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

Let \(\displaystyle z=\)\(20 + 5 i\) and \(\displaystyle w=\)\(3 + 5 i\). 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 \(\displaystyle z=\)\(20 + 5 i\) and \(\displaystyle w=\)\(3 + 5 i\). Work exactly.

Find Im(zw).

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 \(\displaystyle z=\)\(21 + 2 i\) and \(\displaystyle w=\)\(4 + i\). Work exactly.

Find Im(zw).

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 \(\displaystyle z=\)\(21 + 2 i\) and \(\displaystyle w=\)\(4 + i\). Work exactly.

Find \(\displaystyle \operatorname{Re}(\frac{z}{w})\).

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 \(\displaystyle z=\)\(22 + 3 i\) and \(\displaystyle w=\)\(2 + 2 i\). Work exactly.

Find \(\displaystyle \operatorname{Re}(\frac{z}{w})\).

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 \(\displaystyle z=\)\(22 + 3 i\) and \(\displaystyle w=\)\(2 + 2 i\). Work exactly.

Find \(\displaystyle |z|^{2}\).

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 \(\displaystyle z=\)\(23 + 4 i\) and \(\displaystyle w=\)\(3 + 3 i\). Work exactly.

Find \(\displaystyle |z|^{2}\).

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 real part of the specified 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

\(\displaystyle P(z)=\)\(z^{3} - 46 z^{2} + 613 z - 1780\). It is given that \(21 + 2 i\) is a root.

Find the real part of the specified 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

\(\displaystyle P(z)=\)\(z^{3} - 46 z^{2} + 613 z - 1780\). It is given that \(21 + 2 i\) 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

\(\displaystyle P(z)=\)\(z^{3} - 46 z^{2} + 581 z - 986\). It is given that \(22 + 3 i\) 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

\(\displaystyle P(z)=\)\(z^{3} - 46 z^{2} + 581 z - 986\). It is given that \(22 + 3 i\) is a root.

Find the product of the conjugate non-real roots.

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

\(\displaystyle P(z)=\)\(z^{3} - 49 z^{2} + 683 z - 1635\). It is given that \(23 + 4 i\) is a root.

Find the product of the conjugate non-real roots.

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

\(\displaystyle P(z)=\)\(z^{3} - 49 z^{2} + 683 z - 1635\). It is given that \(23 + 4 i\) is a root.

Find the real root of P.

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

\(\displaystyle P(z)=\)\(z^{3} - 52 z^{2} + 793 z - 2404\). It is given that \(24 + 5 i\) is a root.

Find the real root of P.

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 |z|.

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 \(\displaystyle z=3(\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))\). Use principal arguments in (−π,π].

Find |z|.

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 \(\displaystyle z=3(\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))\). Use principal arguments in (−π,π].

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

Find arg z 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 \(\displaystyle z=4(\cos(\frac{\pi}{4})+i\sin(\frac{\pi}{4}))\). 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 \(\displaystyle z=5(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))\). 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 \(\displaystyle z=5(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))\). Use principal arguments in (−π,π].

Find \(\displaystyle |z^{3}|\).

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 \(\displaystyle z=6(\cos(\frac{2 \pi}{3})+i\sin(\frac{2 \pi}{3}))\). Use principal arguments in (−π,π].

Find \(\displaystyle |z^{3}|\).

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

On an Argand diagram let \(\displaystyle c=\)\(22 + 3 i\). Consider \(|z-c|=4\).

Write z=x+iy and \(\displaystyle r=4\).

Find the centre real coordinate.

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

On an Argand diagram let \(\displaystyle c=\)\(23 + 4 i\). Consider \(|z-c|=5\).

Write z=x+iy and \(\displaystyle r=5\).

Find the centre real coordinate.

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

On an Argand diagram let \(\displaystyle c=\)\(23 + 4 i\). Consider \(|z-c|=5\).

Write z=x+iy and \(\displaystyle r=5\).

Find the centre imaginary coordinate.

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

On an Argand diagram let \(\displaystyle c=\)\(24 + 5 i\). Consider \(|z-c|=6\).

Write z=x+iy and \(\displaystyle r=6\).

Find the centre imaginary coordinate.

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

On an Argand diagram let \(\displaystyle c=\)\(24 + 5 i\). Consider \(|z-c|=6\).

Write z=x+iy and \(\displaystyle r=6\).

Find the squared radius.

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

On an Argand diagram let \(\displaystyle c=\)\(25 + 2 i\). Consider \(|z-c|=7\).

