It is known that a particular complex number $z$ is NOT a real number - HSC - SSCE Mathematics Extension 2 - Question 6 - 2022 - Paper 1
Question 6
It is known that a particular complex number $z$ is NOT a real number.
Which of the following could be true for this number $z$?
A. $\overline{z} = iz$
B. $\overl... show full transcript
Worked Solution & Example Answer:It is known that a particular complex number $z$ is NOT a real number - HSC - SSCE Mathematics Extension 2 - Question 6 - 2022 - Paper 1
Step 1
$\overline{z} = iz$
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Answer
This can be true because if we let z=x+iy, then z=x−iy. Setting z=iz gives:
x−iy=i(x+iy)
which simplifies to x−iy=−y+ix.
This leads to two equations:
x=−y
−y=x.
This means that z can be expressed in terms of a purely imaginary number, thus this statement can be true.
Step 2
$\overline{z} = |z|$
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Answer
This cannot be true because ∣z∣ is always a non-negative real number, while z would generally yield a complex number unless z is itself a real number. As z is not a real number, this statement is false.
Step 3
$\text{Re}(iz) = \text{Im}(z)$
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Answer
Letting z=x+iy, we have:
iz=i(x+iy)=−y+ix.
Thus, Re(iz)=−y and Im(z)=y.
This leads to the condition −y=y, which is true if y=0. However, since z is not a real number, this can't generally hold true.
Step 4
$\text{Arg}(\frac{z}{3}) = \text{Arg}(z)$
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Answer
This statement holds true since the argument of a complex number must be invariant to such a scalar multiplication (in this case 3). Therefore, if z is not a real number, this relationship is consistent and valid, confirming that this could be true.
