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KVPY Sample Paper KVPY Stream-SX Model Paper-21

  • 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) 5

    D) 7

    Correct Answer: A

    Solution :

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


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