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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer
    If \[z=x+iy,\,{{z}^{1/3}}=a-ib\] and \[\frac{x}{a}-\frac{y}{b}=k\,({{a}^{2}}-{{b}^{2}})\]  then value of k equals [DCE 2005]

    A) 2

    B) 4

    C) 6

    D) 1

    Correct Answer: B

    Solution :

    \[{{(x+iy)}^{1/3}}=a-ib\] \[x+iy={{(a-ib)}^{3}}=({{a}^{3}}-3a{{b}^{2}})+i({{b}^{3}}-3{{a}^{2}}b)\] Þ   \[x={{a}^{3}}-3a{{b}^{2}},\,y={{b}^{3}}-3{{a}^{2}}b\] Þ  \[\frac{x}{a}={{a}^{2}}-3{{b}^{2}},\,\frac{y}{b}={{b}^{2}}-3{{a}^{2}}\] \[\therefore \]  \[\frac{x}{a}-\frac{y}{b}={{a}^{2}}-3{{b}^{2}}-{{b}^{2}}+3{{a}^{2}}\]    \[\frac{x}{a}-\frac{y}{b}=4({{a}^{2}}-{{b}^{2}})=k({{a}^{2}}-{{b}^{2}})\]     \[\therefore \]  \[k=4\].


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