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Question
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
Solution
\[3 i^3 - 2a i^2 + (1 - a)i + 5\]
\[ = - 3i + 2a + (1 - a)i + 5\]
\[ = (2a + 5) + i(1 - a - 3)\]
\[ = (2a + 5) + i( - 2 - a)\]
Since,
\[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
\[\therefore Im\left[ 3 i^3 - 2a i^2 + (1 - a)i + 5 \right] = 0\]
\[ \Rightarrow - 2 - a = 0\]
\[ \Rightarrow a = - 2\]
Hence, the real value of a for which
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