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

  • question_answer If \[1,\omega ,{{\omega }^{2}},{{\omega }^{3}}.......,{{\omega }^{n-1}}\] are the \[n,{{n}^{th}}\] roots of unity, then \[(1-\omega )(1-{{\omega }^{2}}).....(1-{{\omega }^{n-1}})\] equals [MNR 1992; IIT 1984; DCE 2001; MP PET 2004]

    A) 0

    B) 1

    C) \[n\]

    D) \[{{n}^{2}}\]

    Correct Answer: C

    Solution :

    Since\[1,\omega ,{{\omega }^{2}},{{\omega }^{3}},.....{{\omega }^{n-1}}\]are the \[n,{{n}^{th}}\] roots of unity, therefore, we have the identity \[=(x-1)(x-\omega )(x-{{\omega }^{2}}).....(x-{{\omega }^{n-1}})={{x}^{n}}-1\] or \[(x-\omega )(x-{{\omega }^{2}}).....(x-{{\omega }^{n-1}})=\frac{{{x}^{n}}-1}{x-1}\] =\[{{x}^{n-1}}+{{x}^{n-2}}+.....+x+1\] Putting \[x=1\] on both sides, we get \[(1-\omega )(1-{{\omega }^{2}}).....(1-{{\omega }^{n-1}})=n\]

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