Practice section 4 of 9

Do not use a calculator in answering this question.

Given that $z=1+i$ is a root of $2z^4+az^3+7z^2+bz+2=0$, find the values of the real numbers $a$ and $b$ and the other roots.

Deduce the roots of $2z^4+bz^3+7z^2+az+2=0$.

Without using a calculator, express $(2-i)^3$ in the form $a+ib$.

Hence, find a root of $(z-i)^3=-11-2i$.

Solve (a) $z^2-4z+24=0$ and (b) $(3z+1)^2+4=0$.

Do not use a graphic calculator in answering this question.

(i) By letting $z=a+bi$, solve $z^2=i$, giving your answers in exact form.

(ii) Solve $w^2+2w+(1-8i)=0$, giving the roots in Cartesian form.

(iii) Hence, solve $(1-8i)v^2+2iv-1=0$, giving your answers in Cartesian form.

Determine, in exact form, the modulus and argument of:

(a) $2$; (b) $-3$; (c) $7i$; (d) $-5i$;

(e) $1+i\sqrt3$; (f) $1-i\sqrt3$; (g) $-1+i$; (h) $-1-i$.

Determine, in exact form, the modulus and argument of:

(i) $12$; (ii) $-3$; (iii) $\sqrt7\,i$; (iv) $-13i$; (v) $1+i$; (vi) $1-\sqrt3\,i$; (vii) $-\sqrt{12}+2i$; (viii) $-\sqrt2-\sqrt2\,i$.