KVPY Sample Paper KVPY Stream-SX Model Paper-25

The solution set of the inequality \[\left| {{9}^{x}}-{{3}^{x+1}}-15 \right|<{{2.9}^{x}}-{{3}^{x}}\]is

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

B) \[(1,\infty )\]

C) \[(-\infty ,\text{ }1]\]

D) None of these

Correct Answer: B

Solution :

Let \[{{3}^{x}}=y,\] then the inequality is \[\left| {{y}^{2}}-3y-15 \right|<2{{y}^{2}}-y\] ? (1)

The inequality holds if \[2{{y}^{2}}-y>0\Rightarrow y<0\,\operatorname{or}\,y>\frac{1}{2}\]

\[\because \]\[y={{3}^{x}}0\Rightarrow y>\frac{1}{2}\]

Now the inequality on solving,

\[-(2{{y}^{2}}-y)<{{y}^{2}}-3y-15<2{{y}^{2}}-y\] \[\Rightarrow \]\[3{{y}^{2}}-4y-15>0\]and\[{{y}^{2}}+2y+15>0\]

Solution of first inequality

\[3{{y}^{2}}-4y-15>0\operatorname{is}\,y<\frac{5}{3}\operatorname{or}\,y>3\]

Solution of second inequality

\[{{y}^{2}}+2y+15>0\] is \[y\in R\]

The common solution is

\[y>3\Rightarrow {{3}^{x}}>x\Rightarrow x>1\Rightarrow x\in (1,\infty )\]