Which of the following is equal to $e^{ar{z}}$, where $z = x + iy$ with $x$ and $y$ real numbers?
A - HSC - SSCE Mathematics Extension 2 - Question 8 - 2024 - Paper 1
Question 8
Which of the following is equal to $e^{ar{z}}$, where $z = x + iy$ with $x$ and $y$ real numbers?
A. $\bar{e}$
B. $e^{-z}$
C. $e^{2x}e^{y}$
D. $e^{-2x}e^{\bar{z}}$
Worked Solution & Example Answer:Which of the following is equal to $e^{ar{z}}$, where $z = x + iy$ with $x$ and $y$ real numbers?
A - HSC - SSCE Mathematics Extension 2 - Question 8 - 2024 - Paper 1
Step 1
Identify the expression: $e^{ar{z}}$
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Answer
Given that z=x+iy, the conjugate of z, denoted as ar{z}, is x−iy. Hence, we need to find e^{ar{z}} = e^{x - iy}.
Step 2
Use the properties of exponents
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Answer
Using the exponent rule, we can split this into:
e^{ar{z}} = e^x imes e^{-iy}.
Step 3
Apply Euler's formula for $e^{-iy}$
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Answer
According to Euler's formula, we have:
e^{-iy} = rac{1}{e^{iy}} = rac{1}{ ext{cos}(y) + i ext{sin}(y)} = ext{cos}(y) - i ext{sin}(y). Therefore, we combine this with ex to express:
e^{ar{z}} = e^x( ext{cos}(y) - i ext{sin}(y)).
Step 4
Compare with the given options
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Answer
Now we analyze the answer choices:
A: eˉ does not apply here.
B: e−z=e−x−iy=e−x(e−iy), which is incorrect.
C: e2xey does not match either.
D: e−2xezˉ does not correspond.
Thus, the correct choice is A, eˉ.
