Let R be the region in the complex plane defined by
1 < Re(z) ≤ 3 and
\frac{\pi}{6} ≤ 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) ≤ 3 and
\frac{\pi}{6} ≤ Arg(z) < \frac{\pi}{3}.
Which diagram best represents the region R?
Worked Solution & Example Answer:Let R be the region in the complex plane defined by
1 < Re(z) ≤ 3 and
\frac{\pi}{6} ≤ Arg(z) < \frac{\pi}{3} - HSC - SSCE Mathematics Extension 2 - Question 1 - 2022 - Paper 1
Step 1
Define the constraints for the region R
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Answer
The region R is defined by two constraints:
The real part of the complex number z (denoted as Re(z)) must lie between 1 and 3:
1<Re(z)≤3
The argument of z (denoted as Arg(z)) must lie between \frac{\pi}{6} and \frac{\pi}{3}:
6π≤Arg(z)<3π
Step 2
Sketch the region based on the constraints
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Answer
To visualize this region:
Start by drawing vertical lines at Re(z) = 1 and Re(z) = 3.
The vertical line at Re(z) = 1 is not included (open boundary), while the line at Re(z) = 3 is included (closed boundary).
Next, for the argument:
The angle \frac{\pi}{6} corresponds to 30 degrees, and \frac{\pi}{3} corresponds to 60 degrees, creating a wedge in the upper half of the complex plane.
The intersection of these two regions defines R.
Step 3
Select the correct diagram
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Answer
Based on the analysis above, the diagram that best represents the region R is diagram A, which includes the specified boundaries and angles.
