| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Standard quadratic with real coefficients |
| Difficulty | Moderate -0.3 This is a straightforward Further Pure 1 question testing basic complex number operations: (a) uses the quadratic formula with complex roots (routine), (b)(i) requires multiplying complex numbers (standard manipulation), and (b)(ii) involves conjugates and solving simultaneous equations. All parts are textbook exercises requiring only direct application of techniques with no problem-solving insight, making it slightly easier than average despite being Further Maths content. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(w = \frac{-6 \pm \sqrt{36-4(34)}}{2} = \frac{-6 \pm \sqrt{-100}}{2}\) | M1 | Correct substitution into quadratic formula |
| \(= \frac{-6 \pm 10i}{2}\) | B1 | \(\sqrt{-100} = 10i\) or \(\sqrt{-100}/2 = 5i\) |
| \(= -3 \pm 5i\) | A1 | \(-3 \pm 5i\) (\(p=-3\), \(q=\pm5\)); NMS mark as 3/3 or 0/3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z = i(1+i)(2+i) = i(2+3i+i^2)\) | M1 | Attempt to expand all brackets |
| \(= 2i+3(-1)+i(-1)\) | B1 | \(i^2 = -1\) used at least once |
| \(= -3 + i\) | A1 | \(-3+i\) (\(a=-3\), \(b=1\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z^* = -3 - i\) | B1F | OE. Ft c's \(a - bi\) |
| \(-3+i+m(-3-i) = ni\) | M1 | Equating both real parts and imaginary parts, PI by next line |
| \(\Rightarrow m=-1,\ n=2\) | A1 | Both correct |
## Question 2(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $w = \frac{-6 \pm \sqrt{36-4(34)}}{2} = \frac{-6 \pm \sqrt{-100}}{2}$ | M1 | Correct substitution into quadratic formula |
| $= \frac{-6 \pm 10i}{2}$ | B1 | $\sqrt{-100} = 10i$ or $\sqrt{-100}/2 = 5i$ |
| $= -3 \pm 5i$ | A1 | $-3 \pm 5i$ ($p=-3$, $q=\pm5$); NMS mark as 3/3 or 0/3 |
## Question 2(b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z = i(1+i)(2+i) = i(2+3i+i^2)$ | M1 | Attempt to expand all brackets |
| $= 2i+3(-1)+i(-1)$ | B1 | $i^2 = -1$ used at least once |
| $= -3 + i$ | A1 | $-3+i$ ($a=-3$, $b=1$) |
## Question 2(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z^* = -3 - i$ | B1F | OE. Ft c's $a - bi$ |
| $-3+i+m(-3-i) = ni$ | M1 | Equating **both** real parts and imaginary parts, PI by next line |
| $\Rightarrow m=-1,\ n=2$ | A1 | Both correct |
---2
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $w ^ { 2 } + 6 w + 34 = 0$, giving your answers in the form $p + q \mathrm { i }$, where $p$ and $q$ are integers.
\item It is given that $z = \mathrm { i } ( 1 + \mathrm { i } ) ( 2 + \mathrm { i } )$.
\begin{enumerate}[label=(\roman*)]
\item Express $z$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are integers.
\item Find integers $m$ and $n$ such that $z + m z ^ { * } = n \mathrm { i }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q2 [9]}}