Let R be the region in the complex plane defined by
$1 < Re(z) \\leq 3$ and
$\frac{\pi}{6} \\leq Arg(z) < \frac{\pi}{3}$ - HSC - SSCE Mathematics Extension 2 - Question 1 - 2022 - Paper 1
Question 1
Let R be the region in the complex plane defined by
$1 < Re(z) \\leq 3$ and
$\frac{\pi}{6} \\leq Arg(z) < \frac{\pi}{3}$.
Which diagram best represents the reg... show full transcript
Worked Solution & Example Answer:Let R be the region in the complex plane defined by
$1 < Re(z) \\leq 3$ and
$\frac{\pi}{6} \\leq Arg(z) < \frac{\pi}{3}$ - HSC - SSCE Mathematics Extension 2 - Question 1 - 2022 - Paper 1
Step 1
Define the boundaries for Re(z)
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Answer
The region defined by 1<Re(z)≤3 implies that the 'Real' part of the complex number z must lie between 1 and 3, inclusive of 3 but not including 1. This creates two vertical lines at Re(z)=1 (dashed line, indicating that points on this line are not included) and Re(z)=3 (solid line, indicating that points on this line are included).
Step 2
Define the boundaries for Arg(z)
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Answer
The argument condition 6π≤Arg(z)<3π defines an angular sector in the complex plane. The angle 6π radians corresponds to 30 degrees and 3π radians corresponds to 60 degrees. This means the region is limited to the sector between these two angles.
Step 3
Combine the regions to identify R
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Answer
The region R is therefore the area in the complex plane that is bounded by the vertical lines at Re(z)=1 and Re(z)=3, and within the angular sector defined by the angles 6π and 3π. The best representation of this region will be the diagram that shows this intersection correctly. Based on the marking scheme, this corresponds to diagram A.
