A) 3
B) 2
C) 1
D) zero
Correct Answer: D
Solution :
Since, \[l\], m and n are the \[p\]th, Q th and rth term of a GP whose first term is A and common ratio is R. \[\therefore \] \[\log l=\log A+(p-1)\log R\] Similarly, \[\log m=\log A+(q-1)\log R\] and \[\log n=\log A+(r-1)\log R\] Now, \[\left| \begin{matrix} \log l & p & 1 \\ \log m & q & 1 \\ \log n & r & 1 \\ \end{matrix} \right|=\left| \begin{matrix} \log A+(p-1)\log R & p & 1 \\ \log A+(q-1)\log R & q & 1 \\ \log A+(r-1)\log R & r & 1 \\ \end{matrix} \right|\] Applying \[{{C}_{1}}\to {{C}_{1}}-[{{C}_{3}}\log A+({{C}_{2}}-{{C}_{3}})\log R]\] \[=\left| \begin{matrix} 0 & p & 1 \\ 0 & q & 1 \\ 0 & r & 1 \\ \end{matrix} \right|=0\]
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