| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 8 |
| Topic | Vectors: Lines & Planes |
| Type | Angle between two lines |
| Difficulty | Standard +0.3 This is a standard FP1 vector geometry question testing routine techniques: finding intersection by equating parametric equations, using dot product for angles, and applying perpendicularity conditions. All parts follow textbook methods with no novel insight required, though part (iii) requires visualizing the geometric setup. Slightly easier than average A-level due to straightforward algebraic manipulation. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(-11 + 2\lambda = 5 + 3\mu\) or \(10 - 2\lambda = 2 + \mu\) or \(3 + \lambda = 4 - 2\mu\) | [3] M1 | Any two correct and an attempt to solve simultaneously (eg adding or subtracting two equations even if incorrect/not useful) |
| Answer: \(\lambda = 5\) or \(\mu = -2\) | A1 | |
| Answer: \(\begin{pmatrix} -1 \\ 0 \\ 8 \end{pmatrix}\) | A1 | Condone as coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix} = 2\) | [3] M1 | Condone multiples (even unknown) if answer is correct but must be the direction vectors |
| Answer: \(\cos\theta = \frac{2}{3\sqrt{14}}\) | A1 | or awrt 0.178 |
| Answer: \(\theta = 79.7°\) | A1 | awrt 79.7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(12\tan 79.7\) | M1ft | NB units essential. Accept 66.3 |
| Answer: \(66 \text{ m}\) | A1 |
## (i)
Answer: $-11 + 2\lambda = 5 + 3\mu$ or $10 - 2\lambda = 2 + \mu$ or $3 + \lambda = 4 - 2\mu$ | [3] M1 | Any two correct and an attempt to solve simultaneously (eg adding or subtracting two equations even if incorrect/not useful)
Answer: $\lambda = 5$ or $\mu = -2$ | A1 |
Answer: $\begin{pmatrix} -1 \\ 0 \\ 8 \end{pmatrix}$ | A1 | Condone as coordinates
## (ii)
Answer: $\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix} = 2$ | [3] M1 | Condone multiples (even unknown) if answer is correct but must be the direction vectors
Answer: $\cos\theta = \frac{2}{3\sqrt{14}}$ | A1 | or awrt 0.178
Answer: $\theta = 79.7°$ | A1 | awrt 79.7
## (iii)
Answer: $12\tan 79.7$ | M1ft | NB units essential. Accept 66.3
Answer: $66 \text{ m}$ | A1 |
---Two lines, $l_1$ and $l_2$, have the following equations.
$$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$
$$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$
$P$ is the point of intersection of $l_1$ and $l_2$.
\begin{enumerate}[label=(\roman*)]
\item Find the position vector of $P$. [3]
\item Find, correct to 1 decimal place, the acute angle between $l_1$ and $l_2$. [3]
\end{enumerate}
$Q$ is a point on $l_1$ which is 12 metres away from $P$. $R$ is the point on $l_2$ such that $QR$ is perpendicular to $l_1$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Determine the length $QR$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2017 Q3 [8]}}