| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| 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 FP1 question requiring only direct application of the quadratic formula to two simple quadratics to find complex roots, followed by basic addition. Part (a) involves routine factorization and solving x²=-4 and completing the square or using the formula for x²+8x+25=0. Part (b) is trivial using either direct addition or Vieta's formulas. No problem-solving or insight required beyond standard techniques. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(x^2 + 4 = 0 \Rightarrow x = ki,\quad x = \pm 2i\) | M1, A1 | Just \(x = 2i\) is M1 A0; \(x = \pm 2\) is M0A0 |
| Solving 3-term quadratic: \(x = \frac{-8 \pm \sqrt{64-100}}{2} = -4+3i\) and \(-4-3i\) | M1 | M1 for solving quadratic following usual conventions |
| A1 A1ft | A1 for correct root (simplified); A1ft for conjugate of first answer; accept correct answers with no working |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(2i + (-2i) + (-4+3i) + (-4-3i) = -8\) | M1 A1cso | M1 for adding four roots of which at least two are complex conjugates and getting a real answer; A1 for \(-8\) following correct roots; if any incorrect working in (a) this A mark will be A0 |
| Alternative: Expands \(f(x)\) as quartic and chooses \(\pm\) coefficient of \(x^3\); result \(-8\) | M1, A1 cso |
## Question 3:
### Part (a)
| Working/Answer | Marks | Notes |
|---|---|---|
| $x^2 + 4 = 0 \Rightarrow x = ki,\quad x = \pm 2i$ | M1, A1 | Just $x = 2i$ is M1 A0; $x = \pm 2$ is M0A0 |
| Solving 3-term quadratic: $x = \frac{-8 \pm \sqrt{64-100}}{2} = -4+3i$ and $-4-3i$ | M1 | M1 for solving quadratic following usual conventions |
| | A1 A1ft | A1 for correct root (simplified); A1ft for conjugate of first answer; accept correct answers with no working |
### Part (b)
| Working/Answer | Marks | Notes |
|---|---|---|
| $2i + (-2i) + (-4+3i) + (-4-3i) = -8$ | M1 A1cso | M1 for adding four roots of which at least two are complex conjugates and getting a real answer; A1 for $-8$ following correct roots; if any incorrect working in (a) this A mark will be A0 |
| Alternative: Expands $f(x)$ as quartic and chooses $\pm$ coefficient of $x^3$; result $-8$ | M1, A1 cso | |
---3.
$$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find the four roots of $\mathrm { f } ( x ) = 0$.
\item Find the sum of these four roots.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2009 Q3 [7]}}