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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2000

  • question_answer
    The value of \[{{\left[ -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right]}^{3n}}+{{\left[ -\frac{1}{2}-i\frac{\sqrt{3}}{2} \right]}^{3n}}\]is:

    A)  0                                            

    B)         1                            

    C)         2                                            

    D)         3

    E)         \[-2\]

    Correct Answer: C

    Solution :

    We know\[-\frac{1}{2}+i\frac{\sqrt{3}}{2}=\omega \]and\[-\frac{1}{2}-i\frac{\sqrt{3}}{2}={{\omega }^{2}}\] \[\therefore \]  \[{{\left[ \frac{-1+i\sqrt{3}}{2} \right]}^{3n}}+{{\left[ \frac{1-i\sqrt{3}}{2} \right]}^{3n}}\]                 \[={{\omega }^{3n}}+{{({{\omega }^{2}})}^{3n}}\]                 \[={{({{\omega }^{3}})}^{n}}+{{[{{({{\omega }^{3}})}^{n}}]}^{2}}\]                 \[=1+1=2\]


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