| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Verifying roots satisfy equations |
| Difficulty | Moderate -0.3 This is a straightforward verification question requiring basic complex number arithmetic (squaring, conjugation) and substitution. Part (a) involves routine calculation, parts (b) and (c) are direct verification by substitution. While it's Further Maths content, the mechanical nature and step-by-step guidance make it slightly easier than average A-level difficulty. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a)(i) \(z_1^2 = -\frac{1}{4} - i + i^2 = -\frac{1}{4} - i - 1 = -\frac{5}{4} - i\) | M1A1 | 2 marks |
| 5(a)(ii) LHS \(= -\frac{3}{4} - i + \frac{1}{2} + i + \frac{1}{4} = 0\) | M1A1 | 2 marks |
| 5(b) LHS \(= -\frac{3}{4} + i + \frac{1}{2} - i + \frac{1}{4} = 0\) | M1A1 | 2 marks |
| 5(c) z real \(\Rightarrow z^* = z\) | M1 | 2 marks |
| Discr't zero or correct factorisation | A1 |
**5(a)(i)** $z_1^2 = -\frac{1}{4} - i + i^2 = -\frac{1}{4} - i - 1 = -\frac{5}{4} - i$ | M1A1 | 2 marks | M1 for use of $i^2 = -1$
**5(a)(ii)** LHS $= -\frac{3}{4} - i + \frac{1}{2} + i + \frac{1}{4} = 0$ | M1A1 | 2 marks | AG; M1 for $z^*$ correct
**5(b)** LHS $= -\frac{3}{4} + i + \frac{1}{2} - i + \frac{1}{4} = 0$ | M1A1 | 2 marks | AG; M1 for $z_3^2$ correct
**5(c)** z real $\Rightarrow z^* = z$ | M1 | 2 marks | Clearly stated
Discr't zero or correct factorisation | A1 | | AG
**Total: 8 marks**
---5
\begin{enumerate}[label=(\alph*)]
\item It is given that $z _ { 1 } = \frac { 1 } { 2 } - \mathrm { i }$.
\begin{enumerate}[label=(\roman*)]
\item Calculate the value of $z _ { 1 } ^ { 2 }$, giving your answer in the form $a + b \mathrm { i }$.
\item Hence verify that $z _ { 1 }$ is a root of the equation
$$z ^ { 2 } + z ^ { * } + \frac { 1 } { 4 } = 0$$
\end{enumerate}\item Show that $z _ { 2 } = \frac { 1 } { 2 } + \mathrm { i }$ also satisfies the equation in part (a)(ii).
\item Show that the equation in part (a)(ii) has two equal real roots.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q5 [8]}}