InterviewSolution
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InterviewSolution
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The value of the determinant `|[log_a(x/y), log_a(y/z), log_a(z/x)], [log_b (y/z), log_b (z/x), log_b (x/y)], [log_c (z/x), log_c (x/y), log_c (y/z)]|`A. 1B. -1C. 0D. `(1)/(6) log _(a) xyz` |
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Answer» Correct Answer - C `Delta = |{:(log_(a)((x)/(y)),,log_(a)((y)/(z)),,log_(a)((z)/(x))),(log_(a^(2))((y)/(z)),,log_(a^(2))((z)/(x)),,log_(a^(2))((x)/(y))),(log_(a^(3))((z)/(y)),,log_(a^(3))((x)/(y)),,log_(a^(3))((y)/(z))):}|` `=(1)/(2) xx(1)/(3) |{:(log_(a)((x)/(y)),,log_(a)((y)/(z)),,log_(a)((z)/(x))),(log_(a)((y)/(z)),,log_(a)((z)/(x)),,log_(a)((x)/(y))),(log_(a)((z)/(y)),,log_(a)((x)/(y)),,log_(a)((y)/(z))):}|` Now ,`log_(a)((x)/(y)) +log_(a)((y)/(z))+log_(a)((z)/(y))` `=log_(a)((x)/(y))((y)/(z))((z)/(x))=log_(a)1=0` So applying operation `R_(1) to R_(1)+R_(2)+R_(3) " on " Delta ` we get `Delta =(1)/(2)xx(1)/(3) |{:(0,,0,,0),(log_(a).((y)/(z)),,log_(a).((z)/(x)),,log_(a).((x)/(y))),(log_(a).((z)/(x)),,log_(a).((x)/(y)),,log_(a).((y)/(z))):}|=0` |
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