| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Standard +0.3 This is a standard Further Maths question on geometric interpretation of 3×3 systems. Students must form augmented matrices, row reduce, and interpret the results (unique solution vs special cases). While it requires systematic work and understanding of rank/consistency, the techniques are routine for Further Maths students with no novel insight needed. Slightly easier than average due to straightforward arithmetic and clear structure. |
| Spec | 4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | when k = −1, det M ≠ 0 [so meet at a point] |
| Answer | Marks |
|---|---|
| by calculator, point of intersection is (−1, 3, 2) | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | calculating determinant |
| Answer | Marks |
|---|---|
| 2 | or finding M−1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (b) | when k = 2/3, det M = 0, |
| Answer | Marks |
|---|---|
| and intersected by third plane | M1 |
| Answer | Marks |
|---|---|
| [4] | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | or 2 planes are parallel |
Question 9:
9 | (a) | when k = −1, det M ≠ 0 [so meet at a point]
x 0
y =M−1 −5
z 1
by calculator, point of intersection is (−1, 3, 2) | M1
M1
A1
[3] | 1.1
1.1
1.1 | calculating determinant
solving (soi)
−1
BC, allow 3
2 | or finding M−1
or by sim equations
If correct ans found by
solving simultaneously
SCB3
B1
M1
A1
M1
A1
M1
A1
Attempt to multiply factors
x2 – 4x + 5
Attempt to find other quad
factor
2x2 − 12x + 50
condone sign errors
or long division
9 | (b) | when k = 2/3, det M = 0,
so no unique point of intersection
[coeffts of 2nd plane are 3 times those of first]
so first two planes are parallel
and intersected by third plane | M1
A1
B1
B1
[4] | 3.1a
1.1
3.2a
3.2a | or 2 planes are parallel9 Three planes have equations\\
$k x + y - 2 z = 0$\\
$2 x + 3 y - 6 z = - 5$\\
$3 x - 2 y + 5 z = 1$\\
where $k$ is a constant.
Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.
\begin{enumerate}[label=(\alph*)]
\item $k = - 1$
\item $k = \frac { 2 } { 3 }$
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2020 Q9 [7]}}