KVPY Sample Paper KVPY Stream-SX Model Paper-1

If \[{{\log }_{x}}8=z,\]\[{{\log }_{y}}x=-\,1\]and \[{{\log }_{\frac{1}{4}}}y=-\,1,\] then \[{{\left( \frac{1}{x}+1 \right)}^{\log \sqrt{5}({{y}^{2}}+4{{z}^{2}})}}\] is equal to

A) \[\sqrt{5}\]

B) 5

C) 25

D) 625

Correct Answer: D

Solution :

[d]

We have,

\[{{\log }_{x}}8=z\]\[\Rightarrow \]\[{{x}^{z}}=8\]

\[{{\log }_{y}}x=-1\,\Rightarrow x=\frac{1}{y}\Rightarrow xy=1\]

\[{{\log }_{1/4}}y=-\,1,\]\[y=4,x=\frac{1}{4}\]

\[\Rightarrow \]\[{{\left( \frac{1}{4} \right)}^{z}}=8\]\[\Rightarrow \]\[{{2}^{-\,2z}}={{2}^{3}}\]

\[\Rightarrow \]\[z=-\frac{3}{2}\]

Now, \[{{\left( \frac{1}{x}+1 \right)}^{\log \sqrt{5}({{y}^{2}}+\,4{{z}^{2}})}}\]

\[{{(5)}^{\log \sqrt{5}\,\left( 16\,+\,4\,\times \,\frac{9}{4} \right)}}\]

\[={{(\sqrt{5})}^{2\log \sqrt{5}(16\,+\,9)=}}{{\sqrt{5}}^{^{\log \sqrt{5}{{(25)}^{2}}}}}\]

\[={{25}^{2}}=625\]