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JEE Main & Advanced JEE Main Paper (Held On 10-Jan-2019 Evening)

  • question_answer
    Let \[z={{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}\,+\,{{\left( \frac{\sqrt{3}}{2}-\frac{i}{2} \right)}^{5}}\]. If R(z) and I(z) respectively denote the real and imaginary parts of z, then- [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

    A) \[R\left( z \right)=-3\]

    B) \[R\left( z \right)<0\text{ }and\text{ }I\left( z \right)>0\]

    C) \[I\left( z \right)=0\]

    D) \[R\left( z \right)>0\text{ }and\text{ }I\left( z \right)>0\]

    Correct Answer: C

    Solution :

    \[z={{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}+{{\left( \frac{\sqrt{3}}{2}-\frac{i}{2} \right)}^{5}}\] \[z={{\left( {{e}^{i\frac{\pi }{6}}} \right)}^{5}}+{{\left( {{e}^{-i\frac{\pi }{6}}} \right)}^{5}}\] \[z\,\,=\,{{e}^{i\frac{5\pi }{6}}}+{{e}^{-i\frac{5\pi }{6}}}\] \[z\,\,=\,\,2\cos \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{5\pi }{6}\] \[z\,\,=\,\,2\left( -\frac{\sqrt{3}}{2} \right)\,=-\sqrt{3}\,\,\,\,\,\,\] It means \[Im\left( z \right)=0\]


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