Let R be the region in the complex plane defined by
1 < Re(z) ≤ 3 and
π/6 ≤ Arg(z) < π/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
π/6 ≤ Arg(z) < π/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
π/6 ≤ Arg(z) < π/3 - HSC - SSCE Mathematics Extension 2 - Question 1 - 2022 - Paper 1
Step 1
1 < Re(z) ≤ 3
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Answer
This inequality indicates that the real part of the complex number z (denoted as Re(z)) lies between 1 and 3. In the complex plane, this corresponds to a vertical strip bounded by the vertical lines x = 1 and x = 3, where x represents Re(z).
Step 2
π/6 ≤ Arg(z) < π/3
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Answer
The argument of the complex number z (denoted as Arg(z)) restricts the angle. The angles π/6 and π/3 correspond to 30° and 60°, respectively. Therefore, the region is bounded by the rays originating from the origin at these angles, leading to a triangular shape in the first quadrant.
Step 3
Combining the inequalities
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Answer
The intersection of the vertical strip and the triangular region gives us the required region R. This area will be in the first quadrant between the lines at angles of π/6 and π/3 and between the vertical lines at Re(z) = 1 and Re(z) = 3.
Step 4
Identifying the correct diagram
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Answer
Among the available diagrams, the one that best confines this region, including the boundaries defined by the inequalities, corresponds to option A.
