Maths4U Original · Short practice

Let \(\displaystyle z=\)\(18 + 3 i\) and \(\displaystyle w=\)\(4 + 3 i\). Work exactly.

Find Re(z+w).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(\displaystyle z=\)\(19 + 4 i\) and \(\displaystyle w=\)\(2 + 4 i\). Work exactly.

Find Re(z+w).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

\(\displaystyle P(z)=\)\(z^{3} - 40 z^{2} + 453 z - 754\). It is given that \(19 + 4 i\) is a root.

Find the positive imaginary part of a non-real root.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

\(\displaystyle P(z)=\)\(z^{3} - 43 z^{2} + 545 z - 1275\). It is given that \(20 + 5 i\) is a root.

Find the positive imaginary part of a non-real root.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(\displaystyle z=6(\cos(- \frac{\pi}{6})+i\sin(- \frac{\pi}{6}))\). Use principal arguments in (−π,π].

Find the principal arg(\(\displaystyle z^{2}\)) divided by π.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(\displaystyle z=2(\cos(- \frac{3 \pi}{4})+i\sin(- \frac{3 \pi}{4}))\). Use principal arguments in (−π,π].

Find the principal arg(\(\displaystyle z^{2}\)) divided by π.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(\displaystyle c=\)\(21 + 2 i\) and \(\displaystyle \alpha =\)\(\displaystyle \operatorname{atan}{\left(\frac{2}{3} \right)}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=208\). For the short checks use \(\displaystyle h=21, u=3\).

Write z=x+iy.

Find the x-coordinate of the ray–circle intersection.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(\displaystyle c=\)\(22 + 3 i\) and \(\displaystyle \alpha =\)\(\operatorname{atan}{\left(3 \right)}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=40\). For the short checks use \(\displaystyle h=22, u=1\).

Write z=x+iy.

Find the x-coordinate of the ray–circle intersection.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.