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

  • question_answer If the cube roots of unity be \[1,\omega ,{{\omega }^{2}},\] then the roots of the equation \[{{(x-1)}^{3}}+8=0\]are [IIT 1979; MNR 1986; DCE 2000; AIEEE 2005]

    A) \[-1,\,1+2\omega ,\,1+2{{\omega }^{2}}\]

    B) \[-1,\,1-2\omega ,\,1-2{{\omega }^{2}}\]

    C) \[-1,\,-1,\,-1\]

    D) None of these

    Correct Answer: B

    Solution :

    \[{{(x-1)}^{3}}=-8\Rightarrow x-1={{(-8)}^{1/3}}\] Þ \[x-1=-2,-2\omega ,-2{{\omega }^{2}}\] Þ \[x=-1,1-2\omega ,1-2{{\omega }^{2}}\] Trick: By inspection, we see that (b) satisfies the equation i.e,   \[{{(-1-1)}^{3}}+8=0,{{(1-2\omega -1)}^{3}}+8=0\]\[{{(1-2{{\omega }^{2}}-1)}^{3}}+8=0\].

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