| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Challenging +1.8 This is a Further Maths question requiring systematic analysis of plane configurations using determinants and rank conditions. Part (a) requires finding when the coefficient matrix is singular (det=0), which is computational but straightforward. Part (b) demands deeper understanding of consistency conditions and geometric interpretation of different configurations (parallel planes, sheaf, line of intersection), requiring multiple augmented matrix calculations and conceptual reasoning about solution spaces. |
| Spec | 4.03s Consistent/inconsistent: systems of equations4.03t Plane intersection: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| 12(a) | Recognises the need to set | |
| the determinant = 0 | AO3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| k | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| k | AO1.1b | A1 |
| Answer | Marks |
|---|---|
| 12(b) | Selects an appropriate |
| Answer | Marks | Guidance |
|---|---|---|
| first value of k | AO3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| comment. | AO2.2a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| description with full working. | AO3.2a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| comment. | AO2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| description. | AO3.2a | B1 |
| Total | 8 | |
| Q | Marking Instructions | AO |
Question 12:
--- 12(a) ---
12(a) | Recognises the need to set
the determinant = 0 | AO3.1a | M1 | 9k2 −9k−180=0
k =5 and k = −4
Obtains and solves a three-
term quadratic equation in
k | AO1.1a | M1
Obtains the correct values of
k | AO1.1b | A1
--- 12(b) ---
12(b) | Selects an appropriate
method and substitutes their
first value of k | AO3.1a | M1 | For
k =5
4 −5 1 8
0 23 −23 0
0 35 −35 0
Consistent
Line of intersection (sheaf)
For
k =−4
3x+2y+4z =6
−6x−4y−8z =6
Inconsistent
Two planes parallel and distinct
with third plane crossing both
For k =5 (k must be
correct):
Deduces that equations are
consistent – must have
sufficient working to justify
comment. | AO2.2a | M1
Gives correct geometrical
description with full working. | AO3.2a | A1
For k = −4 (k must be
correct):
Deduces that equations are
inconsistent by comparing
eqs 2 & 3 – must have
comment. | AO2.2a | B1
Gives correct geometrical
description. | AO3.2a | B1
Total | 8
Q | Marking Instructions | AO | Marks | Typical SolutionThree planes have equations
\begin{align}
4x - 5y + z &= 8\\
3x + 2y - kz &= 6\\
(k - 2)x + ky - 8z &= 6
\end{align}
where $k$ is a real constant.
The planes do not meet at a unique point.
\begin{enumerate}[label=(\alph*)]
\item Find the possible values of $k$.
[3 marks]
\item For each value of $k$ found in part (a), identify the configuration of the given planes.
Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system.
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 2019 Q12 [8]}}