Practice section 2 of 9

Given that $(2-i)^2+(3\lambda+i)(\mu-i)+5i=10$, find the exact values of the real numbers $\lambda$ and $\mu$.

Find the real values of $a$ and $b$ given the following:

(a) $(2+ai)(1+i)=5+ib$.

(b) $(a-3i)(1+ib)=-1-5i$.

Find the square roots of the complex number $15+8i$.

Write down the conjugate of (i) $z_1=1+3i$ and (ii) $z_2=-1-4i$.

Let $z=2+3i$. State the values of (a) $z+z^*$, (b) $z-z^*$ and (c) $zz^*$.

Given that $\dfrac{z}{z^*}=-1$ and $zz^*=1$, find the complex number $z$.