If `a gt 0, b gt 0, c gt0` are respectively the pth, qth, rth terms of a G.P., then the value of the determinant
`|(log a,p,1),(log b,q,1),(log c,r,1)|`, is

Options

A. 0
B. log (abc)
C. `-(p+q+r)`
D. none of these

Answer

A

Explanation

Let the first terms of G.P. be A and common ration be R. then
`a=AR^(p-1)`
`rArr" log a=log A+ (p-1) log R.etc "`
`rArr |{:(log a,,p,,1),(log b,,q,,1),(log c,,r,,1):}|=|{:(log A +(p-1)log R,,p,,1),(log A+(q-1) log R,,q,,1),(log A+(r-1) log R,,r,,1):}|`
`= |{:((p-1)log R,,p,,1),((q-1) log R,,q,,1),((r-1) log R,,r,,1):}| " " [C_(1) to C_(1) -( log A) c_(3)]`
`= log R |{:(p,,p,,1),(q,,q,,1),(r,,r,,1):}|=0" "(C_(1) to C_(1)+C_(3))`