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11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer
    For positive integers \[{{n}_{1}},{{n}_{2}}\]the value of the expression  \[{{(1+i)}^{{{n}_{1}}}}+{{(1+{{i}^{3}})}^{{{n}_{1}}}}+{{(1+{{i}^{5}})}^{{{n}_{2}}}}+{{(1+{{i}^{7}})}^{{{n}_{2}}}}\]where \[i=\sqrt{-1}\]  is a real number if and only if [IIT 1996]

    A) \[{{n}_{1}}={{n}_{2}}+1\]

    B) \[{{n}_{1}}={{n}_{2}}-1\]

    C) \[{{n}_{1}}={{n}_{2}}\]

    D) \[{{n}_{1}}>0,{{n}_{2}}>0\]

    Correct Answer: D

    Solution :

    Using \[{{i}^{3}}=-i,{{i}^{5}}=i\] and\[{{i}^{7}}=-i\], we can write the given expression as \[{{(1+i)}^{{{n}_{1}}}}+{{(1-i)}^{{{n}_{1}}}}+{{(1+i)}^{{{n}_{2}}}}+{{(1-i)}^{{{n}_{2}}}}\] \[=2{{[}^{{{n}_{1}}}}{{C}_{0}}{{+}^{{{n}_{1}}}}{{C}_{2}}{{i}^{2}}{{+}^{{{n}_{1}}}}{{C}_{4}}{{i}^{4}}+.....]\]\[+2{{[}^{{{n}_{2}}}}{{C}_{0}}{{+}^{{{n}_{2}}}}{{C}_{2}}{{i}^{2}}{{+}^{{{n}_{2}}}}{{C}_{4}}{{i}^{4}}+.....]\]   \[=2{{[}^{{{n}_{1}}}}{{C}_{0}}{{-}^{{{n}_{1}}}}{{C}_{2}}{{+}^{{{n}_{1}}}}{{C}_{4}}+....]\] \[+2{{[}^{{{n}_{2}}}}{{C}_{0}}{{-}^{{{n}_{2}}}}{{C}_{2}}{{+}^{{{n}_{2}}}}{{C}_{4}}+....]\] This is a real number irrespective of the values of \[{{n}_{1}}\]and \[{{n}_{2}}\].


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