| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Verifying roots satisfy equations |
| Difficulty | Easy -1.2 This is a routine Further Maths question testing basic complex number operations: solving quadratic equations with complex roots (standard formula application), binomial expansion, and direct substitution to verify a root. All parts are straightforward recall and mechanical computation with no problem-solving or insight required, making it easier than average even for Further Maths. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Roots are \(\pm 4i\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Roots are \(1 \pm 4i\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 + x)^3 = 1 + 3x + 3x^2 + x^3\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\ldots = (-2 + 2i) + (2 + 2i) - 0\) | M1, A1 | 2 marks |
### Part (a)(i)
Roots are $\pm 4i$ | M1A1 | 2 marks | M1 for one correct root or two correct factors
### Part (a)(ii)
Roots are $1 \pm 4i$ | M1A1 | 2 marks | M1 for correct method
### Part (b)(i)
$(1 + x)^3 = 1 + 3x + 3x^2 + x^3$ | M1A1 | 2 marks | M1A0 if one small error
### Part (b)(ii)
$(1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i$ | M1A1 | 2 marks | M1 if $i^2 = -1$ used
### Part (b)(iii)
$(1 + i)^3 + 2(1 + i) - 4i$
$\ldots = (-2 + 2i) + (2 + 2i) - 0$ | M1, A1 | 2 marks | with attempt to evaluate; convincingly shown (AG)
### **Total for Question 1: 10 marks**
---1
\begin{enumerate}[label=(\alph*)]
\item Solve the following equations, giving each root in the form $a + b \mathrm { i }$ :
\begin{enumerate}[label=(\roman*)]
\item $x ^ { 2 } + 16 = 0$;
\item $x ^ { 2 } - 2 x + 17 = 0$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Expand $( 1 + x ) ^ { 3 }$.
\item Express $( 1 + \mathrm { i } ) ^ { 3 }$ in the form $a + b \mathrm { i }$.
\item Hence, or otherwise, verify that $x = 1 + \mathrm { i }$ satisfies the equation
$$x ^ { 3 } + 2 x - 4 \mathrm { i } = 0$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q1 [10]}}