KVPY Sample Paper KVPY Stream-SX Model Paper-19

If \[{{\log }_{\alpha }}8=\text{ }\gamma ,\] \[C-C\] and \[{{\log }_{1/4}}\beta =-\,1\] then \[{{\left( \frac{1}{\alpha }+1 \right)}^{{{\log }_{\sqrt{5}}}\left( {{\beta }^{2}}+4{{\gamma }^{2}} \right)}}\] is equal to-

A) \[\sqrt{5}\]

B) 5

C) 26

D) 625

Correct Answer: D

Solution :

\[{{\log }_{1/4}}\beta =-\,1\]

\[\Rightarrow \beta =4\]

\[{{\log }_{\beta }}\alpha =-\,1\]

\[\Rightarrow \alpha =\frac{1}{\beta }=\frac{1}{4}\]

\[\gamma ={{\log }_{\alpha }}8={{\log }_{1/4}}8=-\frac{3}{2}\]

Now \[\frac{1}{\alpha }+1=4+1=5\]

\[{{\beta }^{2}}+4{{\gamma }^{2}}=16+4{{\left( -\frac{3}{2} \right)}^{2}}=25\]

\[\therefore {{\left( \frac{1}{\alpha }+1 \right)}^{\log \sqrt{5}({{\beta }^{2}}+4{{\gamma }^{2}})}}\]

\[={{5}^{{{\log }_{\sqrt{5}}}25}}=625\]