KVPY Sample Paper KVPY Stream-SX Model Paper-2

If \[\underset{x\to \,0}{\mathop{lim}}\,\frac{{{729}^{x}}-{{243}^{x}}-{{81}^{x}}+{{9}^{x}}+{{3}^{x}}-1}{{{x}^{3}}}=K\,{{(\ell n\,3)}^{3}}\] then k is equal to

A) 4

B) 5

C) 6

D) 8

Correct Answer: C

Solution :

\[\underset{x\to 0}{\mathop{\text{Lim}}}\,\frac{{{3}^{6x}}-{{3}^{5x}}-{{3}^{4x}}+{{3}^{2x}}+{{3}^{x}}-1}{{{x}^{3}}}\]

\[\underset{x\to 0}{\mathop{\text{Lim}}}\,\frac{({{3}^{x}}-1)\,\,({{3}^{5x}}-{{3}^{3x}}-{{3}^{2x}}+1)}{{{x}^{3}}}=\]

\[\underset{x\to 0}{\mathop{\text{Lim}}}\,\frac{({{3}^{x}}-1)\,\,({{3}^{2x}}-1)({{3}^{3x}}-1)}{{{x}^{3}}}=\]

\[(\ell \,n\,3)\,\,(2\,\ell \,n\,3)\,\,(3\,\ell \,n\,3)=6\,{{(\ell \,n\,3)}^{3}}\]