| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Verifying roots satisfy equations |
| Difficulty | Standard +0.3 This is a straightforward Further Pure 1 question testing basic complex number operations and standard results about conjugate roots. Part (a) involves routine multiplication and direct substitution to verify a root. Part (b) applies the standard theorem that complex roots of real quadratics come in conjugate pairs, then uses sum/product of roots formulas—all textbook exercises requiring recall and careful arithmetic but no problem-solving insight. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) Full expansion of product | M1 | |
| Use of \(i^2 = -1\) | m1 | |
| \((2 + \sqrt{5i})(\sqrt{5} - i) = 3\sqrt{5} + 3i\) | A1 | \(\sqrt{5} \cdot \sqrt{5} = 5\) must be used – Accept not fully simplified |
| (ii) \(z^* = x - iy = (\sqrt{5} + i)\) | M1 | |
| Hence result | A1 | Convincingly shown (AG) |
| Other root is \(\sqrt{5} + i\) | B1 | |
| (b)(i) Sum of roots is \(2\sqrt{5}\) | B1 | |
| (ii) Product is \(6\) | M1A1 | |
| (iii) \(p = -2\sqrt{5}\), \(q = 6\) | B1; B1√ | ft wrong answers in (ii) |
**(a)(i)** Full expansion of product | M1
Use of $i^2 = -1$ | m1
$(2 + \sqrt{5i})(\sqrt{5} - i) = 3\sqrt{5} + 3i$ | A1 | $\sqrt{5} \cdot \sqrt{5} = 5$ must be used – Accept not fully simplified
**(ii)** $z^* = x - iy = (\sqrt{5} + i)$ | M1
Hence result | A1 | Convincingly shown (AG)
Other root is $\sqrt{5} + i$ | B1 |
**(b)(i)** Sum of roots is $2\sqrt{5}$ | B1
**(ii)** Product is $6$ | M1A1 |
**(iii)** $p = -2\sqrt{5}$, $q = 6$ | B1; B1√ | ft wrong answers in (ii)
**Total for Q5: 11 marks**
---5
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate $( 2 + \mathrm { i } \sqrt { 5 } ) ( \sqrt { 5 } - \mathrm { i } )$.
\item Hence verify that $\sqrt { 5 } - \mathrm { i }$ is a root of the equation
$$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$
where $z ^ { * }$ is the conjugate of $z$.
\end{enumerate}\item The quadratic equation
$$x ^ { 2 } + p x + q = 0$$
in which the coefficients $p$ and $q$ are real, has a complex root $\sqrt { 5 } - \mathrm { i }$.
\begin{enumerate}[label=(\roman*)]
\item Write down the other root of the equation.
\item Find the sum and product of the two roots of the equation.
\item Hence state the values of $p$ and $q$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q5 [11]}}