| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.3 Part (a) is a straightforward simultaneous equations problem with complex numbers requiring basic algebraic manipulation. Part (b) involves standard loci (circle and half-line), sketching them, and finding minimum distance between loci—a routine geometric optimization problem. Both parts use well-practiced techniques with no novel insights required, making this slightly easier than average. |
| Spec | 4.02e 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) EITHER: Solve for \(u\) or for \(v\) | M1 | |
| Obtain \(u = \frac{2i-6}{1-2i}\) or \(v = \frac{5}{1-2i}\), or equivalent | A1 | |
| Either: Multiply a numerator and denominator by conjugate of denominator, or equivalent | M1 | |
| Or: Set \(u\) or \(v\) equal to \(x + iy\), obtain two equations by equating real and imaginary parts and solve for \(x\) or \(y\) | M1 | |
| Using \(a + ib\) and \(c + id\) for \(u\) and \(v\), equate real and imaginary parts and obtain four equations in \(a, b, c\) and \(d\) | M1 | |
| Obtain \(b + 2d = 2, a + 2c = 0, a + d = 0\) and \(-b + c = 3\), or equivalent | A1 | |
| Solve for one unknown | M1 | |
| Obtain final answer \(u = -2-2i\), or equivalent | A1 | |
| Obtain final answer \(v = 1 + 2i\), or equivalent | A1 | [5] |
| (b) Show a circle with centre \(-i\) | B1 | |
| Show a circle with radius \(1\) | B1 | |
| Show correct half line from \(2\) at an angle of \(\frac{3}{4}\pi\) to the real axis | B1 | |
| Use a correct method for finding the least value of the modulus | M1 | |
| Obtain final answer \(\frac{3}{\sqrt{2}}-1\), or equivalent, e.g. \(1.12\) (allow \(1.1\)) | A1 | [5] |
**(a)** **EITHER:** Solve for $u$ or for $v$ | M1 |
Obtain $u = \frac{2i-6}{1-2i}$ or $v = \frac{5}{1-2i}$, or equivalent | A1 |
Either: Multiply a numerator and denominator by conjugate of denominator, or equivalent | M1 |
Or: Set $u$ or $v$ equal to $x + iy$, obtain two equations by equating real and imaginary parts and solve for $x$ or $y$ | M1 |
Using $a + ib$ and $c + id$ for $u$ and $v$, equate real and imaginary parts and obtain four equations in $a, b, c$ and $d$ | M1 |
Obtain $b + 2d = 2, a + 2c = 0, a + d = 0$ and $-b + c = 3$, or equivalent | A1 |
Solve for one unknown | M1 |
Obtain final answer $u = -2-2i$, or equivalent | A1 |
Obtain final answer $v = 1 + 2i$, or equivalent | A1 | [5]
**(b)** Show a circle with centre $-i$ | B1 |
Show a circle with radius $1$ | B1 |
Show correct half line from $2$ at an angle of $\frac{3}{4}\pi$ to the real axis | B1 |
Use a correct method for finding the least value of the modulus | M1 |
Obtain final answer $\frac{3}{\sqrt{2}}-1$, or equivalent, e.g. $1.12$ (allow $1.1$) | A1 | [5]Throughout this question the use of a calculator is not permitted.
\begin{enumerate}[label=(\alph*)]
\item The complex numbers $u$ and $v$ satisfy the equations
$$u + 2v = 2i \quad \text{and} \quad iu + v = 3.$$
Solve the equations for $u$ and $v$, giving both answers in the form $x + iy$, where $x$ and $y$ are real. [5]
\item On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z + i| = 1$ and the locus representing complex numbers $w$ satisfying $\arg(w - 2) = \frac{\pi}{4}$. Find the least value of $|z - w|$ for points on these loci. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q8 [10]}}