If a1, a2,..., a9 are in G.P., then loga1loga2loga3loga4loga5loga6loga7loga8loga9 is equal to

AP EAMCET · Maths · Determinants

If $a_{1}, a_{2}, . . ., a_{9}$ are in $G . P .,$ then $|\begin{array}{l} \log a_{1} & \log a_{2} & \log a_{3} \\ \log a_{4} & \log a_{5} & \log a_{6} \\ \log a_{7} & \log a_{8} & \log a_{9} \end{array}|$ is equal to

A $\log (a_{1} \cdot a_{2} \cdot . . . \cdot a_{n})$
B $1$
C ${(\log a_{9})}^{9}$
D $0$

Verified Solution

Answer & Solution

Correct Answer

(D) $0$

Step-by-step Solution

Detailed explanation

Given a1, a2, a3,..., a9 are in G.P., hence, by the definition of G.P., we have a2a1=a3a2=a4a3=...a9a8=r, where r is the common ratio of the G.P. Now, let △=loga1loga2loga3loga4loga5loga6loga7loga8loga9 Applying…

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.

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