| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Square roots with follow-up application |
| Difficulty | Standard +0.8 This is a Further Maths FP1 question requiring algebraic solution of complex square roots (setting a=x+iy and solving simultaneous equations) followed by geometric interpretation on an Argand diagram. While the square root method is standard FP1 content, it requires careful algebraic manipulation, and part (ii) demands understanding that each locus is a circle with specific center and radius. The multi-step nature and combination of algebraic and geometric reasoning places this moderately above average difficulty. |
| Spec | 4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| \(\pm(3 - 2i)\) | M1 | Attempt to equate real and imaginary parts of \((x+iy)^2\) & \(5 - 12i\) |
| A1 | Obtain both results, a.e.f | |
| M1 | Obtain quadratic in \(x^2\) or \(y^2\) | |
| M1 | Solve to obtain \(x = (\pm)3\) or \(y = (\pm)2\) | |
| A1, 5 | Obtain correct answers as complex nos |
| Answer | Marks | Guidance |
|---|---|---|
| square root | B1ft | Circle with centre at their |
| B1 | Circle passing through origin | |
| B1ft | 2nd circle centre correct relative to 1st | |
| B1, 4 | Circle passing through origin |
**Part (i)**
$x^2 - y^2 = 5$ and $xy = -6$
$\pm(3 - 2i)$ | M1 | Attempt to equate real and imaginary parts of $(x+iy)^2$ & $5 - 12i$
| A1 | Obtain both results, a.e.f
| M1 | Obtain quadratic in $x^2$ or $y^2$
| M1 | Solve to obtain $x = (\pm)3$ or $y = (\pm)2$
| A1, 5 | Obtain correct answers as complex nos
---
**Part (ii)**
square root | B1ft | Circle with centre at their
| B1 | Circle passing through origin
| B1ft | 2nd circle centre correct relative to 1st
| B1, 4 | Circle passing through origin
---8 The complex number $a$ is such that $a ^ { 2 } = 5 - 12 \mathrm { i }$.\\
(i) Use an algebraic method to find the two possible values of $a$.\\
(ii) Sketch on a single Argand diagram the two possible loci given by $| z - a | = | a |$.
\hfill \mbox{\textit{OCR FP1 2010 Q8 [9]}}