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KVPY Sample Paper KVPY Stream-SX Model Paper-26

  • question_answer
    The remainder when \[{{\left( \underset{k=1}{\overset{5}{\mathop{\sum }}}\,{}^{20}{{C}_{2k-1}} \right)}^{6}}\]is divided by 11, is

    A) 1

    B) 3          

    C) 5

    D) 7

    Correct Answer: B

    Solution :

    We have, \[{{\left( {{\sum\limits_{k=1}^{5}{{}^{20}C}}_{2k-1}} \right)}^{6}}\]\[={{({}^{20}{{C}_{1}}+{}^{20}{{C}_{3}}+{}^{20}{{C}_{5}}+{}^{20}{{C}_{7}}+{}^{20}{{C}_{9}})}^{6}}\]
    \[={{\left( {{2}^{18}} \right)}^{6}}\]
    \[=8\times {{\left( 32 \right)}^{21}}=8{{\left( 33-1 \right)}^{21}}\]
    \[=8\times \left( 11\lambda -1 \right)=8\times 11\lambda -11+3\]
    \[=11\left( 8\lambda -1 \right)+3\]
    Remainder is 3 when divided by 11.


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