QJEEM19Ma7268: JEE Main 2019 Mathematics - Class 11 JEEM19Ma7268 Solution: Let If R(z) and 1(z) respectively denote the real and imaginary …

Question

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Let

z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.

If R(z) and 1(z) respectively denote the real and imaginary parts of z, then :

Accepted Answer

z = {\left( {{{\sqrt 3 + i} \over 2}} \right)^5} + {\left( {{{\sqrt 3 - i} \over 2}} \right)^5}

z = {\left( {{e^{i\pi /6}}} \right)^5} + {\left( {{e^{ - i\pi /6}}} \right)^5}

= {e^{i5\pi /6}} + {e^{ - i5\pi /6}}

= \cos {{5\pi } \over 6} + i{{\sin 5\pi } \over 6} + \cos \left( {{{ - 5\pi } \over 6}} \right) + i\sin \left( {{{ - 5\pi } \over 6}} \right)

= 2\cos {{5\pi } \over 6} &lt; 0

{\rm I}(z) = 0

and

{\mathop{\rm Re}\nolimits} (z) &lt; 0

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