| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Complex number loci on Argand diagrams |
| Difficulty | Standard +0.3 Part (a) requires routine complex number division and applying modulus/argument properties (|w²|=|w|², arg(w²)=2arg(w)), which are standard techniques. Part (b) involves sketching a circle and perpendicular bisector (standard loci), then finding intersections algebraically—straightforward application of known methods with no novel insight required. Slightly easier than average due to being mostly procedural. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| Find \(w\) using conjugate of \(1 + 3i\) | M1 |
| Obtain \(\frac{7 - i}{5}\) or equivalent | A1 |
| Square \(x + iy\) form to find \(w^2\) | M1 |
| Obtain \(w^2 = \frac{48 - 14i}{25}\) and confirm modulus is 2 | A1 |
| Use correct process for finding argument of \(w^2\) | M1 |
| Obtain \(-0.284\) radians or \(-16.3°\) | A1 |
| Answer | Marks |
|---|---|
| Find \(w\) using conjugate of \(1 + 3i\) | M1 |
| Obtain \(\frac{7 - i}{5}\) or equivalent | A1 |
| Find modulus of \(w\) and hence of \(w^2\) | M1 |
| Confirm modulus is 2 | A1 |
| Find argument of \(w\) and hence of \(w^2\) | M1 |
| Obtain \(-0.284\) radians or \(-16.3°\) | A1 |
| Answer | Marks |
|---|---|
| Square both sides to obtain \((-8 + 6i)w^2 = -12 + 16i\) | B1 |
| Find \(w^2\) using relevant conjugate | M1 |
| Use correct process for finding modulus of \(w^2\) | M1 |
| Confirm modulus is 2 | A1 |
| Use correct process for finding argument of \(w^2\) | M1 |
| Obtain \(-0.284\) radians or \(-16.3°\) | A1 |
| Answer | Marks |
|---|---|
| Find modulus of LHS and RHS | M1 |
| Find argument of LHS and RHS | M1 |
| Obtain \(\sqrt{10} \, e^{i.249}\) \(w = \sqrt{20} \, e^{i.107}\) or equivalent | A1 |
| Obtain \(w = \sqrt{2} \, e^{i0.1419}\) or equivalent | A1 |
| Use correct process for finding \(w^2\) | M1 |
| Obtain 2 and \(-0.284\) radians or \(-16.3°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Find moduli of \(2 + 4i\) and \(1 + 3i\) | M1 | |
| Obtain \(\sqrt{20}\) and \(\sqrt{10}\) | A1 | |
| Obtain \( | w^2 | = 2\) correctly |
| Find \(\arg(2 + 4i)\) and \(\arg(1 + 3i)\) | M1 | |
| Use correct process for \(\arg(w^2)\) | A1 | |
| Obtain \(-0.284\) radians or \(-16.3°\) | A1 |
| Answer | Marks |
|---|---|
| Let \(w = a + ib\), form and solve simultaneous equations in \(a\) and \(b\) | M1 |
| \(a = \frac{7}{5}\) and \(b = -\frac{1}{5}\) | A1 |
| Find modulus of \(w\) and hence of \(w^2\) | M1 |
| Confirm modulus is 2 | A1 |
| Find argument of \(w\) and hence of \(w^2\) | M1 |
| Obtain \(-0.284\) radians or \(-16.3°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Find \(w\) using conjugate of \(1 + 3i\) | M1 | |
| Obtain \(\frac{7 - i}{5}\) or equivalent | A1 | |
| Use \( | w^2 | = w\bar{w}\) |
| Confirm modulus is 2 | A1 | |
| Find argument of \(w\) and hence of \(w^2\) | M1 | |
| Obtain \(-0.284\) radians or \(-16.3°\) | A1 | [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Draw circle with centre the origin and radius 5 | B1 | |
