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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Self Evaluation Test - Complex Numbers and Quadratic Equations

  • question_answer
    If \[z=x+iy,\,\,{{z}^{1/3}}=a-ib,\] then \[\frac{x}{a}-\frac{y}{b}=k({{a}^{2}}-{{b}^{2}})\] where k is equal to

    A) 1    

    B) 2    

    C) 3                     

    D) 4

    Correct Answer: D

    Solution :

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


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