| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parametric polynomials with root conditions |
| Difficulty | Standard +0.8 This is a Further Maths FP1 question on complex roots of polynomials requiring knowledge that complex roots come in conjugate pairs, then using this with the constant term to find the real root, followed by expanding factors or using Vieta's formulas to find coefficients. While systematic, it requires multiple techniques (conjugate root theorem, factor theorem, polynomial multiplication or symmetric functions) and careful algebraic manipulation across 11 marks, making it moderately challenging but still a standard FP1 exercise. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
The cubic equation $x^3 + Ax^2 + Bx + 15 = 0$, where $A$ and $B$ are real numbers, has a root $x = 1 + 2\mathrm{j}$.
\begin{enumerate}[label=(\roman*)]
\item Write down the other complex root. [1]
\item Explain why the equation must have a real root. [1]
\item Find the value of the real root and the values of $A$ and $B$. [9]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2007 Q9 [11]}}