Let \( \alpha, \beta \) be real and \( \mathrm{z} \) be complex num...

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Let \( \alpha, \beta \) be real and \( \mathrm{z} \) be complex number. If \( \mathrm{z}^{2}+\alpha \mathrm{z}+\beta=0 \) has two distinct roots on the line \( \operatorname{Re} \mathrm{z}=1 \), then it is necessary that
(1) \( \beta \in(0,1) \)
(2) \( \beta \in(-1,0) \)
(3) \( |\beta|=1 \)
(4) \( \beta \in(1, \infty) \)
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