First slide

Introduction to Determinants

Question

If a>0, b>0, c >0, and are the pth, qth, and rth terms, respectively, of a G.P., then $|\begin{matrix} loga & p & 1 \\ logb & q & 1 \\ logc & r & 1 \end{matrix}|$ is

Moderate

A

0
B

log (abc)
C

-(p + q + r)
D

none of these

Solution

Let the first term of G.P. be A and common ratio be R. Then,
$a = {AR}^{p - 1}$
$\begin{array}{l} \Rightarrow \log a = \log A + (p - 1) \log R, etc. \\ \Rightarrow |\begin{array}{lll} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \end{array}| = |\begin{array}{ccc} \log A + (p - 1) \log R & p & 1 \\ \log A + (q - 1) \log R & q & 1 \\ \log A + (r - 1) \log R & r & 1 \end{array}| \end{array}$
$\begin{matrix} = |\begin{array}{lll} (p - 1) \log R & p & 1 \\ (q - 1) \log R & q & 1 \\ (r - 1) \log R & r & 1 \end{array}| [C_{1} \to C_{1} - (\log A) C_{3}] \\ = \log R |\begin{matrix} (p - 1) & p & 1 \\ (q - 1) & q & 1 \\ (r - 1) & r & 1 \end{matrix}| \\ = \log R |\begin{array}{lll} p & p & 1 \\ q & q & 1 \\ r & r & 1 \end{array}| = 0 (C_{1} \to C_{1} + C_{3}) \end{matrix}$

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