Question

If \( z = (1 - i^3)(x + i) \) is a purely imaginary number for \( x = x_1 \), and if \( z \) is a purely real number for \( x = x_2 \), then find \( x_1, x_2 \).

• A. \( -1 \)
• B. 0
• C. 1
• D. 2

Hint:

For solving such problems, always separate the real and imaginary components to find the values of the unknowns.

Answer 1

The Correct Option is A

We are given that \( z = (1 - i^3)(x + i) \) is a purely imaginary number for \( x = x_1 \), and a purely real number for \( x = x_2 \). To solve for \( x_1 \) and \( x_2 \), we analyze the conditions for the real and imaginary parts of \( z \) under these values of \( x \). - When \( z \) is purely imaginary, the real part must be zero, and when \( z \) is purely real, the imaginary part must be zero.