z = 3+2 i 1-2 i , (i= -1 ) will be purely imaginary if =

Step-by-step solution

For z = 3 + 2i 1 - 2i to be purely imaginary, its real part must vanish. Multiplying numerator and denominator by the conjugate 1 + 2i : z = (3 + 2i )(1 + 2i ) (1 - 2i )(1 + 2i ) = 3 - 4 ^2 + 8i 1 + 4 ^2 Separating real and imaginary parts gives: Re (z) = 3 - 4 ^2 1 + 4 ^2 Im (z) = 8 1 + 4 ^2 Setting the real part equal to zero, using that the denominator is always positive: 3 - 4 ^2 = 0 ^2 = 3 4 = 3 2 The general solution is = n 3 for n Z , matching option C .