| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Easy -1.2 This is a routine Further Pure 1 question testing basic complex number operations: solving simple quadratic equations with complex roots, binomial expansion, and computing powers of complex numbers. All parts are standard textbook exercises requiring only direct application of learned techniques with no problem-solving insight needed. While it's Further Maths content, these are foundational FP1 skills, making it easier than average overall. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Mark |
| (a)(i) | \(x = \pm 3i\) | B1 |
| (a)(ii) | \(x = -2 \pm 3i\) | B1F |
| (b)(i) | \((1 + x)^3 = 1 + 3x + 3x^2 + x^3\) | B1 |
| (b)(ii) | \((1 + 2i)^3 = 1 + 3(2i) + 3(2i)^2 + (2i)^3 = 1+3(2i) + 3(4i^2) + (8i^3)\) \(= 1 + 3(2i) + 3(4)(-1) + (8)(-i) = -11 - 2i\) | B1F M1 A1 |
| (b)(iii) | \(z^* - z^3 = 1 - 2i - (-11 - 2i) = 12\) | M1 A1F |
| Part | Answer/Working | Mark | Guidance |
|------|---|---|---|
| (a)(i) | $x = \pm 3i$ | B1 | 1 mark; $\pm 3i$ ($a = 0, b = \pm3$) |
| (a)(ii) | $x = -2 \pm 3i$ | B1F | 1 mark; If not correct, ft wrong answer(s) to (i) provided (i) has a non-zero b value |
| (b)(i) | $(1 + x)^3 = 1 + 3x + 3x^2 + x^3$ | B1 | 1 mark; Terms simplified in any order |
| (b)(ii) | $(1 + 2i)^3 = 1 + 3(2i) + 3(2i)^2 + (2i)^3 = 1+3(2i) + 3(4i^2) + (8i^3)$ $= 1 + 3(2i) + 3(4)(-1) + (8)(-i) = -11 - 2i$ | B1F M1 A1 | 3 marks; Replacing x in (b)(i) by 2i, squaring and cubing correctly, only it on c's wrong non-zero coefficients from (b)(i); Use of $i^2 = -1$ at least once; $-11 - 2i$ ($a = -11, b = -2$) |
| (b)(iii) | $z^* - z^3 = 1 - 2i - (-11 - 2i) = 12$ | M1 A1F | 2 marks; Use of $z^* = 1 - 2i$; If not correct, only ft on $1 - 2i$ - c's (b)(ii) if c's (b)(ii) answer is of the form $a + bi$ with $a \neq 0$ and $b \neq 0$ |3
\begin{enumerate}[label=(\alph*)]
\item Solve the following equations, giving each root in the form $a + b \mathrm { i }$ :
\begin{enumerate}[label=(\roman*)]
\item $x ^ { 2 } + 9 = 0$;
\item $( x + 2 ) ^ { 2 } + 9 = 0$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Expand $( 1 + x ) ^ { 3 }$.
\item Express $( 1 + 2 \mathrm { i } ) ^ { 3 }$ in the form $a + b \mathrm { i }$.
\item Given that $z = 1 + 2 \mathrm { i }$, find the value of
$$z ^ { * } - z ^ { 3 }$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2012 Q3 [8]}}