(a) For each pair of coefficients, write the characteristic polynomial, evaluate the discriminant, determine the characteristic roots, and compute their moduli.

i. \(\phi_{1}=\sqrt{2} ; \phi_{2}=-1 / 2\)

ii. \(\phi_{1}=9 / 8 ; \phi_{2}=-1 / 4\)

iii. \(\phi_{1}=1 / 4 ; \phi_{2}=1 / 2\)

iv. \(\phi_{1}=-1 / 2 ; \phi_{2}=3 / 4\)

v. \(\phi_{1}=-1 / 2 ; \phi_{2}=1 / 2\)

(b) For each pair of roots, write a characteristic polynomial that has those roots, and write the coefficients of an \(\mathrm{AR}(2)\) process that has a characteristic polynomial with the given roots.

i. Roots \(z_{1}=2\) and \(z_{2}=4\).

ii. Roots \(z_{1}=1+2 \mathrm{i}\) and \(z_{2}=1-2 \mathrm{i}\).