| Exam Board | SPS |
|---|---|
| Module | SPS ASFM Statistics (SPS ASFM Statistics) |
| Year | 2021 |
| Session | May |
| Marks | 11 |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Standard +0.3 This is a standard A-level Further Maths vectors question testing routine techniques: dot product for perpendicularity (straightforward recall), solving simultaneous equations for line intersection (mechanical but requires care with 3 equations), and finding a vector perpendicular to two given vectors using dot products (standard procedure). All parts are textbook exercises with no novel insight required, making it slightly easier than average. |
| 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]
\end{enumerate}
Two lines have equations $l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}$ and $l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item 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{SPS SPS ASFM Statistics 2021 Q2 [11]}}