Question

A complex number \(z\) among the following which does not satisfy \[ z^3+27i=0 \] is:

• A. \(\dfrac{3\sqrt{3}-3i}{2}\)
• B. \(-3i\)
• C. \(\dfrac{3\sqrt{3}+3i}{2}\)
• D. \(\dfrac{-3\sqrt{3}+3i}{2}\)

Hint:

For equations of the form \(z^n=w\), convert \(w\) into polar form and then use the \(n\)-th root formula to find all possible roots.

Answer 1

The Correct Option is A

Step 1: Rearrange the equation.
The given equation is z³ + 27i = 0, which implies z³ = -27i. Express -27i in polar form: 27(cos(3π/2) + i sin(3π/2)).

Step 2: Apply De Moivre's formula for cube roots.
The three cube roots are z = 3[cos((3π/2 + 2kπ)/3) + i sin((3π/2 + 2kπ)/3)] for k = 0, 1, 2.

Step 3: Compute each root explicitly.
For k = 0: angle = π/2 → z = 3i. For k = 1: angle = 7π/6 → z = 3(-√3/2 - i/2) = (-3√3 - 3i)/2. For k = 2: angle = 11π/6 → z = 3(√3/2 - i/2) = (3√3 - 3i)/2.

Step 4: Identify the extraneous option.
The three actual roots are 3i, (-3√3 - 3i)/2, and (3√3 - 3i)/2. The value (3√3 + 3i)/2 does not appear among these and therefore does not satisfy the equation.

Step 5: Final conclusion.
The value that fails to satisfy the equation is (3√3 + 3i)/2.