| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a structured Further Maths question that guides students through standard techniques: expanding a complex number, using the conjugate root theorem, and factoring a cubic. While it's Further Maths content (inherently harder), the question is highly scaffolded with clear steps and uses routine methods that FP1 students practice extensively. It's slightly easier than average for an A-level question overall due to the scaffolding, but appropriate difficulty for FP1. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| \((2+i)^3 = (2+i)(2+i)^2\) | M1 | |
| \((2+i)^2 = 4 + 4i + i^2 = 3 + 4i\) | A1 | |
| \((2+i)(3+4i) = 6 + 8i + 3i + 4i^2 = 2 + 11i\) | A1 | So \(b = 11\) |
| Answer | Marks |
|---|---|
| Substituting \(z = 2+i\): \((2+11i) + p(2+i) + q = 0\) | M1 |
| Real: \(2 + 2p + q = 0\) | A1 |
| Imaginary: \(11 + p = 0 \Rightarrow p = -11\) | A1 |
| Substituting back: \(2 - 22 + q = 0 \Rightarrow q = 20\) | A1 |
| Answer | Marks |
|---|---|
| \((z-(2+i))(z-(2-i)) = (z-2)^2+1 = z^2-4z+5\) | B1 |
| Quadratic factor is \(z^2 - 4z + 5\) | B1 |
| Answer | Marks |
|---|---|
| \(z^3 - 11z + 20 = (z^2-4z+5)(z+k)\) | M1 |
| Comparing constant: \(5k = 20 \Rightarrow k = 4\) | |
| Real root is \(z = -4\) | A1 |
# Question 3:
## Part (a)
| $(2+i)^3 = (2+i)(2+i)^2$ | M1 | |
| $(2+i)^2 = 4 + 4i + i^2 = 3 + 4i$ | A1 | |
| $(2+i)(3+4i) = 6 + 8i + 3i + 4i^2 = 2 + 11i$ | A1 | So $b = 11$ |
## Part (b)(i)
| Substituting $z = 2+i$: $(2+11i) + p(2+i) + q = 0$ | M1 | |
| Real: $2 + 2p + q = 0$ | A1 | |
| Imaginary: $11 + p = 0 \Rightarrow p = -11$ | A1 | |
| Substituting back: $2 - 22 + q = 0 \Rightarrow q = 20$ | A1 | |
## Part (b)(ii)
| $(z-(2+i))(z-(2-i)) = (z-2)^2+1 = z^2-4z+5$ | B1 | |
| Quadratic factor is $z^2 - 4z + 5$ | B1 | |
## Part (b)(iii)
| $z^3 - 11z + 20 = (z^2-4z+5)(z+k)$ | M1 | |
| Comparing constant: $5k = 20 \Rightarrow k = 4$ | | |
| Real root is $z = -4$ | A1 | |3
\begin{enumerate}[label=(\alph*)]
\item Show that $( 2 + \mathrm { i } ) ^ { 3 }$ can be expressed in the form $2 + b \mathrm { i }$, where $b$ is an integer.
\item It is given that $2 + \mathrm { i }$ is a root of the equation
$$z ^ { 3 } + p z + q = 0$$
where $p$ and $q$ are real numbers.
\begin{enumerate}[label=(\roman*)]
\item Show that $p = - 11$ and find the value of $q$.
\item Given that $2 - \mathrm { i }$ is also a root of $z ^ { 3 } + p z + q = 0$, find a quadratic factor of $z ^ { 3 } + p z + q$ with real coefficients.
\item Find the real root of the equation $z ^ { 3 } + p z + q = 0$.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-06_1568_1707_1139_155}
\end{center}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2015 Q3 [11]}}