| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus-argument form conversions |
| Difficulty | Standard +0.3 Part (i) is a straightforward conversion from modulus-argument form to Cartesian form, adding the complex numbers, then converting back—routine A-level Further Maths technique. Part (ii) requires recognizing that arg(z-5)=2π/3 defines a ray from (5,0), then finding the minimum distance from origin to this ray using basic geometry or perpendicular distance, which is a standard locus problem. Both parts are textbook exercises requiring no novel insight. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
5. In this question you must show detailed reasoning.\\
(i) Given that
$$z _ { 1 } = 6 \left( \cos \left( \frac { \pi } { 3 } \right) + i \sin \left( \frac { \pi } { 3 } \right) \right) \quad \text { and } \quad z _ { 2 } = 6 \sqrt { 3 } \left( \cos \left( \frac { 5 \pi } { 6 } \right) + i \sin \left( \frac { 5 \pi } { 6 } \right) \right)$$
show that
$$z _ { 1 } + z _ { 2 } = 12 \left( \cos \left( \frac { 2 \pi } { 3 } \right) + i \sin \left( \frac { 2 \pi } { 3 } \right) \right)$$
(ii) Given that
$$\arg ( z - 5 ) = \frac { 2 \pi } { 3 }$$
determine the least value of $| \boldsymbol { z } |$ as $Z$ varies.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2024 Q5 [6]}}