If z=(1+3i)10+(1−3i)10, then arg z is

(1+3i)10+(1−3i)10=210cosπ3+isinπ310+210cosπ3-isinπ310 =210cos10π3+isin10π3+210cos10π3-isin10π3 =2102cos10π3=211cos4π3=-211 is negative real number Arg(z)=π

If z=(1+3i)10+(1−3i)10, then arg z is

(1+3i)10+(1−3i)10=210cosπ3+isinπ310+210cosπ3-isinπ310 =210cos10π3+isin10π3+210cos10π3-isin10π3 =2102cos10π3=211cos4π3=-211 is negative real number Arg(z)=π

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If $z = (1 + \sqrt{3} i)^{10} + (1 - \sqrt{3} i)^{10}$ , then arg z is

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$\frac{π}{2}$

none of these

$\frac{π}{4}$

$π$

answer is B.

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Detailed Solution

$(1 + \sqrt{3} i)^{10} + (1 - \sqrt{3} i)^{10} = 2^{10} {(\cos \frac{π}{3} + isin \frac{π}{3})}^{10} + 2^{10} {(\cos \frac{π}{3} - isin \frac{π}{3})}^{10} = 2^{10} (\cos \frac{10 π}{3} + isin \frac{10 π}{3}) + 2^{10} (\cos \frac{10 π}{3} - isin \frac{10 π}{3}) = 2^{10} (2 \cos \frac{10 π}{3}) = 2^{11} \cos \frac{4 π}{3} = - 2^{11} is negative real number Arg (z) = π$