| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Factored form to roots |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question requiring routine application of solving quadratic equations with complex roots. Both factors are already given, so students simply use the quadratic formula twice and plot the results—no problem-solving insight needed, just mechanical execution of standard techniques. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks |
|---|---|
| (a) \(4x^2 + 9 = 0 \Rightarrow x = ki, \quad x = \pm \frac{3}{2}i\) or equivalent | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \frac{6 \pm \sqrt{36-136}}{2}\) or \((x-3)^2 - 9 + 34 = 0\) | M1 | |
| \(= 3 \pm 5i\) and \(3 - 5i\) | A1 A1ft | (5) |
| (b) | B1ft; B1ft | Two roots on imaginary axis; Two roots – one the conjugate of the other |
| (2) [7] |
(a) $4x^2 + 9 = 0 \Rightarrow x = ki, \quad x = \pm \frac{3}{2}i$ or equivalent | M1, A1 |
Solving 3-term quadratic by formula or completion of the square
$x = \frac{6 \pm \sqrt{36-136}}{2}$ or $(x-3)^2 - 9 + 34 = 0$ | M1 |
$= 3 \pm 5i$ and $3 - 5i$ | A1 A1ft | (5)
(b) | B1ft; B1ft | Two roots on imaginary axis; Two roots – one the conjugate of the other
| (2) [7] |
**Notes:**
(a) Final A follow through conjugate of their first root.
(b) First B award only for first pair imaginary; Second B award only if second pair complex. Complex numbers labelled, scales or coordinates or vectors required for B marks.
---5.
$$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 6 x + 34 \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find the four roots of $\mathrm { f } ( x ) = 0$
Give your answers in the form $x = p + \mathrm { i } q$, where $p$ and $q$ are real.
\item Show these four roots on a single Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2013 Q5 [7]}}