| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Complex Numbers Arithmetic |
| Type | Parametric polynomials with root conditions |
| Difficulty | Challenging +1.2 This is a structured Further Maths question testing standard complex roots theory (conjugate pairs), Vieta's formulas, and De Moivre's theorem. Parts (a)-(c) are routine applications requiring no novel insight. Part (d) requires converting to modulus-argument form and applying De Moivre's theorem—a standard FM technique, though the algebraic manipulation adds some challenge. The multi-part structure and FM content place it above average difficulty, but it follows predictable patterns without requiring creative problem-solving. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
6 You are given that the cubic equation $2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0$, where $p$ and $q$ are real numbers, has a complex root $\alpha = 1 + i \sqrt { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Write down a second complex root, $\beta$.
\item Determine the third root, $\gamma$.
\item Find the value of $p$ and the value of $q$.
\item Show that if $n$ is an integer then $\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }$ where $\tan \theta = \sqrt { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q6 [9]}}