ધારો કે પૂર્ણાંક સંખ્યાઓ \(\mathrm{a}, \mathrm{b} \in[-3,3]\) એવી રીતે છે કે \(\mathrm{a}+\mathrm{b} \neq 0\). તો શક્ય ક્રમયુક્ત જોડ (a, b) ની કુલ સંખ્યા, જેના માટે \(\left|\frac{z-\mathrm{a}}{z+\mathrm{b}}\right|=1\) અને \(\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1, z \in \mathrm{C}\), જ્યાં \(\omega\) અને \(\omega^2\) એ \(x^2+x+1=0\) ના બીજ છે, તે ___ છે.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.