KVPY Sample Paper KVPY Stream-SX Model Paper-23

Let \[g\,(x)=\frac{{{(x-1)}^{n}}}{\log {{\cos }^{m}}(x-1)};0<x<2,m\] and n are integers, \[m\ne 0,\]\[n>0,\] and let P be the left hand derivatives of \[\left| x-1 \right|\] at \[x=1.\] If \[\lim g(x)=P,\] then

A) \[n=1,\]\[m=1\]

B) \[n=1,\]\[m=-\,1\]

C) \[n=2,\]\[m=2\]

D) \[n>2,\]\[m=n\]

Correct Answer: C

Solution :

Given, \[g\left( x \right)=\frac{{{\left( x-1 \right)}^{n}}}{\log {{\cos }^{m}}\left( x-1 \right)}\]P is left bond derivative of \[|x-1|\]

\[\therefore \,\,\,\,\,\,P=-1\]

\[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,g\left( x \right)=P\]\[\Rightarrow \,\,\,\underset{h\to 0}{\mathop{\lim }}\,g\left( 1+h \right)=-1\]\[\Rightarrow \,\,\underset{h\to 0}{\mathop{\lim }}\,\frac{{{h}^{n}}}{\log {{\cos }^{m}}h}=-1\]\[\Rightarrow \,\,\,\underset{h\to 0}{\mathop{\lim }}\,\frac{n{{h}^{n-1}}}{m\left( -\tan \,h \right)}=-1\]\[\Rightarrow \,\,\frac{-n}{n}\underset{h\to 0}{\mathop{\lim }}\,\frac{{{h}^{n-1}}}{\tan \,h}=-1\]

Which holds of \[n-1=1\,\Rightarrow n=2\]and \[-\frac{n}{m}=-1\,\Rightarrow n=m=2\]

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