| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Linear equations in z and z* |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question testing basic complex number manipulation. Parts (a)(i) and (a)(ii) are definitional recall requiring no problem-solving. Part (b)(i) involves solving a linear equation in z and z* by equating real and imaginary parts—a standard textbook exercise. Part (b)(ii) is simple arithmetic once w is found. The entire question requires only routine application of fundamental complex number properties with no novel insight or challenging multi-step reasoning. |
| Spec | 4.01a Mathematical induction: construct proofs4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| \(z^* = x - \text{i}y\) | B1 | States \(x - yi\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(zz^* = (x+\text{i}y)(x-\text{i}y) = x^2 - \text{i}xy + \text{i}xy - \text{i}^2y^2 = x^2 + y^2\) | M1 | Obtains correct expansion and replaces \(\text{i}^2\) with \(-1\) |
| As \(x\) and \(y\) are both real, \(zz^*\) is real for all \(z\in\mathbb{C}\) | R1 | Simplifies to \(x^2+y^2\) and explains \(zz^*\) is real with reference to \(x\) and \(y\) being real; condone \(x^2\) and \(y^2\) for \(x\) and \(y\) in explanation |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(w = x+\text{i}y\); \(3(x+\text{i}y)+10\text{i} = 2(x-\text{i}y)+5\) | M1 | Substitutes \(x+\text{i}y\) for \(w\) and \(x-\text{i}y\) for \(w^*\) |
| \(3x = 2x+5\) and \(3y+10 = -2y\), so \(x=5\), \(y=-2\) | A1 | Obtains \(\text{Re}(w)=5\) or \(\text{Im}(w)=-2\) |
| \(w = 5-2\text{i}\) | A1 | Obtains \(5-2\text{i}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(w^2(w^*)^2 = 841\) | B1F | Obtains 841; FT their \(w\) of the form \(a+\text{i}b\) where \(a\) and \(b\) are non-zero |
## Question 8:
### Part 8(a)(i):
| $z^* = x - \text{i}y$ | B1 | States $x - yi$ |
### Part 8(a)(ii):
| $zz^* = (x+\text{i}y)(x-\text{i}y) = x^2 - \text{i}xy + \text{i}xy - \text{i}^2y^2 = x^2 + y^2$ | M1 | Obtains correct expansion and replaces $\text{i}^2$ with $-1$ |
| As $x$ and $y$ are both real, $zz^*$ is real for all $z\in\mathbb{C}$ | R1 | Simplifies to $x^2+y^2$ and explains $zz^*$ is real with reference to $x$ and $y$ being real; condone $x^2$ and $y^2$ for $x$ and $y$ in explanation |
### Part 8(b)(i):
| Let $w = x+\text{i}y$; $3(x+\text{i}y)+10\text{i} = 2(x-\text{i}y)+5$ | M1 | Substitutes $x+\text{i}y$ for $w$ and $x-\text{i}y$ for $w^*$ |
| $3x = 2x+5$ and $3y+10 = -2y$, so $x=5$, $y=-2$ | A1 | Obtains $\text{Re}(w)=5$ or $\text{Im}(w)=-2$ |
| $w = 5-2\text{i}$ | A1 | Obtains $5-2\text{i}$ |
### Part 8(b)(ii):
| $w^2(w^*)^2 = 841$ | B1F | Obtains 841; FT their $w$ of the form $a+\text{i}b$ where $a$ and $b$ are non-zero |8
\begin{enumerate}[label=(\alph*)]
\item The complex number $z$ is given by $z = x + i y$ where $x , y \in \mathbb { R }$
8 (a) (i) Write down the complex conjugate $z ^ { * }$ in terms of $x$ and $y$
8 (a) (ii) Hence prove that $z z ^ { * }$ is real for all $z \in \mathbb { C }$\\
8
\item The complex number $w$ satisfies the equation
$$3 w + 10 \mathrm { i } = 2 w ^ { \star } + 5$$
8 (b) (i) Find $w$\\
8 (b) (ii) Calculate the value of $w ^ { 2 } \left( w ^ { * } \right) ^ { 2 }$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2024 Q8 [7]}}