SSCE HSC Mathematics Extension 2 Complex numbers: It is known that a particular co

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. $\bar{z} = iz$ B. $\bar{z} = |... 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

$\bar{z} = iz$

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Answer

This equation states that the conjugate of the complex number $z$ is equal to $iz$ . For a complex number $z = x + iy$ , the conjugate is $\bar{z} = x - iy$ . Setting $x - iy = i(x + iy)$ leads to a contradiction unless $y = 0$ , meaning $z$ wouldn't be a non-real number. Therefore, this cannot be true.

Step 2

$\bar{z} = |z|$

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Answer

This statement asserts that the conjugate of $z$ equals the modulus of $z$ . Since $|z|$ is always a non-negative real number and the conjugate indicates that the imaginary component must be zero for equality, this option cannot be true for a non-real $z$ .

Step 3

$\text{Re}(iz) = \text{Im}(z)$

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Answer

For $z = x + iy$ , $iz = i(x + iy) = -y + ix$ . Thus, $\text{Re}(iz) = -y$ and $\text{Im}(z) = y$ . This leads to the equation $-y = y$ , which can hold true when $y = 0$ . However, since $z$ is non-real, this can be true under certain conditions, making it a potential candidate.

Step 4

$\text{Arg}(z^3) = \text{Arg}(z)$

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Answer

The argument of a complex number relates to its angle in the complex plane. For a non-real number $z$ , $\text{Arg}(z^3) = 3 \cdot \text{Arg}(z)$ . This implies that $\text{Arg}(z)$ can equal $\text{Arg}(z^3)$ only if $n \cdot \text{Arg}(z)$ is a multiple of $2\pi$ . Hence, this could potentially be true under certain circumstances.

It is known that a particular complex number $z$ is NOT a real number - HSC - SSCE Mathematics Extension 2 - Question 6 - 2022 - Paper 1

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

$\bar{z} = iz$

$\bar{z} = |z|$

$\text{Re}(iz) = \text{Im}(z)$

$\text{Arg}(z^3) = \text{Arg}(z)$

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