| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 10 |
| Topic | Vectors: Lines & Planes |
| Type | Angle between two lines |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question testing conversion between Cartesian and vector forms of lines, perpendicularity conditions, and angle between lines. Part (a) is routine manipulation, part (b)(i) requires checking dot product ≠ 0, part (b)(ii) needs recognizing direction vectors can't be scalar multiples, and part (b)(iii) applies the standard cosine formula. All techniques 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 angles |
Line $l_1$ has Cartesian equation
$$x - 3 = \frac{2y + 2}{3} = 2 - z$$
\begin{enumerate}[label=(\alph*)]
\item Write the equation of line $l_1$ in the form
$$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$
where $\lambda$ is a parameter and $\mathbf{a}$ and $\mathbf{b}$ are vectors to be found. [2 marks]
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Line $l_2$ passes through the points $P(3, 2, 0)$ and $Q(n, 5, n)$, where $n$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that the lines $l_1$ and $l_2$ are not perpendicular. [3 marks]
\item Explain briefly why lines $l_1$ and $l_2$ cannot be parallel. [2 marks]
\item Given that $\theta$ is the acute angle between lines $l_1$ and $l_2$, show that
$$\cos \theta = \frac{p}{\sqrt{34n^2 + qn + 306}}$$
where $p$ and $q$ are constants to be found. [3 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q7 [10]}}