If $a_{1}, a_{2}, a_{3}, \ldots, a_{n}, \ldots$ are in G.P., then the value of the determinant $\left|\begin{array}{ccc}\log a_{n} & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8}\end{array}\right|$, is

JEE Main · 2004 · 24 Apr 2004 · Offline · Determinants

a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots

are in G.P., then the value of the determinant

lo g a_{n} lo g a_{n + 3} lo g a_{n + 6} lo g a_{n + 1} lo g a_{n + 4} lo g a_{n + 7} lo g a_{n + 2} lo g a_{n + 5} lo g a_{n + 8}

, is

Answer: (A)

In a G.P. with first term

a

and ratio

r

we have

a_{k} = a r^{k - 1}

, so

lo g a_{k} = lo g a + (k - 1) lo g r

is linear in

k

. Every entry of the determinant is therefore a constant plus a multiple of

lo g r

, and along each row the entries step up by the fixed amount

lo g r

. Replacing the second column by (second minus first) and the third by (third minus second) makes those two new columns identical, each being

lo g r

in every row, and a determinant with two equal columns is

0

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option