| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle equations in complex form |
| Difficulty | Standard +0.3 This is a standard FP1 locus question requiring modulus/argument calculations, recognizing |z-a|=|a| as a circle, and reading off specific points. While it involves multiple parts and Further Maths content, it follows a predictable template with straightforward geometric interpretation and no novel problem-solving required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| \( | a | = \sqrt{(3^2+4^2)} = 5\) |
| \(\arg a = -\tan^{-1}\left(\frac{4}{3}\right) = -0.644\) | B1, B1, M1, A1 | For point \(A\) correctly located / For correct value for the modulus / For any correct relevant trig statement / For correct answer (radians or degrees) |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) | B1, B1, B1 | For any indication that locus is a circle / For any indication that the centre is at \(A\) / For a completely correct diagram |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) (a) \(z = -6i\) | B1 | For correct answer |
| (b) \(z = 8 - 6i\) | M1, A1 | For identification of end of diameter thru \(A\) / For correct answer |
**(i)** (See diagram in part (ii) below)
$|a| = \sqrt{(3^2+4^2)} = 5$
$\arg a = -\tan^{-1}\left(\frac{4}{3}\right) = -0.644$ | B1, B1, M1, A1 | For point $A$ correctly located / For correct value for the modulus / For any correct relevant trig statement / For correct answer (radians or degrees)
**Total: 4 marks**
**(ii)** | B1, B1, B1 | For any indication that locus is a circle / For any indication that the centre is at $A$ / For a completely correct diagram
**Total: 3 marks**
**(iii)** **(a)** $z = -6i$ | B1 | For correct answer
**(b)** $z = 8 - 6i$ | M1, A1 | For identification of end of diameter thru $A$ / For correct answer
**Total: 2 marks**
**Question 6 Total: 10 marks**
---6 In an Argand diagram, the variable point $P$ represents the complex number $z = x + \mathrm { i } y$, and the fixed point $A$ represents $a = 4 - 3 \mathrm { i }$.\\
(i) Sketch an Argand diagram showing the position of $A$, and find $| a |$ and $\arg a$.\\
(ii) Given that $| z - a | = | a |$, sketch the locus of $P$ on your Argand diagram.\\
(iii) Hence write down the non-zero value of $z$ corresponding to a point on the locus for which
\begin{enumerate}[label=(\alph*)]
\item the real part of $z$ is zero,
\item $\quad \arg z = \arg a$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 Q6 [10]}}