| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.3 This is a standard Further Maths FP1 complex numbers question with routine procedures: expanding a complex cube (straightforward algebra), substituting to verify a root (direct calculation), and factorizing using the conjugate root theorem. While it requires careful algebraic manipulation and understanding of complex conjugates, it follows a predictable template with no novel problem-solving required, making it slightly easier than an average A-level question overall. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Express $(2 + 3i)^3$ in the form $a + ib$. [3]
\item Hence verify that $2 + 3i$ is a root of the equation $3z^3 - 8z^2 + 23z + 52 = 0$. [3]
\item Express $3z^3 - 8z^2 + 23z + 52$ as the product of a linear factor and a quadratic factor with real coefficients. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2021 Q3 [10]}}