Standard +0.8 This is a Further Maths P4 question requiring students to prove lines are skew for all values except one specific case. It demands understanding of the conditions for skew lines (not parallel and not intersecting), setting up a system of three equations in two parameters, and showing the system is inconsistent. The algebraic manipulation and logical structure of the proof elevate this above routine vector line problems.
8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }
- 1 \\
5 \\
4
\end{array} \right) + \lambda \left( \begin{array} { r }
2 \\
- 1 \\
5
\end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }
2 \\
- 2 \\
- 5
\end{array} \right) + \mu \left( \begin{array} { r }
4 \\
- 3 \\
b
\end{array} \right)$$
where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Prove that for all values of \(b \neq 7\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
8. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }
- 1 \\
5 \\
4
\end{array} \right) + \lambda \left( \begin{array} { r }
2 \\
- 1 \\
5
\end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }
2 \\
- 2 \\
- 5
\end{array} \right) + \mu \left( \begin{array} { r }
4 \\
- 3 \\
b
\end{array} \right)$$
where $\lambda$ and $\mu$ are scalar parameters and $b$ is a constant.\\
Prove that for all values of $b \neq 7$, the lines $l _ { 1 }$ and $l _ { 2 }$ are skew.\\
\hfill \mbox{\textit{Edexcel P4 2021 Q8 [6]}}