12・17 The equation about the complex number $z$ is $10 z^{2}-3 i z-k=0$, where $i^{2}=-1$. Among the following propositions, which one is correct?
(A) For all positive real numbers $k$, both roots are purely imaginary.
(B) For all negative real numbers $k$, both roots are purely imaginary.
(C) For all purely imaginary numbers $k$, both roots are rational.
(D) For all purely imaginary numbers $k$, both roots are irrational.
(E) For all complex numbers $k$, there is no real root.
(44th American High School Mathematics Examination, 1993)
[Sol] Using the quadratic formula, we get
$z=\frac{3 i \pm \sqrt{-9+40 k}}{20}$, discriminant $\Delta=-9+40 k$.
If $k=1$, then $\Delta=31$, so (A) is incorrect.
If $k$ is a negative number, then $\Delta$ is a negative real number, so (B) is correct.
If $k=i$, then $\Delta=-9+40 i=16+40 i-25=(4+5 i)^{2}$, the two roots are $\frac{1}{5}+\frac{2}{5} i$ and $-\frac{1}{5}-\frac{1}{10} i$, so (C) and (D) are both incorrect.
If $k=0$ then the roots are 0 and $\frac{3}{10} i$, so (E) is also incorrect.
Therefore, only (B) is correct.
Hence, the answer is (B).