Consider initial value problem

$$\displaystyle \boldsymbol{ \vec{y}^{\,\prime} } = \left\lbrack \begin{array}{rr} -4 & k \\ k & 1 \end{array} \right\rbrack \boldsymbol{\vec{y}}, \ \ \ \boldsymbol{\vec{y}}(0) = \boldsymbol{\vec{y}_0},$$
where $k$ is a real parameter.

1. Determine all values of $k$ for which the coefficient matrix has distinct real eigenvalues. Enter NONE if there are no values of $k$ for which the coefficient matrix has distinct real eigenvalues.
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2. Determine all values of $k$ for which the coefficient matrix has distinct complex eigenvalues. Enter NONE if there are no values of $k$ for which the coefficient matrix has distinct complex eigenvalues.
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3. For what values of $k$ found in part (a) does $\sqrt{y_1(t)^2 + y_2(t)^2} \to 0$ as $t \to 0$ for every initial vector $\boldsymbol{\vec{y}_0}$? Enter NONE if there are no values of $k$ for which this is true.
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