Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Let <img alt="\mathrm{g(x)=\frac{(x-1)^m}{\ln \left(\cos ^{m t}(x-1)\right)} ; 0<x<2 m}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bg%28x%29%3D%5Cfrac%7B%28x-1%29%5Em%7D

Let \mathrm{g(x)=\frac{(x-1)^m}{\ln \left(\cos ^{m t}(x-1)\right)} ; 0<x<2 m} and \mathrm{n} are integers, \mathrm{m \neq 0, n>0} and let \mathrm{p} be the left hand derivative of \mathrm{|x-1| at, x=1 If \lim _{x \rightarrow 1^{+}} g(x)=p}, then :

Option: 1

\mathrm{n=1, m=1}


Option: 2

\mathrm{n=1, m=-1}


Option: 3

\mathrm{n=2, m=2}


Option: 4

\mathrm{n>2, m=n}


Answers (1)

best_answer

Let,  \mathrm{y=|x-1|=\left\{\begin{array}{r} x-1 ; x \geq 1 \\ -(x-1) ; x<1 \end{array}\right.}

Left hand derivative of  \mathrm{y^{\prime}=|x-1| \text { at } x=1 \text { is }-\mathrm{L}} 

Hence, \mathrm{p=-1}

\mathrm{\left.\lim _{x \rightarrow 1^{*}} g(x)=\lim _{x \rightarrow 1^{*}}\left(\frac{(x-1)^m}{\ln \left(\cos ^n(x-1)\right.}\right) \quad \text { (Put } x=1+h\right)}

\mathrm{=\lim _{h \rightarrow 0^{*}} \frac{h^{m \prime}}{\ln \left(\cos ^n h\right)}}

\mathrm{=\lim _{h \rightarrow 0^{*}} \frac{h^m}{n \ln (\cos h)}=\lim _{h \rightarrow 0^{*}} \frac{h^m}{n \ln \frac{\left(1-2 \sin ^2 \frac{h}{2}\right)}{\left(-2 \sin ^2 \frac{h}{2}\right)}\left(-2 \sin ^2 \frac{h}{2}\right)}}

[So,  \mathrm{m=2}  for limit to have non-/ero finite value]

\mathrm{=\lim _{h \rightarrow 0} \frac{4 \frac{h^2}{4}}{n \ln \frac{\left(1-2 \sin ^{-\frac{h}{2}}\right.}{\left(-2 \sin ^2 \frac{h}{2}\right)}\left(-2 \sin ^2 \frac{h}{2}\right)}+\frac{4}{-(2 n)}=-\frac{2}{n}}

Given, \mathrm{-\frac{2}{n}=-1}  hence  \mathrm{n=2}

Posted by

Rishi

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026: NIT Rourkela records higher closing ranks in CSE, ECE, AI; core engineering branches stable
anam-khan

JEE Main 2026: IIT Delhi’s CSE, EE most opted; last 2 year’s closing ranks for all categories
anam-khan

Must provide CBSE 11th registration numbers to JEE Mains aspirants, board tells schools

Latest Question