Find the real value of a for which $3 i^{3} - 2 a i^{2} + (1 - a) i + 5$ is real.

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Solution

$3 i^{3} - 2 a i^{2} + (1 - a) i + 5 \Rightarrow - 3 i + 2 a + 5 + (1 - a) i \Rightarrow 2 a + 5 - 3 i + i - a i \Rightarrow 2 a + 5 - 2 i - a i \Rightarrow 2 a + 5 + (- a - 2) i which is real If, - a - 2 = 0 \Rightarrow a = - 2$

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