If a - b - c are positive and are the p th - q th and r th terms- respectively- of a G-P- - then Delta-lbegin vmatrix log a - p - 1 log b - q - 1 llog c- r - 1 lendvmatrix is

Let the first term of the G.P. be A and common ratio be R. Then, nbsp; a=AR^p 1 b=AR^q 1 c=AR^r 1 Δ=log a amp;p amp;1 log b amp;q amp;1 log c amp; ...

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Standard XII

Mathematics

Determinant

If a , b , c ...

Question

If $a, b, c$ are positive and are the $p^{th}$ , $q^{th}$ and $r^{th}$ terms, respectively, of a G.P. , then $Δ = ∣ ∣ ∣ ∣ \begin{matrix} log a & p & 1 log b & q & 1 log c & r & 1 \end{matrix} ∣ ∣ ∣ ∣$ is

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Solution

Let the first term of the G.P. be $A$ and common ratio be $R$ . Then,
$a = A R^{p - 1} b = A R^{q - 1} c = A R^{r - 1}$
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} log a & p & 1 log b & q & 1 log c & r & 1 \end{matrix} ∣ ∣ ∣ ∣$

$R_{1} \to R_{1} - R_{2}, R_{2} \to R_{2} - R_{3}$
$= ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} log (\frac{a}{b}) & p - q & 0 log (\frac{b}{c}) & q - r & 0 log c & r & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} (p - q) log R & p - q & 0 (q - r) log R & q - r & 0 log c & r & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$C_{1} \to C_{1} - log R \cdot C_{2}$
$= ∣ ∣ ∣ ∣ \begin{matrix} 0 & p - q & 0 0 & q - r & 0 log c - r log R & r & 1 \end{matrix} ∣ ∣ ∣ ∣$
$= 0$

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Q. If

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If a,b,c are positive and are the pth, qth and rth terms, respectively, of a G.P. , then Δ=∣∣ ∣∣logap1logbq1logcr1∣∣ ∣∣ is

If $a, b, c$ are positive and are the $p^{th}$ , $q^{th}$ and $r^{th}$ terms, respectively, of a G.P. , then $Δ = ∣ ∣ ∣ ∣ \begin{matrix} log a & p & 1 log b & q & 1 log c & r & 1 \end{matrix} ∣ ∣ ∣ ∣$ is