| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Find inverse then solve system |
| Difficulty | Standard +0.3 This is a straightforward Core Pure AS question testing standard matrix inverse and system-solving techniques. Finding a 3×3 inverse using cofactors/adjugate is routine for this level, and applying it to solve a system is direct substitution. The geometric interpretation adds minimal difficulty. Slightly easier than average A-level due to being a textbook-standard procedure with no problem-solving insight required. |
| Spec | 4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix4.03t Plane intersection: geometric interpretation |
| V349 SIHI NI IMIMM ION OC | VJYV SIHIL NI LIIIM ION OO | VJYV SIHIL NI JIIYM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{M}^{-1} = \frac{1}{69}\begin{pmatrix} 1 & 13 & 5 \\ -11 & -5 & 14 \\ -26 & 7 & 8 \end{pmatrix}\) | B1 | Evidence that determinant is \(\pm 69\) (may be implied by matrix entries) |
| Fully correct inverse with all elements in exact form | B1 | Must be seen in part (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{69}\begin{pmatrix} 1 & 13 & 5 \\ -11 & -5 & 14 \\ -26 & 7 & 8 \end{pmatrix}\begin{pmatrix} -4 \\ 9 \\ 5 \end{pmatrix} = \ldots\) | M1 | Any complete method using their inverse to find \(x, y, z\) |
| \(x=2, y=1, z=3\) or \((2,1,3)\) or \(2\mathbf{i}+\mathbf{j}+3\mathbf{k}\) | A1 | Correct coordinates only scores both marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The point where three planes meet | B1ft | Must include both ideas of planes and meet in a point; dependent on unique point in (b) |
# Question 1:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{M}^{-1} = \frac{1}{69}\begin{pmatrix} 1 & 13 & 5 \\ -11 & -5 & 14 \\ -26 & 7 & 8 \end{pmatrix}$ | B1 | Evidence that determinant is $\pm 69$ (may be implied by matrix entries) |
| Fully correct inverse with all elements in **exact** form | B1 | Must be seen in part (a) |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{69}\begin{pmatrix} 1 & 13 & 5 \\ -11 & -5 & 14 \\ -26 & 7 & 8 \end{pmatrix}\begin{pmatrix} -4 \\ 9 \\ 5 \end{pmatrix} = \ldots$ | M1 | Any complete method using **their** inverse to find $x, y, z$ |
| $x=2, y=1, z=3$ or $(2,1,3)$ or $2\mathbf{i}+\mathbf{j}+3\mathbf{k}$ | A1 | Correct coordinates only scores both marks |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| The **point** where three **planes** meet | B1ft | Must include both ideas of **planes** and **meet in a point**; dependent on unique point in (b) |
---1.
$$\mathbf { M } = \left( \begin{array} { r r r }
2 & 1 & - 3 \\
4 & - 2 & 1 \\
3 & 5 & - 2
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathbf { M } ^ { - 1 }$ giving each element in exact form.
\item Solve the simultaneous equations
$$\begin{array} { r }
2 x + y - 3 z = - 4 \\
4 x - 2 y + z = 9 \\
3 x + 5 y - 2 z = 5
\end{array}$$
\item Interpret the answer to part (b) geometrically.
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V349 SIHI NI IMIMM ION OC & VJYV SIHIL NI LIIIM ION OO & VJYV SIHIL NI JIIYM ION OC \\
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\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2018 Q1 [5]}}