= Solution
{c}
\b[Yes: for $k=2$ the stated inequality guarantees <zero-stability>.] Part (ii) puts every <polynomial root> in the closed <unit disc>. The remaining <root condition for a multistep method> is that every root on the <unit circle> has <multiplicity of a root> one.
If this failed for a <quadratic polynomial>, it would have a repeated root $\xi$ with $|\xi|=1$, and
$$
\rho(z)=\alpha_2(z-\xi)^2
=\alpha_2z^2-2\alpha_2\xi z+\alpha_2\xi^2.
$$
Consequently $|\alpha_1|=2|\alpha_2|$ and $|\alpha_0|=|\alpha_2|$, contradicting
$$
|\alpha_2|\geq|\alpha_0|+|\alpha_1|=3|\alpha_2|,
$$
because $\alpha_2\ne0$. Thus all roots on the <unit circle> are simple and
$$
\boxed{\text{The method is zero-stable; no counterexample satisfies the assumption.}}
$$
This conclusion does not assert <consistency of a numerical method>: part (i) rules out <consistency of a numerical method> whenever the coefficient inequality is strict. <Zero-stability> and <consistency of a numerical method> are separate requirements.
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