| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Challenging +1.8 This is a challenging Further Maths AS question requiring sophisticated understanding of complex loci. Students must recognize Cā as a perpendicular bisector, convert the arg condition to a line equation, solve simultaneously for the intersection in terms of parameter d, then interpret a geometric constraint. The multi-step algebraic manipulation with parameters and the conceptual demand of loci intersection place this well above average difficulty. |
| Spec | 4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (a) | C is (represented by) the line x = 2d2 + 18 |
| Answer | Marks |
|---|---|
| 2d2 + 18 + (2d2 ā 12d + 21)i | B1 |
| Answer | Marks |
|---|---|
| A1 | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | Seen or implied in solution |
| Answer | Marks |
|---|---|
| Must be in complex number form | Half line starting at (12d, c) and with |
| Answer | Marks |
|---|---|
| Alternative Method | B1 |
| Answer | Marks |
|---|---|
| A1 | SOI |
| Answer | Marks |
|---|---|
| Must be in complex number form | Could be shown by making angle |
Question 8:
8 | (a) | C is (represented by) the line x = 2d2 + 18
1
C is (represented by) the (half-)line y = x + c
2
3 = 12d + c
y = x + 3 ā 12d
When x = 2d2 + 18, y = 2d2 ā 12d + 21
2d2 + 18 + (2d2 ā 12d + 21)i | B1
M1
M1
A1
M1
A1 | 3.1a
3.1a
3.1a
1.1
1.1
3.2a | Seen or implied in solution
For understanding the C is a line or
2
half-line whose gradient is 1.
Complete line would pass through the
point (12d, 3) or 12d + 3i.
Attempt at y coordinate
Must be in complex number form | Half line starting at (12d, c) and with
angle
šš
4
or eg 2(d2 + 9) + (2(d ā 3)2 + 3)i
Alternative Method | B1
M1
M1
M1
M1
A1 | SOI
Correct half line needed here
Follow through C line and start of C
1 2
half line
Attempt at y coordinate using base of
triangle = height of triangle and adding
on 3i
Must be in complex number form | Could be shown by making angle
with positive x direction šš
4
or eg 2(d2 + 9) + (2(d ā 3)2 + 3)i
C is (represented by) the line x = 2d2 + 18
1
C is (represented by) the (half-)line starting at the
2
point 12d + 3i
C half line has gradient 1
2
Right-angled triangle indicated with base length
(2d2 + 18) ā 12d
y coordinate at 3 + ((2d2 + 18) ā 12d)
POI at 2d2 + 18 + (2d2 ā 12d + 21)i
[6]
B1
M1
M1
M1
M1
A1
SOI
Correct half line needed here
Follow through C line and start of C
1 2
half line
Attempt at y coordinate using base of
triangle = height of triangle and adding
on 3i
Must be in complex number form
Could be shown by making angle
with positive x direction ššTwo loci, $C_1$ and $C_2$, are defined by
$$C_1 = \{z:|z| = |z - 4d^2 - 36|\}$$
$$C_2 = \left\{z:\arg(z - 12d - 3\text{i}) = \frac{1}{4}\pi\right\}$$
where $d$ is a real number.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $d$, the complex number which is represented on an Argand diagram by the point of intersection of $C_1$ and $C_2$.
[You may assume that $C_1 \cap C_2 \neq \emptyset$.] [6]
\item Explain why the solution found in part (a) is not valid when $d = 3$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS 2020 Q8 [8]}}