| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Square roots with follow-up application |
| Difficulty | Standard +0.8 This FP1 question requires solving z² = 3+4i algebraically (equating real/imaginary parts), then using substitution u = w³ to solve a quadratic, followed by finding cube roots. While methodical, it demands careful algebraic manipulation across multiple steps, understanding of complex number properties, and argument constraints—significantly harder than typical A-level pure maths but standard for Further Maths. |
| Spec | 4.02h Square roots: of complex numbers4.02i Quadratic equations: with complex roots4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
| Answer | Marks |
|---|---|
| M1 | Attempt to equate real and imaginary parts |
| A1 | Obtain both results |
| M1 | Eliminate to obtain quadratic in \(x^2\) or \(y^2\) |
| M1 | Solve to obtain \(x\) or \(y\) value |
| A1 5 | Obtain correct answer as a complex no. |
| Answer | Marks |
|---|---|
| B1 1 | Obtain given answer correctly |
| Answer | Marks |
|---|---|
| M1 | Attempt to solve quadratic equation |
| A1 | Obtain correct answers |
| M1 | Choose negative sign |
| M1 | Relate required value to conjugate of (i) |
| A1 5 | Obtain correct answer |
## (i)
| M1 | Attempt to equate real and imaginary parts
| A1 | Obtain both results
| M1 | Eliminate to obtain quadratic in $x^2$ or $y^2$
| M1 | Solve to obtain $x$ or $y$ value
| A1 5 | Obtain correct answer as a complex no.
$x^2 - y^2 = 3$ $xy = 2$
$z = 2 + i$
## (ii)
| B1 1 | Obtain given answer correctly
## (iii)
| M1 | Attempt to solve quadratic equation
| A1 | Obtain correct answers
| M1 | Choose negative sign
| M1 | Relate required value to conjugate of (i)
| A1 5 | Obtain correct answer
$w^3 = 2 \pm 11i$
$w = 2 - i$The complex number $z$, where $0 < \arg z < \frac{1}{2}\pi$, is such that $z^2 = 3 + 4\text{i}$.
\begin{enumerate}[label=(\roman*)]
\item Use an algebraic method to find $z$. [5]
\item Show that $z^3 = 2 + 11\text{i}$. [1]
\end{enumerate}
The complex number $w$ is the root of the equation
$$w^6 - 4w^3 + 125 = 0$$
for which $-\frac{1}{2}\pi < \arg w < 0$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumii}{2}
\item Find $w$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2010 Q10 [11]}}