| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| 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 straightforward Further Maths question requiring standard techniques: finding a circle equation from diameter endpoints using |z - center| = radius, identifying the perpendicular bisector as the imaginary axis through the midpoint, then solving simultaneous equations. While it involves multiple parts and some geometric insight, the methods are routine for FP1 students with no novel problem-solving required. |
| Spec | 4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3+3\text{i}\) | B1 | Obtain centre as a complex number, allow \((3,3)\) but not \((3,3\text{i})\); give if correct in locus equation |
| \(\sqrt{5}\) | B1 | Obtain correct radius a.e.f., condone decimals |
| \( | z-3-3\text{i} | =\sqrt{5}\) |
| A1ft | Obtain correct answer from their centre and radius; \(z=x+\text{i}y\) is acceptable | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Circle, centre in 1st quadrant, not touching or intersecting axes | B1 | Centre need not be \((3,3)\) |
| Straight line with negative slope through \((3,3)\) | B1 | \((3,3)\) may be implied by working from (i); must be longer than the diameter. N.B. No circle drawn, B1 for \(l\) may be earned |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2+5\text{i}\) and \(4+\text{i}\) | M1 | Gradient of \(l\) is \(-2\) used or attempt to solve Cartesian equations for \(C\) and \(l\); M0 if equations for \(C\) and/or \(l\) contain i |
| A1 A1 | Obtain correct answers, must be complex numbers | |
| [3] |
## Question 6:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3+3\text{i}$ | B1 | Obtain centre as a complex number, allow $(3,3)$ but not $(3,3\text{i})$; give if correct in locus equation |
| $\sqrt{5}$ | B1 | Obtain correct radius a.e.f., condone decimals |
| $|z-3-3\text{i}|=\sqrt{5}$ | M1 | Use correct form for locus |
| | A1ft | Obtain correct answer from their centre and radius; $z=x+\text{i}y$ is acceptable |
| **[4]** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Circle, centre in 1st quadrant, not touching or intersecting axes | B1 | Centre need not be $(3,3)$ |
| Straight line with negative slope through $(3,3)$ | B1 | $(3,3)$ may be implied by working from (i); must be longer than the diameter. N.B. No circle drawn, B1 for $l$ may be earned |
| **[2]** | | |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2+5\text{i}$ and $4+\text{i}$ | M1 | Gradient of $l$ is $-2$ used or attempt to solve Cartesian equations for $C$ and $l$; M0 if equations for $C$ and/or $l$ contain i |
| | A1 A1 | Obtain correct answers, must be complex numbers |
| **[3]** | | |
---6 In an Argand diagram the points $A$ and $B$ represent the complex numbers $5 + 4 \mathrm { i }$ and $1 + 2 \mathrm { i }$ respectively.\\
(i) Given that $A$ and $B$ are the ends of a diameter of a circle $C$, find the equation of $C$ in complex number form.
The perpendicular bisector of $A B$ is denoted by $l$.\\
(ii) Sketch $C$ and $l$ on a single Argand diagram.\\
(iii) Find the complex numbers represented by the points of intersection of $C$ and $l$.
\hfill \mbox{\textit{OCR FP1 2016 Q6 [9]}}