| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Matrix conformability and dimensions |
| Difficulty | Easy -1.3 This is a straightforward question testing basic matrix conformability rules and matrix multiplication. Students only need to check dimensions (2×3 and 1×2 matrices), identify which product is valid, then perform routine matrix multiplication. This requires minimal problem-solving beyond recalling standard procedures, making it easier than average. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| B | B1 | Note "B" is that QP is conformable |
| (For matrices to be conformable for multiplication) the number of columns of the first must equal the number of rows of the second, or "the number of rows of P is equal to the number of columns of Q" | E1 [2] | Statement can be general or specific. Allow e.g. \((1\times2)\times(2\times3)=(1\times3)\) provided it is clear which two numbers must be the same; since told exactly one is true it is sufficient to give a reason why one is true or why one is false |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{QP} = ((1+k) \quad -1)\begin{pmatrix}1 & k & 0 \\ -2 & 1 & 3\end{pmatrix}\) | M1 | Correct method for multiplying matrices (can be implied by any one entry correct); if PQ attempted then M0A0 unless explicitly rejected |
| \(= ((1+k)+2 \quad k(1+k)-1 \quad -3)\) | ||
| \(\left((k+3) \quad (k^2+k-1) \quad -3\right)\) | A1 [2] | Accept un-simplified elements |
**Question 2:**
**(a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| B | B1 | Note "B" is that QP is conformable |
| (For matrices to be conformable for multiplication) the number of columns of the first must equal the number of rows of the second, or "the number of rows of P is equal to the number of columns of Q" | E1 [2] | Statement can be general or specific. Allow e.g. $(1\times2)\times(2\times3)=(1\times3)$ provided it is clear which two numbers must be the same; since told exactly one is true it is sufficient to give a reason why one is true or why one is false |
**(b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{QP} = ((1+k) \quad -1)\begin{pmatrix}1 & k & 0 \\ -2 & 1 & 3\end{pmatrix}$ | M1 | Correct method for multiplying matrices (can be implied by any one entry correct); if PQ attempted then M0A0 unless explicitly rejected |
| $= ((1+k)+2 \quad k(1+k)-1 \quad -3)$ | | |
| $\left((k+3) \quad (k^2+k-1) \quad -3\right)$ | A1 [2] | Accept un-simplified elements |2 Matrices $\mathbf { P }$ and $\mathbf { Q }$ are given by $\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)$ and $\mathbf { Q } = ( ( 1 + k ) - 1 )$ where $k$ is a constant.\\
Exactly one of statements A and B is true.\\
Statement A: $\quad \mathbf { P }$ and $\mathbf { Q }$ (in that order) are conformable for multiplication.\\
Statement B: $\quad \mathbf { Q }$ and $\mathbf { P }$ (in that order) are conformable for multiplication.
\begin{enumerate}[label=(\alph*)]
\item State, with a reason, which one of A and B is true.
\item Find either $\mathbf { P Q }$ or $\mathbf { Q P }$ in terms of $k$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS 2019 Q2 [4]}}