| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring determinant calculation using standard methods (cofactor expansion or row operations), recognizing that singular matrices correspond to non-unique solutions, and solving a 3×3 system. All techniques are routine for Further Maths students with no novel insight required, making it slightly easier than average. |
| Spec | 4.03j Determinant 3x3: calculation4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
3 (i) Evaluate, in terms of $k$, the determinant of the matrix $\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)$.
Three planes have equations $x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21$ and $6 x + 12 y + k z = 31$.\\
(ii) State the value of $k$ for which these three planes do not meet at a single point.\\
(iii) Find the coordinates of the point of intersection of the three planes when $k = 7$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q3}}