| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.8 This is a Further Maths question requiring students to find the intersection of a circle and a half-line in the complex plane. While sketching the loci is routine, finding exact coordinates requires setting up and solving a system involving both Cartesian and polar forms, demanding careful algebraic manipulation and geometric understanding—moderately challenging but within standard Further Maths scope. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | (i) |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | circle |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | (ii) |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | line at 45° to real axis |
| Answer | Marks | Guidance |
|---|---|---|
| correct half-line indicated | with evidence | |
| 7 | (b) | circle is (x−1)2 +(y−3)2 =9 |
| Answer | Marks |
|---|---|
| 2 2 2 2 | B1ft |
| Answer | Marks |
|---|---|
| [6] | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | ft their centre |
Question 7:
7 | (a) | (i) | 4i | M1
A1
A1
[3] | 3.1a
1.1
1.1 | circle
centre 1 + 3i indicated
touching real axis
7 | (a) | (ii) | B1
B1
B1
[3] | 3.1a
1.1
1.1 | line at 45° to real axis
through −4
correct half-line indicated | with evidence
7 | (b) | circle is (x−1)2 +(y−3)2 =9
line is y = x+4
⇒(x−1)2 +(x+1)2 =9
⇒2x2 =7⇒ x=± 7
2
⇒ y =4± 7
2
( ) ( )
so z =− 7 + 4− 7 i or 7 + 4+ 7 i
2 2 2 2 | B1ft
B1ft
M1
A1
A1
A1
[6] | 3.1a
3.1a
1.1
1.1
1.1
3.2a | ft their centre
ft their −4
eliminating y (or x)7
\begin{enumerate}[label=(\alph*)]
\item Sketch on a single Argand diagram
\begin{enumerate}[label=(\roman*)]
\item the set of points for which $| z - 1 - 3 i | = 3$,
\item the set of points for which $\arg ( z + 4 ) = \frac { 1 } { 4 } \pi$.
\end{enumerate}\item Find, in exact form, the two values of $z$ for which $| z - 1 - 3 i | = 3$ and $\arg ( z + 4 ) = \frac { 1 } { 4 } \pi$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2019 Q7 [12]}}