A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 3 - 2023 - Paper 1
Question 3
A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram.
Which of the following complex numbers is equal to $ar{z}$ ?
A. $-z$ ... show full transcript
Worked Solution & Example Answer:A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 3 - 2023 - Paper 1
Step 1
Identify the Meaning of $ar{z}$
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Answer
The notation ar{z} refers to the complex conjugate of the complex number z. If z=x+yi, then ar{z} = x - yi. On the unit circle, the complex conjugate is obtained by reflecting z across the real axis.
Step 2
Evaluate Possible Options
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Answer
Now, we need to analyze each option:
A. −z: This is the negative of z and does not equal ar{z}.
B. −z2: This does not represent the complex conjugate as it squares z and then takes the negative.
C. −z3: Similarly, negative of the cube of z does not yield the conjugate.
D. z4: While this represents another power of z, it does not equal ar{z} either.
In summary, negating z does not reflect the corresponding operation for finding ar{z}.
Step 3
Select the Correct Answer
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Answer
The correct answer is therefore option B. −z2, as z2 has the possibility of forming the related geometry necessary to compute ar{z} on the argument associated with the unit circle differences.
