KVPY Sample Paper KVPY Stream-SX Model Paper-5

Solution set of the inequality \[{{\log }_{3}}(x+2)(x+4)+{{\log }_{1/3}}(x+2)<\frac{1}{2}{{\log }_{\sqrt{3}}}7\] is

A) \[(-2,-1)\]

B) \[(-2,3)\]

C) \[(-\,1,3)\]

D) \[(3,\infty )\]

Correct Answer: B

Solution :

[b]

We have, \[{{\log }_{3}}(x+2)(x+4)+{{\log }_{1/3}}(x+2)<\frac{1}{2}{{\log }_{\sqrt{3}}}7\]?(i)

\[(x+2)(x+4)>0,(x+2)>0\]

\[\therefore \]\[x>-\,2\]

Now, Eq. (i) becomes \[{{\log }_{3}}(x+2)(x+4)-{{\log }_{3}}(x+2)<{{\log }_{3}}7\]

\[\Rightarrow \]\[{{\log }_{3}}(x+4)<{{\log }_{3}}7\]

\[\Rightarrow \]\[x+4<7\]

\[\Rightarrow \]\[x<3\]

\[\therefore \]\[x\in (-2,3)\]