| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 6 |
| 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 straightforward Argand diagram question requiring students to sketch a circle centered at (4,3) with radius 2, intersect it with the half-plane Re(z)≤3, then find the maximum argument geometrically. Part (a) is routine visualization, and part (b) requires identifying the tangent point and calculating an inverse tangent—standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(4 + 3i\). Accept a curved shape with correct point roughly in the middle | B1 | |
| Show a circle with radius 2 and centre not at the origin. The shape should be consistent with their scales | B1 | |
| Show correct vertical line. Enough to meet correct circle twice or complete line for any other circle | B1 | |
| Shade the correct region on a correct diagram. Any other shading must be accompanied by words to explain which region is required | B1 | Need some indication of scale e.g. label the centre, mark key points on the axes or dashes on the axes. Condone dotted lines in place of solid lines. Condone correct shaded shape but not an entire circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a complete method for finding the greatest value of \(\arg(z)\), e.g. \(\tan^{-1}\frac{3}{4} + \sin^{-1}\frac{2}{5}\) \((0.6435 + 0.4115)\) | M1 | Complete method for finding the greatest value of \(\arg(z)\) |
| Obtain answer 1.06 (accept 1.055 or 1.056) radians or \(60.45°\) (accept \(60.4°\) or \(60.5°\)) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Tangent to circle passing through origin has equation \(y = mx\). The equation \((x-4)^2 + (y-3)^2 = 4\) will have one root. Hence \((1+m^2)x^2 - (8+6m)x + 21 = 0\), discriminant \(= 0 = 48m^2 - 96m + 20\) and \(m = \frac{6 \pm \sqrt{21}}{6}\) with the larger value needed to give greatest \(\arg(z)\). Required angle is \(\tan^{-1}m\) | M1 | Complete method for finding the greatest value of \(\arg(z)\) |
| Obtain answer 1.06 radians or \(60.45°\) | A1 | Accept 1.055 or 1.056 radians. Accept \(60.4°\) or \(60.5°\) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $4 + 3i$. Accept a curved shape with correct point roughly in the middle | B1 | |
| Show a circle with radius 2 and centre not at the origin. The shape should be consistent with their scales | B1 | |
| Show correct vertical line. Enough to meet correct circle twice or complete line for any other circle | B1 | |
| Shade the correct region on a correct diagram. Any other shading must be accompanied by words to explain which region is required | B1 | Need some indication of scale e.g. label the centre, mark key points on the axes or dashes on the axes. Condone dotted lines in place of solid lines. Condone correct shaded shape but not an entire circle |
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## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a complete method for finding the greatest value of $\arg(z)$, e.g. $\tan^{-1}\frac{3}{4} + \sin^{-1}\frac{2}{5}$ $(0.6435 + 0.4115)$ | M1 | Complete method for finding the greatest value of $\arg(z)$ |
| Obtain answer 1.06 (accept 1.055 or 1.056) radians or $60.45°$ (accept $60.4°$ or $60.5°$) | A1 | |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Tangent to circle passing through origin has equation $y = mx$. The equation $(x-4)^2 + (y-3)^2 = 4$ will have one root. Hence $(1+m^2)x^2 - (8+6m)x + 21 = 0$, discriminant $= 0 = 48m^2 - 96m + 20$ and $m = \frac{6 \pm \sqrt{21}}{6}$ with the larger value needed to give greatest $\arg(z)$. Required angle is $\tan^{-1}m$ | M1 | Complete method for finding the greatest value of $\arg(z)$ |
| Obtain answer 1.06 radians or $60.45°$ | A1 | Accept 1.055 or 1.056 radians. Accept $60.4°$ or $60.5°$ |4
\begin{enumerate}[label=(\alph*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z - 4 - 3 \mathrm { i } | \leqslant 2$ and $\operatorname { Re } z \leqslant 3$.
\item Find the greatest value of $\arg z$ for points in this region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q4 [6]}}