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KVPY Sample Paper KVPY Stream-SX Model Paper-4

  • question_answer
    Let \[{{(1+{{x}^{2}})}^{2}}{{(1+x)}^{n}}={{A}_{0}}+{{A}_{1}}x+{{A}_{2}}{{x}^{2}}+.....\] If \[{{A}_{0}},\] \[{{A}_{1}},\] \[{{A}_{2}}\] are in A. P. then the value of n is-

    A) 2         

    B) 4    

    C) 6                                 

    D) 7

    Correct Answer: A

    Solution :

    [A] \[{{(1+{{x}^{2}})}^{2}}\,\,{{(1+x)}^{n}}\]\[=(1+2{{x}^{2}}+{{x}^{4}})\,\,{{(}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}\,x{{+}^{n}}{{C}_{2}}\,{{x}^{2}}+....)\]\[=\,{{\,}^{n}}{{C}_{0}}+{{\,}^{n}}{{C}_{1}}x+({{\,}^{n}}{{C}_{2}}+2.{{\,}^{n}}{{C}_{0}})\,\,{{x}^{2}}+......)\] Hence \[{{A}_{0}}=1;\,\,{{A}_{1}}{{=}^{n}}{{C}_{1}};\]\[{{A}_{2}}=\,{{\,}^{n}}{{C}_{2}}+2\] which are in A.P.                 


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