Question

If the roots of the equation z² - i = 0 are α and β, then | Arg β - Arg α | =

• A. 2π
• B. π/2
• C. π
• D. π/4

Answer 1

The Correct Option is C

To solve the problem, we need to find the difference in arguments (angles) of the roots of the complex equation \( z^2 - i = 0 \).

1. Solve the Equation:
We are given the equation:
\( z^2 = i \)
To find \( z \), we take square roots of \( i \):
In polar form, \( i = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) \), so:
\( z = \sqrt{i} = \sqrt{r} \left[ \cos\left(\frac{\theta}{2}\right) + i\sin\left(\frac{\theta}{2}\right) \right] \)
where \( r = 1 \), \( \theta = \frac{\pi}{2} \).
So the roots are:
\( z_1 = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \)
\( z_2 = \cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \)

2. Find Arguments of Roots:
\( \arg(\alpha) = \arg(z_1) = \frac{\pi}{4} \)
\( \arg(\beta) = \arg(z_2) = \frac{5\pi}{4} \)

3. Calculate the Difference:
\( |\arg(\beta) - \arg(\alpha)| = \left| \frac{5\pi}{4} - \frac{\pi}{4} \right| = \pi \)

Final Answer:
The value of \( |\arg(\beta) - \arg(\alpha)| \) is \( \boxed{\pi} \).