Question

If $z(3 - i) = 2 + i$, then $z^2 =$

• A. $\frac{i}{2}$
• B. $-\frac{i}{2}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$
• E. $\frac{1+i}{2}$

Hint:

Always multiply by conjugate when dividing complex numbers.

Answer 1

The Correct Option is A

Concept: Use complex division by multiplying conjugate.

Step 1: Find $z$
\[ z = \frac{2 + i}{3 - i} \] Multiply numerator and denominator by conjugate $(3+i)$: \[ z = \frac{(2+i)(3+i)}{(3-i)(3+i)} \]

Step 2: Simplify
\[ (2+i)(3+i) = 6 + 2i + 3i + i^2 = 5 + 5i \] \[ (3-i)(3+i) = 9 + 1 = 10 \] \[ z = \frac{5+5i}{10} = \frac{1+i}{2} \]

Step 3: Find $z^2$
\[ z^2 = \left(\frac{1+i}{2}\right)^2 = \frac{1 + 2i + i^2}{4} = \frac{2i}{4} = \frac{i}{2} \] Final Conclusion:
Option (A)