If zn=(1+i3)n and the value of 3Im⁡z5z¯4=λ then the integral part of λ100 is

z5z¯4=(1+i3)5(1−i3)4=(1+i3)(1+3)4=(1+i3)44⇒3Im⁡z5z¯4=328=768=λ⇒λ100=7

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If $z_{n} = (1 + i \sqrt{3})^{n} and the value of \sqrt{3} Im (z_{5} {\bar{z}}_{4}) = λ then the integral part of \frac{λ}{100} is$

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answer is 7.

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Detailed Solution

$\begin{array}{l} z_{5} {\bar{z}}_{4} = (1 + i \sqrt{3})^{5} (1 - i \sqrt{3})^{4} = (1 + i \sqrt{3}) (1 + 3)^{4} = (1 + i \sqrt{3}) 4^{4} \\ \Rightarrow \sqrt{3} Im (z_{5} {\bar{z}}_{4}) = 3 (2^{8}) = 768 = λ \Rightarrow [\frac{λ}{100}] = 7 \end{array}$