| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 Part (a) requires substituting w = x + iy into an equation involving conjugates and solving a system, which is standard A-level technique. Part (b) involves sketching standard loci (circle and sector) and finding maximum modulus through geometric reasoning or calculus - routine for Further Maths students. Both parts are multi-step but use well-practiced methods without requiring novel insight. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Substitute \(w = x + iy\) and state a correct equation in \(x\) and \(y\) | B1 | |
| Use \(i^2 = -1\) and equate real parts | M1 | |
| Obtain \(y = -2\) | A1 | |
| Equate imaginary parts and solve for \(x\) | M1 | |
| Obtain \(x = 2\sqrt{2}\), or equivalent, only | A1 | [5] |
| (b) Show a circle with centre \(2i\) | B1 | |
| Show a circle with radius 2 | B1 | |
| Show half line from \(-2\) at \(\frac{1}{4}\pi\) to real axis | B1 | |
| Shade the correct region | B1 | |
| Carry out a complete method for calculating the greatest value of \( | z | \) |
| Obtain answer 3.70 | A1 | [6] |
**(a)** Substitute $w = x + iy$ and state a correct equation in $x$ and $y$ | B1 |
Use $i^2 = -1$ and equate real parts | M1 |
Obtain $y = -2$ | A1 |
Equate imaginary parts and solve for $x$ | M1 |
Obtain $x = 2\sqrt{2}$, or equivalent, only | A1 | [5]
**(b)** Show a circle with centre $2i$ | B1 |
Show a circle with radius 2 | B1 |
Show half line from $-2$ at $\frac{1}{4}\pi$ to real axis | B1 |
Shade the correct region | B1 |
Carry out a complete method for calculating the greatest value of $|z|$ | M1 |
Obtain answer 3.70 | A1 | [6]\begin{enumerate}[label=(\alph*)]
\item The complex number $w$ is such that $\text{Re } w > 0$ and $w + 3w^* = iw^2$, where $w^*$ denotes the complex conjugate of $w$. Find $w$, giving your answer in the form $x + iy$, where $x$ and $y$ are real. [5]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ which satisfy both the inequalities $|z - 2i| \leq 2$ and $0 \leq \arg(z + 2) \leq \frac{1}{4}\pi$. Calculate the greatest value of $|z|$ for points in this region, giving your answer correct to 2 decimal places. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q9 [11]}}