4MA0 Higher January 2016 Paper 3R Q19
19 The diagram shows a grid of equally spaced parallel lines.
The point \(P\) and the vectors a and b are shown on the grid.
\(\overrightarrow{PQ} = 3\mathbf{a} + 4\mathbf{b}\)
(a) On the grid, mark the vector \(\overrightarrow{PQ}\) (1)
\(\overrightarrow{PR} = -4\mathbf{a} + 2\mathbf{b}\)
(b) On the grid, mark the vector \(\overrightarrow{PR}\) (1)
(c) Find, in terms of a and b, the vector \(\overrightarrow{QR}\) (1)
The point \(M\) lies on \(PR\) such that \(PM = \dfrac{2}{3}PR\)
The point \(N\) lies on \(PQ\) such that \(PN = \dfrac{1}{3}PQ\)
(d) Show that \(\overrightarrow{MN} = k\mathbf{a}\) where \(k\) is a constant.
State the value of \(k\). (3)
State the value of \(k\). (3)
| Scheme | Marks |
|---|---|
| Vector, with arrow, marked from \(P\) to point 3 to the right and 8 up. | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| Vector, with arrow, marked from \(P\) to point 4 to the left and 4 up. | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(-7\mathbf{a} - 2\mathbf{b}\) | B1 |
| (1) |
Notes
B1: oe eg \(-7\mathbf{a} + -2\mathbf{b}\)
| Scheme | Marks |
|---|---|
| Eg \((\overrightarrow{MP} =)\;\tfrac{2}{3}(4\mathbf{a} - 2\mathbf{b})\) or \((\overrightarrow{PN} =)\;\tfrac{1}{3}(3\mathbf{a} + 4\mathbf{b})\) | M1 |
| Eg \((\overrightarrow{MN} =)\;\tfrac{2}{3}(4\mathbf{a} - 2\mathbf{b}) + \tfrac{1}{3}(3\mathbf{a} + 4\mathbf{b})\) or \(\tfrac{1}{3}(3\mathbf{a} + 4\mathbf{b}) - \tfrac{2}{3}(-4\mathbf{a} + 2\mathbf{b})\) or \(\tfrac{8}{3}\mathbf{a} - \tfrac{4}{3}\mathbf{b} + \mathbf{a} + \tfrac{4}{3}\mathbf{b}\) | M1 |
| \(\dfrac{11}{3}\) | A1 |
| (3) |
Notes
M1: Accept \(\pm\tfrac{2}{3}(4\mathbf{a} - 2\mathbf{b})\) or \(\pm\tfrac{1}{3}(3\mathbf{a} + 4\mathbf{b})\) or \(\overrightarrow{MN} = \overrightarrow{MP} + \overrightarrow{PN}\) oe
M1: For a correct expression for \(\overrightarrow{MN}\)
A1: Dep on M2
cso
Accept exact value only
Eg \(3\dfrac{2}{3}\) or \(3.\dot{6}\)