| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a standard Further Maths loci question requiring students to interpret modulus inequalities geometrically and find a perpendicular bisector equation. Part (a) involves recognizing |z+i| ≤ |z-2| as a half-plane and converting to Cartesian form (routine algebraic manipulation). Part (b) requires shading the intersection of a disc and exterior of a circle—straightforward geometric interpretation. While this is Further Maths content, these are textbook-standard loci techniques with no novel problem-solving required, making it slightly easier than an average A-level question overall. |
| Spec | 4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Line drawn, perpendicular to line segment joining \((0,-1)\) and \((2,0)\) | M1 | Line needs negative gradient with \( |
| Region below line indicated as being the required region | A1 | Exact perpendicularity not needed, but should be approximately perpendicular |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(m = -1/(\frac{1}{2}) = -2\) | M1 | |
| \(4x + 2y - 3 = 0\) | A1 | Explicitly stated. Note must be in required form \(ax+by+c=0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Circle centre \((-1, 0)\) radius \(3\) or circle centre \((0, 2)\) radius \(2\). Both circles correct | M1 | Radius can be implied by axis labels or tick-marks. If M0A0 then SC1 for two circles with correct radii but centres \((1,0)\) and \((0,-2)\) |
| Both circles correct | A1 | |
| Correct region shaded or otherwise indicated | A1 | Region inside circle with radius \(3\) but outside circle with radius \(2\) |
## Question 4:
### Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Line drawn, perpendicular to line segment joining $(0,-1)$ and $(2,0)$ | M1 | Line needs negative gradient with $|\text{gradient}|>1$ and to intersect $y$-axis at positive value. If "shading out" used, indication that required region is below line needed, e.g. "R" or "This region" |
| Region below line indicated as being the required region | A1 | Exact perpendicularity not needed, but should be approximately perpendicular |
### Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $m = -1/(\frac{1}{2}) = -2$ | M1 | |
| $4x + 2y - 3 = 0$ | A1 | Explicitly stated. Note must be in required form $ax+by+c=0$ |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Circle centre $(-1, 0)$ radius $3$ or circle centre $(0, 2)$ radius $2$. Both circles correct | M1 | Radius can be implied by axis labels or tick-marks. If M0A0 then SC1 for two circles with correct radii but centres $(1,0)$ and $(0,-2)$ |
| Both circles correct | A1 | |
| Correct region shaded or otherwise indicated | A1 | Region inside circle with radius $3$ but outside circle with radius $2$ |4
\begin{enumerate}[label=(\alph*)]
\item A locus $C _ { 1 }$ is defined by $C _ { 1 } = \{ \mathrm { z } : | \mathrm { z } + \mathrm { i } | \leqslant \mid \mathrm { z } - 2 \}$.
\begin{enumerate}[label=(\roman*)]
\item Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing $C _ { 1 }$.
\item Find the cartesian equation of the boundary line of the region representing $C _ { 1 }$, giving your answer in the form $a x + b y + c = 0$.
\end{enumerate}\item A locus $C _ { 2 }$ is defined by $C _ { 2 } = \{ \mathrm { z } : | \mathrm { z } + 1 | \leqslant 3 \} \cap \{ \mathrm { z } : | \mathrm { z } - 2 \mathrm { i } | \geqslant 2 \}$.
Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing $C _ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS 2021 Q4 [7]}}