| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Standard +0.3 This is a standard FP1 complex numbers question testing routine techniques: modulus-argument form (straightforward calculation), complex multiplication (algebraic manipulation), and using geometric/algebraic conditions to find unknowns. All parts follow textbook methods with no novel insight required, though it's slightly above average difficulty due to being Further Maths content and requiring careful algebraic manipulation across multiple parts. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation |
In this question you must show detailed reasoning.
The complex numbers $z_1$ and $z_2$ are given by $z_1 = 2 - 3i$ and $z_2 = a + 4i$ where $a$ is a real number.
\begin{enumerate}[label=(\alph*)]
\item Express $z_1$ in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
\item Find $z_1z_2$ in terms of $a$, writing your answer in the form $c + id$. [2]
\item The real and imaginary parts of a complex number on an Argand diagram are $x$ and $y$ respectively. Given that the point representing $z_1z_2$ lies on the line $y = x$, find the value of $a$. [2]
\item Given instead that $z_1z_2 = (z_1z_2)^*$ find the value of $a$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2021 Q2 [9]}}