First slide

Introduction to Determinants

Question

Suppose $a, b, c > 0$ and $a, b, c$ c are the $p t h, q t h, r t h$ terms of a G.P. Let $Δ = |\begin{array}{lll} 1 p \log a \\ 1 q \log b \\ 1 r \log c \end{array}|$ then numerical value of $∆$ is

Moderate

A

-1
B

2
C

$0$
D

none of these

Solution

Let $a = A R^{p - 1}, b = A R^{q - 1}$ and $c = A R^{r - 1}$
$\Rightarrow \log a = α + (p - 1) β, \log b = α + (q - 1) β$ and
$\log c = α + (r - 1) β$
where $α = \log a, β = \log R$
Now,
$Δ = |\begin{array}{ccc} 1 & p & α + (p - 1) β \\ 1 & q & α + (q - 1) β \\ 1 & r & α + (r 1) β \end{array}|$
Using $C_{3} \to C_{3} - (α - β) C_{1} - β C_{3},$ we get $Δ = 0$

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