બે શૂન્યેતર સંકર સંખ્યાઓ \(z_1\) અને \(z_2\) માટે, જો \(\operatorname{Re}\left(z_1 z_2\right)=0\) અને \(\operatorname{Re}\left(z_1+z_2\right)=0\) હોય, તો \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) અને \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) અને \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) અને \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) અને \(\operatorname{Im}\left( z _2\right) < 0\) નીચે આપેલ પૈકી ક્યાં વિકલ્પ સાચા છે ?

A \(B\) અને \(D\)
B \(B\) અને \(C\)
C \(A\) અને \(B\)
D \(A\) અને \(C\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(B\) અને \(C\)

Step-by-step Solution

Detailed explanation

\(z _1= x _1+ i y _1\) \(z _2= x _2+ iy _2\) \(\operatorname{Re}\left(z_1 z_2\right)=x_1 x_2-y_1 y_2=0\) \(\operatorname{Re}\left(z_1+z_2\right)=x_1+x_2=0\) \(x_1\) and \(x_2\) are of opposite sign \(y_1\) and \(y_2\) are of opposite sign

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.

Same subject

Explore more questions on app

From JEE Mains

Explore more questions on app