| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Topic | 3x3 Matrices |
| Type | Volume/area scale factors |
| Difficulty | Standard +0.3 This is a straightforward application of the determinant-volume relationship for 3×3 matrices. Part (a) requires computing det(A) when x=5 and multiplying by the original volume (standard procedure), plus checking the sign for orientation. Part (b) requires setting det(A)=0 and solving a cubic equation. While it involves a 3×3 determinant with a parameter, the techniques are routine for FP1 students and require no novel insight—just careful algebraic manipulation. |
| Spec | 4.03j Determinant 3x3: calculation4.03k Determinant 3x3: volume scale factor4.03l Singular/non-singular matrices |
4 A transformation A is represented by the matrix $\mathbf { A }$ where $\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)$.\\
The tetrahedron $H$ has vertices at $O , P , Q$ and $R$. The volume of $H$ is 6 units. $P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }$ and $H ^ { \prime }$ are the images of $P , Q , R$ and $H$ under A .
\begin{enumerate}[label=(\alph*)]
\item In the case where $x = 5$
\begin{itemize}
\item find the volume of $H ^ { \prime }$,
\item determine whether A preserves the orientation of $H$.
\item Find the values of $x$ for which $O , P ^ { \prime } , Q ^ { \prime }$ and $R ^ { \prime }$ are coplanar (i.e. the four points lie in the same plane).
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2021 Q4 [7]}}