NDA 2 2024 Maths, question 41

Topic: Algebra

Passage · Q41 to Q42

Let $Z_{1}$ and $Z_{2}$ be any two complex numbers such that $Z_{1}^{2}+Z_{2}^{2}+Z_{1}Z_{2}=0$ .

Q41

What is the value of $\left| \frac{Z_{1}}{Z_{2}} \right|$ ?

(a) $1$ Correct answer
(b) $2$
(c) $3$
(d) $4$

Solution.

Answer: (a) $1$

The requested ratio requires Z₂≠0.
Dividing the equation by Z₂² gives w²+w+1=0, where w=Z₁/Z₂.
Thus w=(−1±i√3)/2 and |w|=1.
The pair Z₁=Z₂=0 satisfies the shared equation but is excluded by the displayed ratio's domain.