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Question

If z3+(3+2i)z+(1+ia)=0 has one real root, then the value of a lies in the interval
(aR)

A
(2,1)
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B
(1,0)
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C
(0,1)
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D
(2,3)
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Solution

The correct option is D (2,3)
Let z=α be a real root. Then,
α3+(3+2i)α+(1+ia)=0
(α3+3α1)+i(a+2α)=0
α3+3α1=0 and α=a2
a383a21=0
a3+12a+8=0
Let f(a)=a3+12a+8\
f(1)<0,f(0)>0,f(2)<0,f(1)>0
and f(3)>0.
Hence, a(1,0) or a(2,1) or a(2,3)

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