| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | January |
| Marks | 13 |
| Topic | Vectors 3D & Lines |
| Type | Foot of perpendicular from origin to line |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question with routine techniques: part (a) requires substituting point B into the line equation to find constants; part (b) uses the perpendicularity condition (dot product = 0) to find a specific point on the line; part (c) applies the cross product formula for triangle area. All methods are textbook exercises requiring careful calculation but no novel insight or complex problem-solving. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
The points $A$ and $B$ have position vectors $5\mathbf{j} + 11\mathbf{k}$ and $c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}$ respectively, where $c$ and $d$ are constants.
The line $l$, through the points $A$ and $B$, has vector equation $\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})$, where $\lambda$ is a parameter.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $c$ and the value of $d$. [3]
\end{enumerate}
The point $P$ lies on the line $l$, and $\overrightarrow{OP}$ is perpendicular to $l$, where $O$ is the origin.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the position vector of $P$. [6]
\item Find the area of triangle $OAB$, giving your answer to 3 significant figures. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q4 [13]}}