Questions 1 - 9#

These questions examine various properties of complex numbers.

Q1#

If $$z_1 =2+i$$ and $$z_2 =-1-3i/2$$ calculate (a) $$z_1 + z_2$$ (b) $$z_1 - z_2$$
(c) $$-iz_1z_2$$

Q2#

(a) If $$i^2=-1$$, what are $$i^3,\;i^4,\;i^5$$, and $$i^6$$?

(b) What relationship links positive powers of i?

Q3#

If $$z = a - ib$$ what is $$i^3z$$?

Q4#

If $$z=3+4i$$ find $$z^2$$ and the modulus and argument of $$z^2$$.

Q5#

Calculate $$z = (2 - 5i)(3 + i) + 3i$$, and find the modulus and argument of the result.

Q6#

Express the number $$\displaystyle \frac{5 - i}{2-3i}$$ in the form $$z = a + bi$$ and find its modulus and argument.

Q7#

Simplify $$z = (2 - 5i)(3 + i)/(3 - i)$$ and find the modulus and argument of the result.

Q8#

Find the modulus and argument of

(a) $$\cos(\theta) - i \sin(\theta)$$,

(b) $$1-i\tan(\theta)$$, where $$0 \lt \theta \lt \pi/2$$ in both cases.

Q9#

If $$w=z^2$$ and $$z=x+iy$$ and $$w=u+it$$, find $$u$$ and $$t$$.