| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Perpendicular bisector locus |
| Difficulty | Standard +0.3 Part (a) is a routine quadratic equation in complex numbers using the quadratic formula. Part (b)(i) requires recognizing that |z| = |z - (4+3i)| represents the perpendicular bisector of the origin and point (4,3), which is standard locus work. Part (b)(ii) involves finding the minimum modulus on this line, requiring geometric insight but only basic optimization. This is a multi-part question with standard techniques and moderate problem-solving, slightly easier than average A-level. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a) EITHER: Use quadratic formula to solve for \(z\) | M1 | |
| Use \(i^2 = -1\) | M1 | |
| Obtain a correct answer in any form, simplified as far as \((-2 \pm i\sqrt{8})/2i\) | A1 | |
| Multiply numerator and denominator by \(i\), or equivalent | M1 | |
| Obtain final answers \(\sqrt{2} + i\) and \(-\sqrt{2} + i\) | A1 | |
| OR: Substitute \(x + iy\) and equate real and imaginary parts to zero | M1 | |
| Use \(i^2 = -1\) | M1 | |
| Obtain \(-2xy + 2x = 0\) and \(x^2 - y^2 + 2y - 3 = 0\), or equivalent | A1 | |
| Solve for \(x\) and \(y\) | M1 | |
| Obtain final answers \(\sqrt{2} + i\) and \(-\sqrt{2} + i\) | A1 | [5] |
| (b)(i) EITHER: Show the point representing \(4 + 3i\) in relatively correct position | B1 | |
| Show the perpendicular bisector of the line segment joining this point to the origin | B1♦ | [2] |
| OR: Obtain correct Cartesian equation of the locus in any form, e.g. \(8x + 6y = 25\) | B1 | |
| Show this line | B1♦ | |
| (ii) State or imply the relevant point is represented by \(2 + 1.5i\) or is at \((2, 1.5)\) | B1 | |
| Obtain modulus \(2.5\) | B1♦ | |
| Obtain argument \(0.64\) (or \(36.9°\)) (allow decimals in [0.64, 0.65] or [36.8, 36.9]) | B1♦ | [3] |
**(a)** **EITHER:** Use quadratic formula to solve for $z$ | M1 |
Use $i^2 = -1$ | M1 |
Obtain a correct answer in any form, simplified as far as $(-2 \pm i\sqrt{8})/2i$ | A1 |
Multiply numerator and denominator by $i$, or equivalent | M1 |
Obtain final answers $\sqrt{2} + i$ and $-\sqrt{2} + i$ | A1 |
**OR:** Substitute $x + iy$ and equate real and imaginary parts to zero | M1 |
Use $i^2 = -1$ | M1 |
Obtain $-2xy + 2x = 0$ and $x^2 - y^2 + 2y - 3 = 0$, or equivalent | A1 |
Solve for $x$ and $y$ | M1 |
Obtain final answers $\sqrt{2} + i$ and $-\sqrt{2} + i$ | A1 | [5]
**(b)(i)** **EITHER:** Show the point representing $4 + 3i$ in relatively correct position | B1 |
Show the perpendicular bisector of the line segment joining this point to the origin | B1♦ | [2]
**OR:** Obtain correct Cartesian equation of the locus in any form, e.g. $8x + 6y = 25$ | B1 |
Show this line | B1♦ | |
**(ii)** State or imply the relevant point is represented by $2 + 1.5i$ or is at $(2, 1.5)$ | B1 |
Obtain modulus $2.5$ | B1♦ |
Obtain argument $0.64$ (or $36.9°$) (allow decimals in [0.64, 0.65] or [36.8, 36.9]) | B1♦ | [3]10
\begin{enumerate}[label=(\alph*)]
\item Showing all necessary working, solve the equation $\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.
\item \begin{enumerate}[label=(\roman*)]
\item On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation $| z | = | z - 4 - 3 \mathrm { i } |$.
\item Find the complex number represented by the point on the locus where $| z |$ is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2016 Q10 [10]}}