| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| 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 coefficients. Parts (b) and (c) involve routine calculations with no novel problem-solving required. While it's a Further Maths topic, the execution is mechanical and below average difficulty for FP1 standard. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02h Square roots: of complex numbers4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1+5i\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\big(x-(1+5i)\big)\big(x-(1-5i)\big) = x^2-2x+26\) | M1A1 | M1: Attempt to expand or use sum and product of complex roots. A1: Correct expression |
| \((x-2)\big(x-(1\pm5i)\big) = x^2-(3\pm5i)x+2(1\pm5i)\) | ||
| \((x^2-2x+26)(x-2) = x^3+px^2+30x+q\) | M1 | Uses their third factor with their quadratic with at least 4 terms in the expansion |
| \(p = -4, \quad q = -52\) | A1, A1 | May be seen in cubic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(1+5i)=0\) or \(f(1-5i)=0\) | M1 | Substitute one complex root to achieve 2 equations in \(p\) and/or \(q\) |
| \(q - 24p - 44 = 0\) and \(10p + 40 = 0\) | A1 | Both equations correct oe |
| Solving for \(p\) and \(q\) | M1 | |
| \(p = -4, \quad q = -52\) | A1, A1 | May be seen in cubic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Argand diagram with \(1+5i\) and \(1-5i\) plotted as conjugate pair | B1 | Conjugate pair correctly positioned and labelled with \(1+5i\), \(1-5i\) or \((1,5)\), \((1,-5)\) or axes labelled 1 and 5 |
| Point at \(x=2\) on real axis also plotted | B1 | The 2 correctly positioned relative to conjugate pair and labelled |
# Question 3:
$$x^3 + px^2 + 30x + q = 0$$
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1+5i$ | B1 | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\big(x-(1+5i)\big)\big(x-(1-5i)\big) = x^2-2x+26$ | M1A1 | M1: Attempt to expand or use sum and product of complex roots. A1: Correct expression |
| $(x-2)\big(x-(1\pm5i)\big) = x^2-(3\pm5i)x+2(1\pm5i)$ | | |
| $(x^2-2x+26)(x-2) = x^3+px^2+30x+q$ | M1 | Uses their third factor with their quadratic with at least 4 terms in the expansion |
| $p = -4, \quad q = -52$ | A1, A1 | May be seen in cubic |
**OR alternative method:**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(1+5i)=0$ or $f(1-5i)=0$ | M1 | Substitute one complex root to achieve 2 equations in $p$ and/or $q$ |
| $q - 24p - 44 = 0$ and $10p + 40 = 0$ | A1 | Both equations correct oe |
| Solving for $p$ and $q$ | M1 | |
| $p = -4, \quad q = -52$ | A1, A1 | May be seen in cubic |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Argand diagram with $1+5i$ and $1-5i$ plotted as conjugate pair | B1 | Conjugate pair correctly positioned and labelled with $1+5i$, $1-5i$ or $(1,5)$, $(1,-5)$ or axes labelled 1 and 5 |
| Point at $x=2$ on real axis also plotted | B1 | The 2 correctly positioned relative to conjugate pair and labelled |
---3. Given that 2 and $1 - 5 \mathrm { i }$ are roots of the equation
$$x ^ { 3 } + p x ^ { 2 } + 30 x + q = 0 , \quad p , q \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item write down the third root of the equation.
\item Find the value of $p$ and the value of $q$.
\item Show the three roots of this equation on a single Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2014 Q3 [8]}}