| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Standard +0.8 This question requires understanding of determinants to determine solution existence, geometric interpretation of singular systems (sheaf of planes through origin for k=0), and recognizing a prism configuration for k=1. It combines matrix theory with 3D geometry requiring conceptual understanding beyond routine calculation, placing it moderately above average difficulty. |
| Spec | 4.03j Determinant 3x3: calculation4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | 1 2 − 3 |
| Answer | Marks |
|---|---|
| on all three planes | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 2.4 | det = −5 |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (b) | (i) |
| Answer | Marks |
|---|---|
| Inconsistent equations, so planes have no common point | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 2.2a | Can eliminate any of x, y, z |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (b) | (ii) |
| [1] | 1.2 | accept ‘prism’, ‘triangular prism’. Accept a diagram |
Question 7:
7 | (a) | 1 2 − 3
− 1 3 − 2 = − 5
1 − 2 0
Non-zero value means there are no other points that lie
on all three planes | B1
B1
[2] | 1.1
2.4 | det = −5
correct conclusion with reference to non-zero determinant
7 | (b) | (i) | Eliminate one variable, e.g. x
5y – 5z = 0, y – z = 1
Inconsistent equations, so planes have no common point | M1
A1
A1
[3] | 3.1a
1.1
2.2a | Can eliminate any of x, y, z
oe
from correct working
7 | (b) | (ii) | The planes form a prismatic intersection | B1
[1] | 1.2 | accept ‘prism’, ‘triangular prism’. Accept a diagram7 Three planes have equations
$$\begin{array} { r }
x + 2 y - 3 z = 0 \\
- x + 3 y - 2 z = 0 \\
x - 2 y + k z = k
\end{array}$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item For the case $k = 0$, the origin lies on all three planes.
Use a determinant to explain whether there are any other points that lie on all three planes in this case.
\item You are now given that $k = 1$.
\begin{enumerate}[label=(\roman*)]
\item Show that there are no points that lie on all three planes.
\item Describe the geometrical arrangement of the three planes.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2024 Q7 [6]}}