KVPY Sample Paper KVPY Stream-SX Model Paper-13

If \[{{\log }_{245}}175=a\]and \[{{\log }_{1715}}875=b,\] then \[\frac{1-ab}{a-b}\]is equal to

A) 2

B) 3

C) 5

D) 7

Correct Answer: C

Solution :

We have, \[{{\log }_{245}}175=a\]
\[\Rightarrow \] \[\frac{\log 175}{\log 245}=a\]
\[\Rightarrow \] \[\frac{2\log 5+\log 7}{\log 5+2\log 7}=a\]
\[\Rightarrow \] \[\frac{2\log 5+\log 7+\log 5+2\log 7}{2\log 5+2\log 7-\log 5-2\log 7}=\frac{a+1}{a-1}\]
\[\Rightarrow \] \[3\left( \frac{\log 5+\log 7}{\log 5-\log 7} \right)=\frac{a+1}{a-1}\]	? (i)
and \[{{\log }_{175}}875=b\]

\[\Rightarrow \] \[\frac{\log 875}{\log 1715}=b\]\[\Rightarrow \]\[\frac{3\log 5+\log 7}{\log 5+3\log 7}=b\]

Apply componendo and dividendo, we get \[2\left( \frac{\log 5+\log 7}{\log 5-\log 7} \right)=\frac{b+1}{b-1}\]

From Eqs. (i) and (ii), we get \[\frac{1}{3}\left( \frac{a+1}{a-1} \right)=\frac{1}{2}\left( \frac{b+1}{b-1} \right)\]

\[\Rightarrow \] \[2\,(a+1)(b-1)=3\,(b+1)(a-1)\]

\[\Rightarrow \] \[2ab+2b-2a-2=3ab+3a-3b-3\]

\[\Rightarrow \] \[1-ab=5\,(a-b)\]

\[\Rightarrow \] \[\frac{1-ab}{a-b}=5\]