| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Direct modulus and argument |
| Difficulty | Moderate -0.3 Part (i) is straightforward application of modulus and argument formulas for a simple complex number. Part (ii) requires substituting z and its conjugate, then equating real and imaginary parts to solve simultaneous equations—a standard FP1 technique with minimal problem-solving demand. The question is slightly easier than average A-level due to its routine nature, though the Further Maths context prevents it from being significantly negative. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \( | z | =\sqrt{5}\) and \(\arg z = -26.6°\) or \(-0.464\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(a+b=2, b-a=-8\) and \(a=5, b=-3\) | B1, M1, A1, M1, A1 [5] | \(z^n=2+i\) stated or used; Obtain two equations from real and imaginary parts; Obtain correct equations; Attempt to solve 2 linear equations; Obtain correct answers |
### (i)
Answer: $|z|=\sqrt{5}$ and $\arg z = -26.6°$ or $-0.464$ | B1; B1 [2] | Allow 2.2; Allow -27° or -0.46(3)
### (ii)
Answer: $a+b=2, b-a=-8$ and $a=5, b=-3$ | B1, M1, A1, M1, A1 [5] | $z^n=2+i$ stated or used; Obtain two equations from real and imaginary parts; Obtain correct equations; Attempt to solve 2 linear equations; Obtain correct answersThe complex number $2 - i$ is denoted by $z$.
\begin{enumerate}[label=(\roman*)]
\item Find $|z|$ and $\arg z$. [2]
\item Given that $az + bz^* = 4 - 8i$, find the values of the real constants $a$ and $b$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2013 Q3 [7]}}