| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Given one complex root of cubic or quartic, find all roots |
| Difficulty | Moderate -0.3 This is a standard FP1 question requiring polynomial division and use of complex conjugate root theorem. Given one complex root of a real polynomial, students find its conjugate, then determine the real root via factorization. The Argand diagram is routine plotting. Straightforward application of well-practiced techniques with no novel insight required. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((z_2) = 3 - i\) | B1 | |
| \((z-(3+i))(z-(3-i)) = z^2 - 6z + 10\) | M1 | Attempt to expand \((z-(3+i))(z-(3-i))\) or any valid method to establish the quadratic factor |
| \((z^2 - 6z + 10)(z - 2) = 0\) | M1 | Attempt at linear factor with their \(cd\) in \((z^2 + az + c)(z + d) = \pm 20\); or \((z^2 - 6z + 10)(z+a) \Rightarrow 10a = -20\); or attempts \(f(2)\) |
| \((z_3) = 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Plot \((3,1)\) and \((3,-1)\) correctly with scale indicated or coordinates labelled | B1 | Allow \(i/-i\) for \(1/-1\) marked on imaginary axis |
| Plot \((2,0)\) correctly relative to conjugate pair with scale or labelled | B1 |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(z_2) = 3 - i$ | B1 | |
| $(z-(3+i))(z-(3-i)) = z^2 - 6z + 10$ | M1 | Attempt to expand $(z-(3+i))(z-(3-i))$ or any valid method to establish the quadratic factor |
| $(z^2 - 6z + 10)(z - 2) = 0$ | M1 | Attempt at linear factor with their $cd$ in $(z^2 + az + c)(z + d) = \pm 20$; or $(z^2 - 6z + 10)(z+a) \Rightarrow 10a = -20$; or attempts $f(2)$ |
| $(z_3) = 2$ | A1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Plot $(3,1)$ and $(3,-1)$ correctly with scale indicated or coordinates labelled | B1 | Allow $i/-i$ for $1/-1$ marked on imaginary axis |
| Plot $(2,0)$ correctly relative to conjugate pair with scale or labelled | B1 | |
---5. The roots of the equation
$$z ^ { 3 } - 8 z ^ { 2 } + 22 z - 20 = 0$$
are $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Given that $z _ { 1 } = 3 + \mathrm { i }$, find $z _ { 2 }$ and $z _ { 3 }$.
\item Show, on a single Argand diagram, the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2012 Q5 [6]}}