| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 7 |
| Topic | Vectors 3D & Lines |
| Type | Line-plane intersection and related angle/perpendicularity |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring understanding of when three planes don't meet at a unique point (determinant = 0), solving for parameter values, then analyzing two different geometric configurations (likely parallel planes vs. sheaf). It demands matrix/determinant work, algebraic manipulation, and geometric interpretation—significantly above average difficulty but standard for FM Pure. |
| Spec | 4.03s Consistent/inconsistent: systems of equations4.03t Plane intersection: geometric interpretation |
Three planes have equations
\begin{align}
(4k + 1)x - 3y + (k - 5)z &= 3 \\
(k - 1)x + (3 - k)y + 2z &= 1 \\
7x - 3y + 4z &= 2
\end{align}
\begin{enumerate}[label=(\alph*)]
\item The planes do not meet at a unique point.
Show that $k = 4.5$ is one possible value of $k$, and find the other possible value of $k$. [3 marks]
\item For each value of $k$ found in part (a), identify the configuration of the given planes.
In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q11 [7]}}