| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Transformation mapping problems |
| Difficulty | Standard +0.8 This is a Further Maths question involving 3×3 matrix inversion (requiring cofactor method with working shown), understanding of transformation mappings between planes, and substitution to find the image plane equation. While systematic, it requires multiple techniques (matrix inversion, coordinate transformation, algebraic substitution) and conceptual understanding of how linear transformations map planes. The multi-step nature and Further Maths context place it moderately above average difficulty. |
| Spec | 4.03o Inverse 3x3 matrix4.04b Plane equations: cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| \(\det\begin{pmatrix}2&0&0\\0&1&4\\3&-2&-3\end{pmatrix} = 2\times(-3+8) = 10\) | B1 | Correct value for determinant, seen or stated and not just in a final answer |
| Minors matrix with at least 6 correct elements leading to cofactors matrix | M1 | Attempts cofactor matrix with at least 6 correct elements |
| \(\frac{1}{10}\begin{pmatrix}5&0&0\\12&-6&-8\\-3&4&2\end{pmatrix}\) or equivalent fractions/decimals | A1ft | Correct inverse but allow ft on their "10"; A0 if clearly obtained incorrectly |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{10}\begin{pmatrix}5&0&0\\12&-6&-8\\-3&4&2\end{pmatrix}\begin{pmatrix}u\\v\\w\end{pmatrix}=\ldots\) | M1 | Multiplies their \(\mathbf{M}^{-1}\) by \(\begin{pmatrix}u\\v\\w\end{pmatrix}\); must use matrix other than \(\mathbf{M}\); condone \(\mathbf{vM}^{-1}=\ldots\) but must not be clearly incorrect multiplication |
| \(\begin{pmatrix}x\\y\\z\end{pmatrix}=\frac{1}{10}\begin{pmatrix}5u\\12u-6v-8w\\-3u+4v+2w\end{pmatrix}\) | A1ft | Two correct vector components, ft their \(d\neq 0\) |
| Full solution with all three correct, ft their non-zero \(d\neq 0\) | A1ft | Must be exact; ft marks not available for incorrect Adj(M) |
| Answer | Marks | Guidance |
|---|---|---|
| \(3x-7y+2z=-3 \Rightarrow 3\left(\frac{1}{2}u\right)-7\left(\frac{6}{5}u-\frac{3}{5}v-\frac{4}{5}w\right)+2\left(-\frac{3}{10}u+\frac{2}{5}v+\frac{1}{5}w\right)=-3\) | M1 | Substitutes their expressions into the equation for \(\Pi_1\) |
| \(-15u+10v+12w=-6\) | A1 | Correct equation, terms in any order but constant isolated; accept any integer multiples |
# Question 2(a):
$\det\begin{pmatrix}2&0&0\\0&1&4\\3&-2&-3\end{pmatrix} = 2\times(-3+8) = 10$ | B1 | Correct value for determinant, seen or stated and not just in a final answer
Minors matrix with at least 6 correct elements leading to cofactors matrix | M1 | Attempts cofactor matrix with at least 6 correct elements
$\frac{1}{10}\begin{pmatrix}5&0&0\\12&-6&-8\\-3&4&2\end{pmatrix}$ or equivalent fractions/decimals | A1ft | Correct inverse but allow ft on their "10"; A0 if clearly obtained incorrectly
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# Question 2(b):
$\frac{1}{10}\begin{pmatrix}5&0&0\\12&-6&-8\\-3&4&2\end{pmatrix}\begin{pmatrix}u\\v\\w\end{pmatrix}=\ldots$ | M1 | Multiplies their $\mathbf{M}^{-1}$ by $\begin{pmatrix}u\\v\\w\end{pmatrix}$; must use matrix other than $\mathbf{M}$; condone $\mathbf{vM}^{-1}=\ldots$ but must not be clearly incorrect multiplication
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\frac{1}{10}\begin{pmatrix}5u\\12u-6v-8w\\-3u+4v+2w\end{pmatrix}$ | A1ft | Two correct vector components, ft their $d\neq 0$
Full solution with all three correct, ft their non-zero $d\neq 0$ | A1ft | Must be exact; ft marks not available for incorrect Adj(**M**)
---
# Question 2(c):
$3x-7y+2z=-3 \Rightarrow 3\left(\frac{1}{2}u\right)-7\left(\frac{6}{5}u-\frac{3}{5}v-\frac{4}{5}w\right)+2\left(-\frac{3}{10}u+\frac{2}{5}v+\frac{1}{5}w\right)=-3$ | M1 | Substitutes their expressions into the equation for $\Pi_1$
$-15u+10v+12w=-6$ | A1 | Correct equation, terms in any order but constant isolated; accept any integer multiples
---\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.
$$\mathbf { M } = \left( \begin{array} { r r r }
2 & 0 & 0 \\
0 & 1 & 4 \\
3 & - 2 & - 3
\end{array} \right)$$
(a) Determine $\mathbf { M } ^ { - 1 }$
The transformation represented by $\mathbf { M }$ maps the plane $\Pi _ { 1 }$ to the plane $\Pi _ { 2 }$ The point $( x , y , z )$ on $\Pi _ { 1 }$ maps to the point $( u , v , w )$ on $\Pi _ { 2 }$\\
(b) Determine $x , y$ and $z$ in terms of $u , v$ and $w$ as appropriate.
The plane $\Pi _ { 1 }$ has equation
$$3 x - 7 y + 2 z = - 3$$
(c) Find a Cartesian equation for $\Pi _ { 2 }$
Give your answer in the form $a u + b v + c w = d$ where $a , b , c$ and $d$ are integers to be determined.
\hfill \mbox{\textit{Edexcel F3 2023 Q2 [8]}}