Write z=x+iy and \(\displaystyle r=7\).

Find the squared radius.

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

On an Argand diagram let \(\displaystyle c=\)\(25 + 2 i\). Consider \(|z-c|=7\).

Write z=x+iy and \(\displaystyle r=7\).

Find the greatest real coordinate on the locus.

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

On an Argand diagram let \(\displaystyle c=\)\(26 + 3 i\). Consider \(|z-c|=3\).

Write z=x+iy and \(\displaystyle r=3\).

Find the greatest real coordinate on the locus.

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

On an Argand diagram let \(\displaystyle a=\)\(1 + 16 i\) and \(\displaystyle b=\)\(3 + 26 i\). Write z=x+iy.

Find the x-coordinate of the segment midpoint.

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

On an Argand diagram let \(\displaystyle a=\)\(2 + 17 i\) and \(\displaystyle b=\)\(6 + 21 i\). Write z=x+iy.

Find the x-coordinate of the segment midpoint.

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

On an Argand diagram let \(\displaystyle a=\)\(2 + 17 i\) and \(\displaystyle b=\)\(6 + 21 i\). Write z=x+iy.

Find its y-coordinate.

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

On an Argand diagram let \(\displaystyle a=\)\(3 + 18 i\) and \(\displaystyle b=\)\(9 + 24 i\). Write z=x+iy.

Find its y-coordinate.

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

On an Argand diagram let \(\displaystyle a=\)\(3 + 18 i\) and \(\displaystyle b=\)\(9 + 24 i\). Write z=x+iy.

Find the gradient of the equal-distance locus.

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

On an Argand diagram let \(\displaystyle a=\)\(4 + 19 i\) and \(\displaystyle b=\)\(6 + 27 i\). Write z=x+iy.

Find the gradient of the equal-distance locus.

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

On an Argand diagram let \(\displaystyle a=\)\(4 + 19 i\) and \(\displaystyle b=\)\(6 + 27 i\). Write z=x+iy.

Find its y-intercept.

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

On an Argand diagram let \(\displaystyle a=\)\(5 + 20 i\) and \(\displaystyle b=\)\(9 + 30 i\). Write z=x+iy.

Find its y-intercept.

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 \(\displaystyle c=\)\(17 + 2 i\) and \(\displaystyle \alpha =\)\(\displaystyle \frac{\pi}{4}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=72\). For the short checks use \(\displaystyle h=17, u=2\).

Write z=x+iy.

Find the gradient of the supporting line.

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 \(\displaystyle c=\)\(19 + 4 i\) and \(\displaystyle \alpha =\)\(\operatorname{atan}{\left(4 \right)}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=68\). For the short checks use \(\displaystyle h=19, u=1\).

Write z=x+iy.

Find the gradient of the supporting line.

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 \(\displaystyle c=\)\(18 + 3 i\) and \(\displaystyle \alpha =\)\(\displaystyle \frac{\pi}{4}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=288\). For the short checks use \(\displaystyle h=18, u=3\).

Write z=x+iy.

Find its y-intercept.

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 \(\displaystyle c=\)\(19 + 4 i\) and \(\displaystyle \alpha =\)\(\operatorname{atan}{\left(4 \right)}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=68\). For the short checks use \(\displaystyle h=19, u=1\).

Write z=x+iy.

Find its y-intercept.

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 \(\displaystyle c=\)\(19 + 4 i\) and \(\displaystyle \alpha =\)\(\operatorname{atan}{\left(4 \right)}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=68\). For the short checks use \(\displaystyle h=19, u=1\).

Write z=x+iy.

Find the ray height at \(\displaystyle x=h+u\).

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 \(\displaystyle c=\)\(20 + 5 i\) and \(\displaystyle \alpha =\)\(\displaystyle \operatorname{atan}{\left(\frac{5}{2} \right)}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=261\). For the short checks use \(\displaystyle h=20, u=2\).

Write z=x+iy.

Find the ray height at \(\displaystyle x=h+u\).

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 \(\displaystyle c=\)\(20 + 5 i\) and \(\displaystyle \alpha =\)\(\displaystyle \operatorname{atan}{\left(\frac{5}{2} \right)}\). Consider arg(z−c)=α, and its intersection with the circle \(|z-c|^2=261\). For the short checks use \(\displaystyle h=20, u=2\).

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 \(\displaystyle c=\)\(21 + 2 i\) 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.