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KVPY Sample Paper KVPY Stream-SX Model Paper-1

  • question_answer
    For all complex number \[\left| {{z}_{1}} \right|=12\] and \[\left| {{z}_{2}}-3-4i \right|=5\] the minimum value of \[\left| {{z}_{1}}-{{z}_{2}} \right|\] is

    A) 0                                 

    B) 2

    C) 7                                 

    D) 17

    Correct Answer: B

    Solution :

    (b)
    \[\therefore \left| {{z}_{1}}-{{z}_{2}}\left| \,\ge \, \right|\,\left| {{z}_{1}} \right|-\left| {{z}_{2}} \right| \right|\]
    \[\therefore \left| {{z}_{2}}-3-4i\left| \,\ge \, \right. \right|\left| {{z}_{2}} \right|\left. - \right|\left. \left. 3+4i \right| \right|\]
    \[\Rightarrow 5\ge \left| \left| {{z}_{2}} \right|-54 \right|\] \[\Rightarrow -5\le \left| {{z}_{2}} \right|-5\le 5\Rightarrow \left| {{z}_{2}} \right|\le 10\] \[\Rightarrow -\left| {{z}_{2}} \right|\ge -10\]
    \[\therefore \left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|\ge 12-10=2\]
    But\[\left| {{z}_{1}}-{{z}_{2}} \right|\ge \left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|\ge 2\] \[\Rightarrow \]Minimum value of \[\left| {{z}_{1}}-{{z}_{2}} \right|\,\,\operatorname{is}\,\,2.\].
     


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