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KVPY Sample Paper KVPY Stream-SX Model Paper-31

  • question_answer
    The sum of coefficients of even powers of \[x\] in is: \[{{\left( x+\sqrt{{{x}^{3}}-1} \right)}^{6}}+{{\left( x-\sqrt{{{x}^{3}}-1} \right)}^{6}}\]

    A) 23

    B) 24

    C) 18

    D) 21

    Correct Answer: B

    Solution :

    \[{{\left( x+\sqrt{{{x}^{3}}-1} \right)}^{6}}+{{\left( x+\sqrt{{{x}^{3}}-1} \right)}^{6}}\]\[=2{{(}^{6}}{{C}_{0}}{{x}^{6}}{{+}^{6}}{{C}_{2}}{{x}^{4}}({{x}^{3}}-1){{+}^{6}}{{C}_{4}}.{{x}^{2}}{{({{x}^{3}}-1)}^{2}}\]\[{{+}^{6}}{{C}_{6}}{{({{x}^{2}}-1)}^{3}})\]
    Terms with even powers of \[x\]
    \[=2{{(}^{6}}{{C}_{0}}.{{x}^{6}}{{-}^{6}}{{C}_{2}}.{{x}^{4}}{{+}^{6}}{{C}_{4}}{{x}^{2}}{{+}^{6}}{{C}_{4}}{{x}^{8}}{{+}^{6}}{{C}_{6}}(-1-3{{x}^{6}})\]
    coefficients\[=2{{(}^{6}}{{C}_{0}}{{-}^{6}}{{C}_{2}}{{+}^{6}}{{C}_{4}}{{+}^{6}}{{C}_{4}}{{-}^{6}}{{C}_{6}}-{{3.}^{6}}{{C}_{6}})\]
    \[=2(15-3)=24.\]


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