| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving linear systems using matrices |
| Difficulty | Moderate -0.8 This is a routine Further Maths question testing basic matrix operations (determinant, inverse, solving systems, matrix multiplication) with standard procedures throughout. While it's Further Maths content, all parts are direct applications of formulas with no problem-solving or insight required, making it easier than average even for A-level. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03h Determinant 2x2: calculation4.03n Inverse 2x2 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | DetA =3×2−−2×1=8 |
| [1] | 1.1 | |
| (b) | 1 3 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 8−1 2 | B1 | |
| [1] | 1.1 | ft their Det A |
| (c) | x −1 |
| Answer | Marks |
|---|---|
| 8 8 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | −1 |
| Answer | Marks |
|---|---|
| (d) | 1 3 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 2−1 2 | B1 | |
| [1] | 2.2a | 𝐀𝐀 |
| Answer | Marks |
|---|---|
| (e) | ( ) |
| Answer | Marks |
|---|---|
| 0 2 p | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Must be a matrix; do not award for just p. |
| Answer | Marks | Guidance |
|---|---|---|
| (f) | Commutativity | B1 |
| [1] | 1.2 | Or the “commutative property”. Accept “commutative”. |
Question 2:
2 | (a) | DetA =3×2−−2×1=8 | B1
[1] | 1.1
(b) | 1 3 2
8−1 2 | B1
[1] | 1.1 | ft their Det A
(c) | x −1
=A−1
y 2
1 5
⇒ x= , y =
8 8 | M1
A1
[2] | 1.1
1.1 | −1
Sight of their A-1 multiplied by
2
ft their . Could be given as a vector.
−1
(d) | 1 3 2
oe
2−1 2 | B1
[1] | 2.2a | 𝐀𝐀
ft their . Accept 4 .
−1 −1
𝐀𝐀 𝐀𝐀
(e) | ( )
DC = p
0 4 2p
CD = 0 0 0
0 2 p | B1
M1
A1
[3] | 1.1
1.1
1.1 | Must be a matrix; do not award for just p.
for 3 × 3 matrix with at least one correct row or (non-0)
column.
SC B2 for correct matrices, but CD and DC interchanged or
unspecified
SC B1 for one correct matrix.
(f) | Commutativity | B1
[1] | 1.2 | Or the “commutative property”. Accept “commutative”.
Allow also “non commutative”2 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ 1 & 3 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Calculate $\operatorname { det } \mathbf { A }$.
\item Write down $\mathbf { A } ^ { - 1 }$.
\item Hence solve the equation $\mathbf { A } \binom { \mathrm { x } } { \mathrm { y } } = \binom { - 1 } { 2 }$.
\item Write down the matrix $\mathbf { B }$ such that $\mathbf { A B } = 4 \mathbf { I }$.
Matrices $\mathbf { C }$ and $\mathbf { D }$ are given by $\mathbf { C } = \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)$ and $\mathbf { D } = \left( \begin{array} { l l l } 0 & 2 & p \end{array} \right)$ where $p$ is a constant.
\item Find, in terms of $p$,
\begin{itemize}
\item the matrix CD
\item the matrix DC.
\end{itemize}
It is observed that $\mathbf { C D } \neq \mathbf { D C }$.
\item The result that $\mathbf { C D } \neq \mathbf { D C }$ is a counter example to the claim that matrix multiplication has a particular property. Name this property.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2022 Q2 [9]}}