Question

If the roots of the equation \[ Z^3 + iZ^2 + 2i = 0 \] are the vertices of a triangle ABC, then that triangle ABC is:

• A. A right-angled triangle
• B. An equilateral triangle
• C. An isosceles triangle
• D. A right-angled isosceles triangle

Hint:

For complex number-based triangle problems, convert roots into Cartesian coordinates and use distance formulas.

Answer 1

The Correct Option is C

Step 1: Finding the roots of the given equation We solve \( Z^3 + iZ^2 + 2i = 0 \) using factorization or numerical methods and obtain three roots.
Step 2: Geometrical interpretation The roots represent vertices of a triangle. Calculating the pairwise distances between the roots, we check for equality: \[ \text{If two sides are equal, the triangle is isosceles.} \] Verifying the squared distances, we confirm: \[ \text{Triangle ABC is isosceles.} \]