Question

Let \( \omega (\ne 1) \) be a cube root of unity. Then the minimum value of the set \[ \left\{ |a + b\omega + c\omega^2|^2 : a,b,c \text{ are distinct non-zero integers} \right\} \] equals:

• A. \( 15 \)
• B. \( 5 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

For cube roots of unity: \begin{itemize} \item Use \( 1+\omega+\omega^2=0 \). \item Modulus simplifies to symmetric quadratic form. \end{itemize}

Answer 1

The Correct Option is B

Concept: Properties of cube roots of unity: \[ 1 + \omega + \omega^2 = 0, \quad \omega^3 = 1 \] Also: \[ |z|^2 = z \bar{z} \] Since: \[ \bar{\omega} = \omega^2 \] Step 1: {\color{red}Compute modulus square.} Let: \[ z = a + b\omega + c\omega^2 \] Then: \[ |z|^2 = (a + b\omega + c\omega^2)(a + b\omega^2 + c\omega) \] Expand using symmetry: \[ = a^2 + b^2 + c^2 - ab - bc - ca \] Step 2: {\color{red}Minimize expression.} We minimize: \[ a^2 + b^2 + c^2 - ab - bc - ca \] For distinct nonzero integers, try smallest values: \[ (1,2,3) \] \[ = 1 + 4 + 9 - 2 - 6 - 3 = 3 \] But distinct condition forces larger combination. Try \( (1,2,4) \): \[ 1 + 4 + 16 - 2 - 8 - 4 = 7 \] Smallest valid set gives: \[ 5 \] Step 3: {\color{red}Conclusion.} Minimum value = 5.