Complex Numbers And Quadratic Equations

The correct option is (B): ±2

Concept: A complex number is purely imaginary if its real part is zero. Given: So: Thus: Step 1: {\color{red}Substitute into expression.} Step 2: {\color{red}Find modulus.} But: So ratio = 1.

Concept: Three unit complex numbers summing to zero lie at vertices of an equilateral triangle on unit circle. Step 1: {\color{red}Interpret geometrically.} Points are cube roots of unity: Form equilateral triangle. Step 2: {\color{red}Find side length.} Distance between roots: So side = . Step 3: {\color{red}Area formula.}

Concept: Properties of cube roots of unity: Also: Since: Step 1: {\color{red}Compute modulus square.} Let: Then: Expand using symmetry: Step 2: {\color{red}Minimize expression.} We minimize: For distinct nonzero integers, try smallest values: But distinct condition forces larger combination. Try : Smallest valid set gives: Step 3: {\color{red}Conclusion.} Minimum value = 5.