JEE MAIN - Mathematics (2002 - No. 50)
l, m, n are the $${p^{th}}$$, $${q^{th}}$$ and $${r^{th}}$$ term of a G.P all positive, $$then\,\left| {\matrix{
{\log \,l} & p & 1 \cr
{\log \,m} & q & 1 \cr
{\log \,n} & r & 1 \cr
} } \right|\,equals$$
- 1
2
1
0
Explanation
$$l = A{R^{p - 1}}$$
$$ \Rightarrow \log 1 = \log A + \left( {p - 1} \right)\log R$$
$$m = A{R^{q - 1}}$$
$$ \Rightarrow \log m = \log A + \left( {q - 1} \right)\log R$$
$$n = A{R^{r - 1}}$$
$$ \Rightarrow \log n = \log A + \left( {r - 1} \right)\log R$$
Now, $$\left| {\matrix{ {\log l} & p & 1 \cr {\log m} & q & 1 \cr {\log n} & r & 1 \cr } } \right|$$
$$ = \left| {\matrix{ {\log A + \left( {p - 1} \right)\log R} & p & 1 \cr {\log A + \left( {q - 1} \right)\log R} & q & 1 \cr {\log A + \left( {r - 1} \right)\log R} & r & 1 \cr } } \right|$$
Operating $${C_1} - \left( {\log R} \right){C_2} + \left( {\log R - \log A} \right){C_3}$$
$$ = \left| {\matrix{ 0 & p & 1 \cr 0 & q & 1 \cr 0 & r & 1 \cr } } \right| = 0$$
$$ \Rightarrow \log 1 = \log A + \left( {p - 1} \right)\log R$$
$$m = A{R^{q - 1}}$$
$$ \Rightarrow \log m = \log A + \left( {q - 1} \right)\log R$$
$$n = A{R^{r - 1}}$$
$$ \Rightarrow \log n = \log A + \left( {r - 1} \right)\log R$$
Now, $$\left| {\matrix{ {\log l} & p & 1 \cr {\log m} & q & 1 \cr {\log n} & r & 1 \cr } } \right|$$
$$ = \left| {\matrix{ {\log A + \left( {p - 1} \right)\log R} & p & 1 \cr {\log A + \left( {q - 1} \right)\log R} & q & 1 \cr {\log A + \left( {r - 1} \right)\log R} & r & 1 \cr } } \right|$$
Operating $${C_1} - \left( {\log R} \right){C_2} + \left( {\log R - \log A} \right){C_3}$$
$$ = \left| {\matrix{ 0 & p & 1 \cr 0 & q & 1 \cr 0 & r & 1 \cr } } \right| = 0$$
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