| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Complex number matrices |
| Difficulty | Standard +0.8 This is a Further Maths question involving complex number matrix operations. Part (a) requires computing a 2×2 determinant with complex entries (routine but requires careful arithmetic). Part (b) requires matrix multiplication with complex numbers and solving a system to find multiple unknowns such that the product is a scalar natural number—this demands systematic algebraic manipulation and understanding of when complex matrices yield real scalar multiples of identity. More demanding than standard A-level but typical for Further Maths. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3 \times 2 - i(i-1) = 6 - i^2 + i\) | M1 | Writes a correct unsimplified expression for det A |
| \(= 7 + i\) | R1 | Completes fully correct proof; must have \(a = 7\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| B \(= p\)A\(^{-1}\) | M1 | Recognises B is equal to a multiple of A\(^{-1}\), or multiplies A and B to find at least one correct unsimplified element of AB |
| \(\begin{bmatrix} 14-2i & b \\ c & d \end{bmatrix} = \frac{p}{7+i}\begin{bmatrix} 2 & 1-i \\ -i & 3 \end{bmatrix}\) | M1 | Sets up at least one correct non-matrix equation in one or two unknowns |
| \(14 - 2i = \frac{2p}{7+i}\), so \((7-i)(7+i) = p\), \(p = 50\) | M1, A1 | Sets up another correct equation in one unknown; obtains at least one correct value |
| \(b = \frac{50(1-i)}{7+i} = 6 - 8i\) | A1 | Obtains at least two correct values of \(p, b, c\) or \(d\) |
| \(c = \frac{-50i}{7+i} = -1 - 7i\) | ||
| \(d = \frac{150}{7+i} = 21 - 3i\) | A1 | Obtains all four correct values; accept \(p=50\), B \(= \begin{bmatrix} 14-2i & 6-8i \\ -1-7i & 21-3i \end{bmatrix}\) |
## Question 10(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3 \times 2 - i(i-1) = 6 - i^2 + i$ | M1 | Writes a correct unsimplified expression for det **A** |
| $= 7 + i$ | R1 | Completes fully correct proof; must have $a = 7$ |
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## Question 10(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **B** $= p$**A**$^{-1}$ | M1 | Recognises **B** is equal to a multiple of **A**$^{-1}$, or multiplies **A** and **B** to find at least one correct unsimplified element of **AB** |
| $\begin{bmatrix} 14-2i & b \\ c & d \end{bmatrix} = \frac{p}{7+i}\begin{bmatrix} 2 & 1-i \\ -i & 3 \end{bmatrix}$ | M1 | Sets up at least one correct non-matrix equation in one or two unknowns |
| $14 - 2i = \frac{2p}{7+i}$, so $(7-i)(7+i) = p$, $p = 50$ | M1, A1 | Sets up another correct equation in one unknown; obtains at least one correct value |
| $b = \frac{50(1-i)}{7+i} = 6 - 8i$ | A1 | Obtains at least two correct values of $p, b, c$ or $d$ |
| $c = \frac{-50i}{7+i} = -1 - 7i$ | | |
| $d = \frac{150}{7+i} = 21 - 3i$ | A1 | Obtains all four correct values; accept $p=50$, **B** $= \begin{bmatrix} 14-2i & 6-8i \\ -1-7i & 21-3i \end{bmatrix}$ |
---10
\begin{enumerate}[label=(\alph*)]
\item Show that $\operatorname { det } \mathbf { A } = a + \mathrm { i }$ where $a$ is an integer to be determined.
10 Matrix A is given by
10
\item Matrix B is given by
$$\mathbf { B } = \left[ \begin{array} { c c }
14 - 2 \mathrm { i } & b \\
c & d
\end{array} \right] \quad \text { and } \quad \mathbf { A B } = p$$
where $b , c , d \in \mathbb { C }$ and $p \in \mathbb { N }$\\
Find $b , c , d$ and $p$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2021 Q10 [8]}}