| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 9 |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a standard two-part vectors question requiring equating components to find intersection (routine algebraic manipulation) and then using the scalar product formula for angle between direction vectors. Both techniques are core A-level/Pre-U material with no novel insight required, making it slightly easier than average. |
| Spec | 4.04e Line intersections: parallel, skew, or intersecting |
**(i)** Form 3 simultaneous equations in $\lambda$ and $\mu$ M1
All three correct equations A1
Solving any two equations M1
$\lambda + 3\mu = 7$
$2\lambda - \mu = 0$
$6\lambda + 2\mu = 10$
Obtain $\lambda = 1$ and $\mu = 2$ from two equations A1
Substitute into the third equation to show LHS and RHS equal M1
State point of intersection $= (-2, 3, 1)$ A1 **[6]**
**(ii)** Obtain $(\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}).(-3\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = -13$ or equiv B1
Attempt correct scalar product
$\cos\theta = \frac{\begin{pmatrix}1\\2\\6\end{pmatrix}\cdot\begin{pmatrix}-3\\1\\-2\end{pmatrix}}{\sqrt{1^2+2^2+6^2}\sqrt{(-3)^2+1^2+(-2)^2}}$ or equiv M1
$\theta = 123°$ or better A1 **[3]**8 (i) Show that the lines
$$\mathbf { r } = - 3 \mathbf { i } + \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + \mathbf { 6 } \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } + \mu ( - 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$
intersect and write down the coordinates of their point of intersection.\\
(ii) Find in degrees the obtuse angle between the two lines.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q8 [9]}}