| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Moderate -0.3 This is a straightforward application of the complex conjugate root theorem and Vieta's formulas. Part (a) requires recalling that complex roots come in conjugate pairs for real polynomials. Parts (b) and (c) involve routine calculations with no novel problem-solving required. While it's Further Maths content, it's a standard textbook exercise testing basic understanding rather than deeper insight. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks |
|---|---|
| \(5 - 2i\) is a root | B1 (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-(5+2i))(x-(5-2i)) = x^2 - 10x + 29\) | M1 M1 | 1st M: form brackets using \((x-\alpha)(x-\beta)\) and expand; 2nd M: achieve 3-term quadratic with no \(i\)'s |
| \(x^3 - 12x^2 + cx + d = (x^2 - 10x + 29)(x-2)\) | M1 | |
| \(c = 49\), \(d = -58\) | A1, A1 (5) |
| Answer | Marks |
|---|---|
| Conjugate pair in 1st and 4th quadrants (symmetrical about real axis) | B1 |
| Fully correct, labelled diagram | B1 (2) |
## Question 6:
**(a)**
$5 - 2i$ is a root | B1 (1) |
**(b)**
$(x-(5+2i))(x-(5-2i)) = x^2 - 10x + 29$ | M1 M1 | 1st M: form brackets using $(x-\alpha)(x-\beta)$ and expand; 2nd M: achieve 3-term quadratic with no $i$'s
$x^3 - 12x^2 + cx + d = (x^2 - 10x + 29)(x-2)$ | M1 |
$c = 49$, $d = -58$ | A1, A1 (5) |
**(c)**
Conjugate pair in 1st and 4th quadrants (symmetrical about real axis) | B1 |
Fully correct, labelled diagram | B1 (2) |6. Given that 2 and $5 + 2 \mathrm { i }$ are roots of the equation
$$x ^ { 3 } - 12 x ^ { 2 } + c x + d = 0 , \quad c , d \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item write down the other complex root of the equation.
\item Find the value of $c$ and the value of $d$.
\item Show the three roots of this equation on a single Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2010 Q6 [8]}}