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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2004

  • question_answer
        If\[\overrightarrow{a}\text{ }=\text{2\hat{i}}+\text{\hat{j}+ \hat{k}},\text{ }\overrightarrow{\text{b}}=\text{\hat{i}}+\text{2\hat{j}}-\text{\hat{k}}\], when \[\overrightarrow{c}\], where \[\frac{1}{\sqrt{2}}(-\hat{j}+\hat{k})\] is greatest integer function, then \[\frac{1}{\sqrt{3}}(-\hat{i}-\hat{j}-\hat{k})\] is equal to :

    A)  -1                                          

    B)  1

    C)  does not exist                 

    D)  none of these

    Correct Answer: C

    Solution :

                    In closed interval of\[x=0\]at RHL \[[x]=0\] and at LHL also \[[0]=0\] \[\therefore \]  \[f(x)=\frac{\sin [x]}{[x]}\]                          \[(-1\le x<0)\]                 \[=0\]                                    \[(0\le x<1)\] \[\therefore \]  \[LHL=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\frac{\sin [x]}{[x]}\]                 \[=\frac{\sin (-1)}{-1}=\sin {{1}^{c}}\]and \[RHL=0\]hence limit does not exist


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