Question 6 - A-Level Maths

AQA Further AS Paper 1 2022 June — Question 6 5 marks

Exam Board	AQA
Module	Further AS Paper 1 (Further AS Paper 1)
Year	2022
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Matrices
Type	Matrix multiplication
Difficulty	Moderate -0.3 This is a straightforward Further Maths matrices question testing standard procedures: determinant calculation (1 step), matrix inversion using the formula (routine), and solving for B using A^{-1} then computing M (2-3 steps). All techniques are direct applications of learned methods with no problem-solving insight required. Slightly easier than average A-level due to its purely procedural nature, though the multi-part structure and Further Maths context keep it close to baseline.
Spec	4.03a Matrix language: terminology and notation 4.03h Determinant 2x2: calculation 4.03n Inverse 2x2 matrix

6 The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left[ \begin{array} { c c } 5 & 2 \\ - 3 & 4 \end{array} \right]$$ 6

$\quad$ Find $\operatorname { det } \mathbf { A }$ 6
Find $\mathbf { A } ^ { - 1 }$ 6
Given that $\mathbf { A B } = \left[ \begin{array} { c c } 9 & 6 \\ 5 & 12 \end{array} \right]$ and $\mathbf { M } = 2 \mathbf { A } + \mathbf { B }$ find the matrix $\mathbf { M }$

Show mark scheme Show mark scheme source

Question 6(a):

Answer	Marks	Guidance
$\det \mathbf{A} = 5\times4 - (-3)\times2 = 26$	B1	Obtains the correct determinant

Question 6(b):

Answer	Marks	Guidance
$\mathbf{A}^{-1} = \frac{1}{26}\begin{bmatrix}4 & -2\\3 & 5\end{bmatrix}$	B1F	Obtains the correct inverse matrix; FT their determinant

Question 6(c):

Answer	Marks	Guidance
$\mathbf{B} = \mathbf{A}^{-1}\mathbf{AB}$; selects method to find $\mathbf{B}$ or $\mathbf{AM}$, e.g. calculates $\mathbf{A}^{-1}\mathbf{AB}$, or calculates $2\mathbf{A}^2+\mathbf{AB}$, or writes four simultaneous equations	M1	Selects a valid method; 3.1a
$\mathbf{B} = \frac{1}{26}\begin{bmatrix}4 & -2\\3 & 5\end{bmatrix}\begin{bmatrix}9 & 6\\5 & 12\end{bmatrix} = \frac{1}{26}\begin{bmatrix}26 & 0\\52 & 78\end{bmatrix} = \begin{bmatrix}1 & 0\\2 & 3\end{bmatrix}$	A1F	Obtains correct matrix for $\mathbf{B}$ or $\mathbf{AM}$; PI by correct $\mathbf{M}$; FT their $\mathbf{A}^{-1}$
$\mathbf{M} = 2\begin{bmatrix}5 & 2\\-3 & 4\end{bmatrix} + \begin{bmatrix}1 & 0\\2 & 3\end{bmatrix} = \begin{bmatrix}11 & 4\\-4 & 11\end{bmatrix}$	A1F	Obtains matrix $\mathbf{M}$; FT their $\mathbf{A}^{-1}$

## Question 6(a):

$\det \mathbf{A} = 5\times4 - (-3)\times2 = 26$ | B1 | Obtains the correct determinant

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## Question 6(b):

$\mathbf{A}^{-1} = \frac{1}{26}\begin{bmatrix}4 & -2\\3 & 5\end{bmatrix}$ | B1F | Obtains the correct inverse matrix; FT their determinant

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## Question 6(c):

$\mathbf{B} = \mathbf{A}^{-1}\mathbf{AB}$; selects method to find $\mathbf{B}$ or $\mathbf{AM}$, e.g. calculates $\mathbf{A}^{-1}\mathbf{AB}$, or calculates $2\mathbf{A}^2+\mathbf{AB}$, or writes four simultaneous equations | M1 | Selects a valid method; 3.1a

$\mathbf{B} = \frac{1}{26}\begin{bmatrix}4 & -2\\3 & 5\end{bmatrix}\begin{bmatrix}9 & 6\\5 & 12\end{bmatrix} = \frac{1}{26}\begin{bmatrix}26 & 0\\52 & 78\end{bmatrix} = \begin{bmatrix}1 & 0\\2 & 3\end{bmatrix}$ | A1F | Obtains correct matrix for $\mathbf{B}$ or $\mathbf{AM}$; PI by correct $\mathbf{M}$; FT their $\mathbf{A}^{-1}$

$\mathbf{M} = 2\begin{bmatrix}5 & 2\\-3 & 4\end{bmatrix} + \begin{bmatrix}1 & 0\\2 & 3\end{bmatrix} = \begin{bmatrix}11 & 4\\-4 & 11\end{bmatrix}$ | A1F | Obtains matrix $\mathbf{M}$; FT their $\mathbf{A}^{-1}$

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Show LaTeX source

6 The matrix $\mathbf { A }$ is given by

$$\mathbf { A } = \left[ \begin{array} { c c } 
5 & 2 \\
- 3 & 4
\end{array} \right]$$

6
\begin{enumerate}[label=(\alph*)]
\item $\quad$ Find $\operatorname { det } \mathbf { A }$\\

6
\item Find $\mathbf { A } ^ { - 1 }$\\

6
\item Given that $\mathbf { A B } = \left[ \begin{array} { c c } 9 & 6 \\ 5 & 12 \end{array} \right]$ and $\mathbf { M } = 2 \mathbf { A } + \mathbf { B }$ find the matrix $\mathbf { M }$
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 1 2022 Q6 [5]}}

This paper (15 questions)

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Q1 1 Q2 1 Q3 1 Q4 1 Q5 3 Q6 5 Q7 9 Q8 7 Q9 5 Q10 6 Q11 4 Q12 6 Q13 10 Q14 15 Q15 6

Answer	Marks	Guidance
\(\mathbf{B} = \mathbf{A}^{-1}\mathbf{AB}\); selects method to find \(\mathbf{B}\) or \(\mathbf{AM}\), e.g. calculates \(\mathbf{A}^{-1}\mathbf{AB}\), or calculates \(2\mathbf{A}^2+\mathbf{AB}\), or writes four simultaneous equations	M1	Selects a valid method; 3.1a
\(\mathbf{B} = \frac{1}{26}\begin{bmatrix}4 & -2\\3 & 5\end{bmatrix}\begin{bmatrix}9 & 6\\5 & 12\end{bmatrix} = \frac{1}{26}\begin{bmatrix}26 & 0\\52 & 78\end{bmatrix} = \begin{bmatrix}1 & 0\\2 & 3\end{bmatrix}\)	A1F	Obtains correct matrix for \(\mathbf{B}\) or \(\mathbf{AM}\); PI by correct \(\mathbf{M}\); FT their \(\mathbf{A}^{-1}\)
\(\mathbf{M} = 2\begin{bmatrix}5 & 2\\-3 & 4\end{bmatrix} + \begin{bmatrix}1 & 0\\2 & 3\end{bmatrix} = \begin{bmatrix}11 & 4\\-4 & 11\end{bmatrix}\)	A1F	Obtains matrix \(\mathbf{M}\); FT their \(\mathbf{A}^{-1}\)