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
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
Show Hint
Logarithms of GP terms form an AP. A determinant with an AP and units column is zero.
Updated On: Apr 10, 2026
- log xyz
- (p-1)(q-1)(r-1)
- pqr
- 0
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
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)
\( x = aR^{p-1}, \; y = aR^{q-1}, \; z = aR^{r-1} \)
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)
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