - 2 \times \dfrac{3}{5}n = 2n\) or \(2 \times \dfrac{5}{3}m = 2 - 2m\) oe or eg \(2n = 2x\) or \(n = x\) or \(3n = 5 - 5x\) oe and \(3x = 5 - 5x\) or \(3n = 5 - 5n\) oe | M1 |
| eg \(m = \dfrac{3}{8}\) or \(n = \dfrac{5}{8}\) or \(x = \dfrac{5}{8}\) oe | M1 |
Working is required Answer: \(\dfrac{5}{4}\mathbf{a} + \dfrac{15}{8}\mathbf{b}\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for a vector equation for \(\overrightarrow{OP}\)
M1: 2 vector equations for \(\overrightarrow{OP}\) that can be used to find \(\overrightarrow{OP}\) - must be in terms of a and b and a scalar
M1: Writing one equation in terms of only one scalar eg one of \(n\) or \(m\) or \(x\) etc
M1: for a correct value for one scalar
| Scheme | Marks |
|---|---|
| eg \(\overrightarrow{OP} = n(2\mathbf{a} + 3\mathbf{b})\) | M1 |
eg \(CP : OP = 3 : 5\) or \(CP : CO = 3 : 8\) or \(\dfrac{CP}{OP} = \dfrac{3}{5}\) or \(\dfrac{CP}{CO} = \dfrac{3}{8}\) oe | M2 |
| \(\overrightarrow{OP} = \dfrac{5}{8}\overrightarrow{OC}\) or \(n = \dfrac{5}{8}\) | M1 |
Working is required Answer: \(\dfrac{5}{4}\mathbf{a} + \dfrac{15}{8}\mathbf{b}\) | A1 |
Notes
M1: for a vector equation for \(\overrightarrow{OP}\)
M2: for a correct ratio for two sides in triangle \(ACP\) and triangle \(BOP\) that help to find \(\overrightarrow{OP}\) as a fraction of \(\overrightarrow{OC}\) (could be seen on the diagram)