| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a straightforward 3D vectors question requiring standard techniques: finding position vectors using given ratios, using parallelogram properties (opposite sides equal), writing vector equations of lines, and finding intersection by equating parameters. While it has multiple steps, each is routine application of A-level vector methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.10e Position vectors: and displacement1.10g Problem solving with vectors: in geometry4.04a Line equations: 2D and 3D, cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\mathbf{AX} = (10, 2, 14)\) or \((2a+4b)\) seen | M1 | |
| \(\mathbf{r} = (1, -1, 1) + \lambda(5, 1, 7)\) aef | A1 | |
| \(\mathbf{CD} = (5, 7, 9)\) or \(-3a+4b\) seen | B1 | |
| \(\mathbf{r} = (3, -3, 3) + \mu(5, 7, 9)\) aef | B1 | |
| \(\mathbf{r} =\) included in an eqn | B1 | [5] |
| (ii) At least two components equated with different scalars. Allow any notation. | M1 | |
| Attempt to find at least one of \(\lambda\) or \(\mu\) | M1 | |
| \(\mu = 0.4\) OR \(\lambda = 0.8\) | A1 | |
| Substitute \(\lambda\) and \(\mu\) into the third eqn to check consistency and state \((5, -0.2, 6.6)\) | A1 | [4] |
(i) $\mathbf{AX} = (10, 2, 14)$ or $(2a+4b)$ seen | M1 |
$\mathbf{r} = (1, -1, 1) + \lambda(5, 1, 7)$ aef | A1 |
$\mathbf{CD} = (5, 7, 9)$ or $-3a+4b$ seen | B1 |
$\mathbf{r} = (3, -3, 3) + \mu(5, 7, 9)$ aef | B1 |
$\mathbf{r} =$ included in an eqn | B1 | [5]
(ii) At least two components equated with different scalars. Allow any notation. | M1 |
Attempt to find at least one of $\lambda$ or $\mu$ | M1 |
$\mu = 0.4$ OR $\lambda = 0.8$ | A1 |
Substitute $\lambda$ and $\mu$ into the third eqn to check consistency and state $(5, -0.2, 6.6)$ | A1 | [4]The points $A$ and $B$ have position vectors $\mathbf{i} - \mathbf{j} + \mathbf{k}$ and $2\mathbf{i} + \mathbf{j} + 3\mathbf{k}$ respectively, relative to the origin $O$. The point $C$ is on the line $OA$ extended so that $\overrightarrow{AC} = 2\overrightarrow{OA}$ and the point $D$ is on the line $OB$ extended so that $\overrightarrow{BD} = 3\overrightarrow{OB}$. The point $X$ is such that $OCXD$ is a parallelogram.
\begin{enumerate}[label=(\roman*)]
\item Show that a vector equation of the line $AX$ is $\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})$ and find an equation of the line $CD$ in a similar form. [5]
\item Prove that the lines $AX$ and $CD$ intersect and find the position vector of their point of intersection. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q8 [9]}}