| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Topic | Vectors 3D & Lines |
| Type | Line intersection verification |
| Difficulty | Moderate -0.3 This is a standard procedural question on 3D line relationships requiring equating components and solving a system of linear equations. While it involves multiple steps and careful algebraic manipulation, it follows a well-established algorithm taught in all A-level Further Maths courses with no novel insight required. |
| Spec | 4.04e Line intersections: parallel, skew, or intersecting |
Set up at least 2 equations: $4 + 2\mu = 35 - 5\lambda$, $7 + 3\mu = 6 + 2\lambda$, $3 + 7\mu = 14 + 3\lambda$ **M1**
Find a value for $\lambda$ or $\mu$ from two of them **M1**
Obtain $\mu = 3$, $\lambda = 5$ from the first two ($\mu = 5$, $\lambda = 8$ from last two; $\mu = 3.61$, $\lambda = 4.76$ from the first and last) **A1**
Demonstrate inconsistency in third eqn, e.g. $7 \times 3 - 3 \times 5 = 6 \neq 11$ **and** state do not intersect. This requires correct values for $\lambda$ and $\mu$ $(3 + 7(3) = 24 \neq 14 + 3(5) = 29$ or $14 \neq -5)$ **M1\***
Show the direction vectors are not multiples of each other **and** state they are not parallel **B1\***
**OR** find angle between direction vectors ($= 69.498°$) **and** state not parallel **OR** find dot product ($= 17$) **and** state is not equal to 1 and therefore not parallel)
State skew (requires accurate previous working) **depB1**
**[6]**9 Determine whether the lines whose equations are
$$\mathbf { r } = ( 4 + 2 \mu ) \mathbf { i } + ( 7 + 3 \mu ) \mathbf { j } + ( 3 + 7 \mu ) \mathbf { k } \quad \text { and } \quad \mathbf { r } = ( 35 - 5 \lambda ) \mathbf { i } + ( 6 + 2 \lambda ) \mathbf { j } + ( 14 + 3 \lambda ) \mathbf { k }$$
intersect, are parallel or are skew.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q9 [6]}}