Question 3 - A-Level Maths

OCR FP1 2014 June — Question 3 7 marks

Exam Board	OCR
Module	FP1 (Further Pure Mathematics 1)
Year	2014
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Matrices
Type	Matrix arithmetic operations
Difficulty	Moderate -0.8 This is a straightforward matrix arithmetic question testing basic operations: scalar multiplication, addition, finding a 2×2 inverse using the standard formula, and applying inverse properties. All parts are routine calculations with no problem-solving or conceptual challenge, making it easier than average but not trivial since it's Further Maths content requiring multiple accurate computations.
Spec	4.03b Matrix operations: addition, multiplication, scalar 4.03n Inverse 2x2 matrix 4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1)

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } 2 & 1 \\ - 4 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find

\(4 \mathbf { A } - \mathbf { B } + 2 \mathbf { I }\),
\(\mathrm { A } ^ { - 1 }\),
\(\left( \mathbf { A B } ^ { - 1 } \right) ^ { - 1 }\).

Show mark scheme Show mark scheme source

Question 3(i):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\frac{1}{17}(25+19i) \cdot \mathbf{3}\) — 2 elements correct	B1	Condone missing brackets in (i), (ii) & (iii)
\(\begin{pmatrix} 7 & 3 \\ -18 & 19 \end{pmatrix}\) — all elements correct	B1
[2]

Question 3(ii):

Answer	Marks	Guidance
Answer	Marks	Guidance
Both diagonals correct, ignore determinant	B1	\(\frac{\begin{pmatrix}5 & -1\\4 & 2\end{pmatrix}}{14}\) is OK for 2nd B1
\(\frac{1}{14}\begin{pmatrix} 5 & -1 \\ 4 & 2 \end{pmatrix}\) or equivalent	B1
[2]

Question 3(iii):

Answer	Marks	Guidance
Answer	Marks	Guidance
\((AB^{-1})^{-1} = BA^{-1}\) or \(B^{-2} = \frac{1}{7}\begin{pmatrix}3 & -1\\-2 & 3\end{pmatrix}\) — correct result seen or used	B1
Multiplication attempt for any pair of \(2\times 2\) matrices, 2 elements correct, but not \(\mathbf{I}\)	M1
\(\frac{1}{14}\begin{pmatrix}19 & -1\\22 & 4\end{pmatrix}\) — correct answer a.e.f.	A1
[3]

# Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{17}(25+19i) \cdot \mathbf{3}$ — 2 elements correct | B1 | Condone missing brackets in (i), (ii) & (iii) |
| $\begin{pmatrix} 7 & 3 \\ -18 & 19 \end{pmatrix}$ — all elements correct | B1 | |
| **[2]** | | |

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# Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Both diagonals correct, ignore determinant | B1 | $\frac{\begin{pmatrix}5 & -1\\4 & 2\end{pmatrix}}{14}$ is OK for 2nd B1 |
| $\frac{1}{14}\begin{pmatrix} 5 & -1 \\ 4 & 2 \end{pmatrix}$ or equivalent | B1 | |
| **[2]** | | |

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# Question 3(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(AB^{-1})^{-1} = BA^{-1}$ or $B^{-2} = \frac{1}{7}\begin{pmatrix}3 & -1\\-2 & 3\end{pmatrix}$ — correct result seen or used | B1 | |
| Multiplication attempt for any pair of $2\times 2$ matrices, 2 elements correct, but not $\mathbf{I}$ | M1 | |
| $\frac{1}{14}\begin{pmatrix}19 & -1\\22 & 4\end{pmatrix}$ — correct answer a.e.f. | A1 | |
| **[3]** | | |

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Show LaTeX source

3 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r l } 2 & 1 \\ - 4 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 3 \end{array} \right)$ and $\mathbf { I }$ is the $2 \times 2$ identity matrix. Find\\
(i) $4 \mathbf { A } - \mathbf { B } + 2 \mathbf { I }$,\\
(ii) $\mathrm { A } ^ { - 1 }$,\\
(iii) $\left( \mathbf { A B } ^ { - 1 } \right) ^ { - 1 }$.

\hfill \mbox{\textit{OCR FP1 2014 Q3 [7]}}

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