| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 6 |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a standard two-part question on line intersection requiring students to equate vector equations and solve simultaneous equations for parameters λ and μ, then use the result to find the unknown constant a. While it involves multiple steps and algebraic manipulation, it follows a routine procedure taught in FP1 with no novel insight required, making it slightly easier than average. |
| Spec | 4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3 + \lambda = 1 + 2\mu\) and \(-5 - 2\lambda = 1 - 3\mu\) | M1 | Attempt to solve simultaneously (eg adding or subtracting two equations even if incorrect) |
| eg \(6 + 2\lambda = 2 + 4\mu\) and or \(6 + 2\lambda - 5 - 2\lambda = 2 + 4\mu + 1 - 3\mu\) | M1 | |
| \(\lambda = -6\) | A1 | |
| \(\begin{pmatrix} -3 \\ -15 \\ 7 \end{pmatrix}\) | A1 | Condone as coordinates |
| [4] | ||
| (ii) \(a + 2\mu = -15\) and \(\mu = -2\) | M1 | Must be seen in (ii) |
| \(-11\) | A1 | |
| [2] |
**(i)** $3 + \lambda = 1 + 2\mu$ and $-5 - 2\lambda = 1 - 3\mu$ | M1 | Attempt to solve simultaneously (eg adding or subtracting two equations even if incorrect)
eg $6 + 2\lambda = 2 + 4\mu$ and or $6 + 2\lambda - 5 - 2\lambda = 2 + 4\mu + 1 - 3\mu$ | M1 |
$\lambda = -6$ | A1 |
$\begin{pmatrix} -3 \\ -15 \\ 7 \end{pmatrix}$ | A1 | Condone as coordinates
| [4] |
**(ii)** $a + 2\mu = -15$ and $\mu = -2$ | M1 | Must be seen in (ii)
$-11$ | A1 |
| [2] |
---$$\begin{aligned}
l _ { 1 } : \mathbf { r } = \left( \begin{array} { c }
4 \\
2 \\
5
\end{array} \right) + \lambda \left( \begin{array} { c }
1 \\
3 \\
- 2
\end{array} \right) \\
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
1 \\
a \\
1
\end{array} \right) + \mu \left( \begin{array} { c }
2 \\
2 \\
- 3
\end{array} \right)
\end{aligned}$$
(i) Find the position vector of the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.\\
(ii) Determine the value of $a$.
\hfill \mbox{\textit{OCR FP1 AS 2018 Q3 [6]}}