Question

If \( x, y, z \) are all positive and are the \( p \)th, \( q \)th and \( r \)th terms of a geometric progression respectively, then the value of the determinant \( \begin{vmatrix} \log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1 \end{vmatrix} \) equals

• A. log xyz
• B. (p-1)(q-1)(r-1)
• C. pqr
• D. 0

Hint:

Logarithms of GP terms form an AP. A determinant with an AP and units column is zero.

Answer 1

The Correct Option is D

Step 1: GP General Terms
\( x = aR^{p-1}, \; y = aR^{q-1}, \; z = aR^{r-1} \)
Step 2: Logarithmic Form
\( \log x = \log a + (p-1)\log R \) \( \log y = \log a + (q-1)\log R \) \( \log z = \log a + (r-1)\log R \)
Step 3: Determinant Structure
The determinant is \( \log a \begin{vmatrix} 1 & p & 1 \\ 1 & q & 1 \\ 1 & r & 1 \end{vmatrix} + \log R \begin{vmatrix} p-1 & p & 1 \\ q-1 & q & 1 \\ r-1 & r & 1 \end{vmatrix} \)
Step 4: Zero Value
In both determinants, columns become identical after simple column operations, so the value is 0.
Final Answer: (d)