| Draw straight line parallel to imaginary axis in correct position | B1 | |
| Use relevant trigonometry on a correct diagram to find argument(s) | M1 | |
| Obtain \(5e^{\pm i \frac{2\pi}{3}}\) or equivalents in required form | A1 | [4] |
**(a)**
**Either**
Find $w$ using conjugate of $1 + 3i$ | M1 |
Obtain $\frac{7 - i}{5}$ or equivalent | A1 |
Square $x + iy$ form to find $w^2$ | M1 |
Obtain $w^2 = \frac{48 - 14i}{25}$ and confirm modulus is 2 | A1 |
Use correct process for finding argument of $w^2$ | M1 |
Obtain $-0.284$ radians or $-16.3°$ | A1 |
**Or 1**
Find $w$ using conjugate of $1 + 3i$ | M1 |
Obtain $\frac{7 - i}{5}$ or equivalent | A1 |
Find modulus of $w$ and hence of $w^2$ | M1 |
Confirm modulus is 2 | A1 |
Find argument of $w$ and hence of $w^2$ | M1 |
Obtain $-0.284$ radians or $-16.3°$ | A1 |
**Or 2**
Square both sides to obtain $(-8 + 6i)w^2 = -12 + 16i$ | B1 |
Find $w^2$ using relevant conjugate | M1 |
Use correct process for finding modulus of $w^2$ | M1 |
Confirm modulus is 2 | A1 |
Use correct process for finding argument of $w^2$ | M1 |
Obtain $-0.284$ radians or $-16.3°$ | A1 |
**Or 3**
Find modulus of LHS and RHS | M1 |
Find argument of LHS and RHS | M1 |
Obtain $\sqrt{10} \, e^{i.249}$ $w = \sqrt{20} \, e^{i.107}$ or equivalent | A1 |
Obtain $w = \sqrt{2} \, e^{i0.1419}$ or equivalent | A1 |
Use correct process for finding $w^2$ | M1 |
Obtain 2 and $-0.284$ radians or $-16.3°$ | A1 |
**Or 4**
Find moduli of $2 + 4i$ and $1 + 3i$ | M1 |
Obtain $\sqrt{20}$ and $\sqrt{10}$ | A1 |
Obtain $|w^2| = 2$ correctly | A1 |
Find $\arg(2 + 4i)$ and $\arg(1 + 3i)$ | M1 |
Use correct process for $\arg(w^2)$ | A1 |
Obtain $-0.284$ radians or $-16.3°$ | A1 |
**Or 5**
Let $w = a + ib$, form and solve simultaneous equations in $a$ and $b$ | M1 |
$a = \frac{7}{5}$ and $b = -\frac{1}{5}$ | A1 |
Find modulus of $w$ and hence of $w^2$ | M1 |
Confirm modulus is 2 | A1 |
Find argument of $w$ and hence of $w^2$ | M1 |
Obtain $-0.284$ radians or $-16.3°$ | A1 |
**Or 6**
Find $w$ using conjugate of $1 + 3i$ | M1 |
Obtain $\frac{7 - i}{5}$ or equivalent | A1 |
Use $|w^2| = w\bar{w}$ | M1 |
Confirm modulus is 2 | A1 |
Find argument of $w$ and hence of $w^2$ | M1 |
Obtain $-0.284$ radians or $-16.3°$ | A1 | [6]
**(b)**
Draw circle with centre the origin and radius 5 | B1 |
Draw straight line parallel to imaginary axis in correct position | B1 |
Use relevant trigonometry on a correct diagram to find argument(s) | M1 |
Obtain $5e^{\pm i \frac{2\pi}{3}}$ or equivalents in required form | A1 | [4]9
\begin{enumerate}[label=(\alph*)]
\item It is given that $( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }$. Showing all necessary working, prove that the exact value of $\left| w ^ { 2 } \right|$ is 2 and find $\arg \left( w ^ { 2 } \right)$ correct to 3 significant figures.
\item On a single Argand diagram sketch the loci $| z | = 5$ and $| z - 5 | = | z |$. Hence determine the complex numbers represented by points common to both loci, giving each answer in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2015 Q9 [10]}}