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

  • question_answer \[\omega \] is an imaginary cube root of unity. If \[{{(1+{{\omega }^{2}})}^{m}}=\] \[{{(1+{{\omega }^{4}})}^{m}},\] then least positive integral value of m is  [IIT Screening 2004]

    A) 6

    B) 5

    C) 4

    D) 3

    Correct Answer: D

    Solution :

    We have, \[{{(1+{{\omega }^{2}})}^{m}}={{(1+{{\omega }^{4}})}^{m}}\] \[(\because \,\,{{\omega }^{3}}=1)\] \[{{(1+{{\omega }^{2}})}^{m}}={{(1+\omega )}^{m}}\] \[{{(-\omega )}^{m}}={{(-{{\omega }^{2}})}^{m}}\] \[\Rightarrow {{\left( \frac{\omega }{{{\omega }^{2}}} \right)}^{m}}=1\]\[\Rightarrow {{({{\omega }^{2}})}^{m}}=1\]\[={{(\omega )}^{2m}}=({{\omega }^{3}})\]\[\Rightarrow m=\frac{3}{2}\] Hence least positive integral value of m is 3.


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