| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | Specimen |
| Marks | 7 |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a standard vectors question testing parallel lines (direction vector proportionality) and line intersection (solving simultaneous equations). Part (i) is routine; part (ii) requires solving a system but with straightforward arithmetic. Slightly easier than average A-level as the calculations are manageable and the concepts are core syllabus material. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
**(i)**
- Indication that relevant vectors are parallel e.g. $L_3 = -\frac{1}{3}L_1$ [M1]
- $c = -3$ [A1]
**(ii)**
- Produce 2/3 equations containing $t$, $u$ (and $c$), e.g. $2-3t=3-2u$, $3-8t=-1+cu$, $6t=4+u$ [M1]
- Solve 2 equations not containing '$c$' [M1]
- $t=1,\ u=2$ [A1]
- Substitute their $t$ and $u$ into equation containing $c$ [M1]
- $c = -2$ [A1]
**Total: 7 marks**10 Lines $L _ { 1 } , L _ { 2 }$ and $L _ { 3 }$ have vector equations
$$\begin{aligned}
& L _ { 1 } = ( 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + s ( 6 \mathbf { i } + 9 \mathbf { j } - 3 \mathbf { k } ) , \\
& L _ { 2 } = ( 2 \mathbf { i } + 3 \mathbf { j } ) + t ( - 3 \mathbf { i } - 8 \mathbf { j } + 6 \mathbf { k } ) , \\
& L _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } ) + u ( - 2 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) .
\end{aligned}$$
In each of the following cases, find the value of $c$.\\
(i) $\quad L _ { 1 }$ and $L _ { 3 }$ are parallel.\\
(ii) $\quad L _ { 2 }$ and $L _ { 3 }$ intersect.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q10 [7]}}