4MA0 Higher June 2016 Paper 4 Q22
22
Diagram NOT accurately drawn
\(OMN\) is a triangle.
\(P\) is the point on \(OM\) such that \(OP = \dfrac{1}{4}OM\)
\(Q\) is the midpoint of \(ON\)
\(R\) is the midpoint of \(PN\)
\(\overrightarrow{OP} = \mathbf{p}\) \(\overrightarrow{OQ} = \mathbf{q}\)
(a) Find, in terms of \(\mathbf{p}\) and \(\mathbf{q}\),
(i) \(\overrightarrow{MN}\)
(ii) \(\overrightarrow{PR}\)
(2)(b) Use a vector method to prove that \(QR\) is parallel to \(OP\) (2)
| Scheme | Marks |
|---|---|
| (i) Answer: \(2\mathbf{q} - 4\mathbf{p}\) oe | B1 |
(ii) Answer: \(\mathbf{q} - \dfrac{1}{2}\mathbf{p}\) oe | B1 |
| (2) |
Notes
B1: Eg \(2(\mathbf{q} - 2\mathbf{p})\)
B1: Eg \(0.5(-\mathbf{p} + 2\mathbf{q})\)
| Scheme | Marks |
|---|---|
| Eg (\(\overrightarrow{QR}\) =) \(-\mathbf{q} + \mathbf{p} + \mathbf{q} - \dfrac{1}{2}\mathbf{p}\) or \(\dfrac{1}{2}\mathbf{p}\) oe | M1 |
Eg (\(\overrightarrow{QR}\) =) \(\dfrac{1}{2}\mathbf{p}\) and \(\overrightarrow{QR} = 0.5\overrightarrow{OP}\) or (\(\overrightarrow{QR}\) =) \(\dfrac{1}{2}\mathbf{p}\) and \(\overrightarrow{OP} = 2\overrightarrow{QR}\) Answer: Shown | A1 |
| (2) |
Notes
M1: For (\(\overrightarrow{QR}\)) = \(\dfrac{1}{2}\mathbf{p}\) or
For (\(\overrightarrow{QR}\)) = \(-\mathbf{q} + \mathbf{p}\) + “their a(ii)”
or (\(\overrightarrow{QR}\)) = \(\mathbf{q}\) – “their a(ii)”
A1: For (\(\overrightarrow{QR}\)) = \(\dfrac{1}{2}\mathbf{p}\) and a valid conclusion such as:
\(\overrightarrow{QR} = 0.5\overrightarrow{OP}\) or \(\overrightarrow{OP} = 2\overrightarrow{QR}\) or
\(\mathbf{p}\) is a multiple of \(\dfrac{1}{2}\mathbf{p}\) or
They have the same direction but \(OP\) is twice as long or
They have the same vector component.