| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring students to sketch two standard loci (a half-line from arg and a perpendicular bisector from modulus equality) and find their intersection algebraically. While it requires multiple techniques and careful algebraic manipulation, the loci types are standard Further Pure content and the solution path is straightforward once the geometric setup is understood. The 'show detailed reasoning' requirement and exact form answer add some rigour, but this remains a typical Further Maths exercise rather than requiring novel insight. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | 1st locus: line at 135° to +ve real axis |
| Answer | Marks |
|---|---|
| where P represents −2+i | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1.1 | Allow M1 for errors in P | |
| 9 | (b) | DR |
| Answer | Marks |
|---|---|
| 6 6 | B1 |
| Answer | Marks |
|---|---|
| [5] | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | oe |
Question 9:
9 | (a) | 1st locus: line at 135° to +ve real axis
half line only starting at 2
2nd locus: perp bisector of OP
where P represents −2+i | M1
A1
M1
A1
[4] | 1.1
1.1
1.1
1.1 | Allow M1 for errors in P
9 | (b) | DR
1st locus is line y=2−x
Midpoint of OP is (−1, 1)
2
Gradient of perpendicular bisector is 2
Equation is y−1 =2(x+1)
2
⇒ y=2x+21
2
Hence 2−x=2x+21
2
⇒x=−1, y=13
6 6
⇒w=−1+13i
6 6 | B1
B1FT
B1FT
M1
A1
[5] | 3.1a
1.1
3.1a
1.1
3.2a | oe
midpoint and gradient ft their (−2, 1)
oe
Solving simultaneously
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\begin{enumerate}[label=(\alph*)]
\item On a single Argand diagram, sketch the loci defined by
\begin{itemize}
\item $\arg ( z - 2 ) = \frac { 3 } { 4 } \pi$,
\item $\quad | z | = | z + 2 - i |$.
\item In this question you must show detailed reasoning.
\end{itemize}
The point of intersection of the two loci in part (a) represents the complex number $w$.
Find $w$, giving your answer in exact form.
\section*{END OF QUESTION PAPER}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2021 Q9 [9]}}