| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2017 |
| Session | Specimen |
| Marks | 11 |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Standard +0.3 This is a standard FP1 vectors question testing perpendicularity via dot products and line intersection. Part (i) is routine (dot product = 0), part (ii) is a standard textbook exercise (equate components, solve simultaneous equations), and part (iii) requires setting up two dot product equations and solving. While it's Further Maths content, these are foundational vector techniques with no novel insight required, making it slightly easier than average overall. |
| 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 |
\begin{enumerate}[label=(\roman*)]
\item Find the value of $k$ such that $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}$ are perpendicular. [2]
\item Two lines have equations $l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}$ and $l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$.
Find the point of intersection of $l_1$ and $l_2$. [4]
\item The vector $\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}$ is perpendicular to the lines $l_1$ and $l_2$.
Find the values of $a$ and $b$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2017 Q9 [11]}}