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KVPY Sample Paper KVPY Stream-SX Model Paper-7

  • question_answer
    For each\[t\in R,\]let \[[t]\] be the greatest integer less than or equal to \[t.\] Then, \[\underset{x\to 1+}{\mathop{\lim }}\,\frac{(1-|x|+\sin |1-x|)sin\left( \frac{\pi }{2}[1-x] \right)}{|1-x\left\| 1-x \right.|}\]

    A) equals 1            

    B)  equals 0

    C) equals\[-1\]                    

    D) does not exist

    Correct Answer: B

    Solution :

    \[\underset{x\to 1+}{\mathop{\lim }}\,\frac{(1-\left| x \right|+\sin \left| 1-x \right|)\sin \left( [1-x]\frac{\pi }{2} \right)}{\left| 1-x \right|[1-x]}\]
    \[=\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{(1-x+\sin (x-1)\sin \left( -\frac{\pi }{2} \right)}{(x-1)(-1)}\]
    \[=\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{-(x-1)+\sin (x-1)}{(x-1)}=-1+1=0.\]


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