| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving linear systems using matrices |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard matrix operations (multiplication, finding inverses, solving linear systems). Part (a) requires routine matrix multiplication with a parameter, parts (b)(i-ii) involve recognizing that MN = λI implies N = λM^(-1), part (c) is straightforward matrix equation solving using the inverse found earlier, and part (d) asks for basic geometric interpretation of three planes intersecting at a point. All techniques are standard Core Pure AS content with clear scaffolding between parts. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix4.03t Plane intersection: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(\mathbf{MN} = \begin{pmatrix} 2k-24 & 0 & 0 \\ k^2-7k+10 & 6k-44 & -10k+50 \\ 4k-20 & 0 & -14 \end{pmatrix}\) | B1, B1 | 1.1b – B1: 2 correct rows or columns (unsimplified); B1: fully correct simplified matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(\mathbf{MN} = \begin{pmatrix} -14 & 0 & 0 \\ 0 & -14 & 0 \\ 0 & 0 & -14 \end{pmatrix}\) | B1ft | 1.1b – Follow through from part (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(\mathbf{M}^{-1} = -\frac{1}{14}\begin{pmatrix} -2 & 6 & -10 \\ 2 & -20 & 24 \\ -3 & 2 & -1 \end{pmatrix}\) | B1 | 1.1b – Correct inverse matrix, may use calculator |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(\mathbf{M}^{-1} = -\frac{1}{14}\begin{pmatrix} -2 & 6 & -10 \\ 2 & -20 & 24 \\ -3 & 2 & -1 \end{pmatrix}\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} = \ldots\) | M1 | 1.1b – Any complete method using their inverse |
| \(\left(-\frac{12}{7},\ \frac{40}{7},\ -\frac{1}{14}\right)\) | A1 | 1.1b – Correct exact coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| The coordinates of the only point at which the planes represented by the equations in (c) meet. | B1 | 2.2a – Describes correct geometrical configuration of the planes |
# Question 4:
## Part (a):
| Working | Mark | Notes |
|---------|------|-------|
| $\mathbf{MN} = \begin{pmatrix} 2k-24 & 0 & 0 \\ k^2-7k+10 & 6k-44 & -10k+50 \\ 4k-20 & 0 & -14 \end{pmatrix}$ | B1, B1 | 1.1b – B1: 2 correct rows or columns (unsimplified); B1: fully correct simplified matrix |
## Part (b)(i):
| Working | Mark | Notes |
|---------|------|-------|
| $\mathbf{MN} = \begin{pmatrix} -14 & 0 & 0 \\ 0 & -14 & 0 \\ 0 & 0 & -14 \end{pmatrix}$ | B1ft | 1.1b – Follow through from part (a) |
## Part (b)(ii):
| Working | Mark | Notes |
|---------|------|-------|
| $\mathbf{M}^{-1} = -\frac{1}{14}\begin{pmatrix} -2 & 6 & -10 \\ 2 & -20 & 24 \\ -3 & 2 & -1 \end{pmatrix}$ | B1 | 1.1b – Correct inverse matrix, may use calculator |
## Part (c):
| Working | Mark | Notes |
|---------|------|-------|
| $\mathbf{M}^{-1} = -\frac{1}{14}\begin{pmatrix} -2 & 6 & -10 \\ 2 & -20 & 24 \\ -3 & 2 & -1 \end{pmatrix}\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} = \ldots$ | M1 | 1.1b – Any complete method using their inverse |
| $\left(-\frac{12}{7},\ \frac{40}{7},\ -\frac{1}{14}\right)$ | A1 | 1.1b – Correct exact coordinates |
## Part (d):
| Working | Mark | Notes |
|---------|------|-------|
| The coordinates of the only point at which the **planes** represented by the equations in (c) meet. | B1 | 2.2a – Describes correct geometrical configuration of the planes |
**Total: 7 marks**
---4.
$$\mathbf { M } = \left( \begin{array} { r r r }
2 & 1 & 4 \\
k & 2 & - 2 \\
4 & 1 & - 2
\end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r r }
k - 7 & 6 & - 10 \\
2 & - 20 & 24 \\
- 3 & 2 & - 1
\end{array} \right)$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Determine, in simplest form in terms of $k$, the matrix $\mathbf { M N }$.
\item Given that $k = 5$
\begin{enumerate}[label=(\roman*)]
\item write down $\mathbf { M N }$
\item hence write down $\mathbf { M } ^ { - 1 }$
\end{enumerate}\item Solve the simultaneous equations
$$\begin{aligned}
& 2 x + y + 4 z = 2 \\
& 5 x + 2 y - 2 z = 3 \\
& 4 x + y - 2 z = - 1
\end{aligned}$$
\item Interpret the answer to part (c) geometrically.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2021 Q4 [7]}}