KVPY Sample Paper KVPY Stream-SX Model Paper-26

Suppose \[{{\log }_{a}}b+{{\log }_{b}}a=c\], The smallest positive integer value of c for all, \[a,b>1\]

A) 4

B) 3

C) 2

D) 1

Correct Answer: C

Solution :

We have, \[{{\log }_{a}}b={{\log }_{b}}a=c\] \[\Rightarrow \,\,\,{{\log }_{a}}b+\frac{1}{{{\log }_{a}}b}=c\]

By \[AM\ge GM\]

\[\therefore \,{{\log }_{a}}b+\frac{1}{{{\log }_{a}}b}\ge 2\]

\[\therefore \,\,\,\,\,\,\,\,\,c\ge 2\]