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Further Maths Complex Numbers Question

Question:
a)Given that the complex numbers

w_{1}

and

w_{2}

are the roots of the equation

z^2-5-12i=0

, express

w_{1}

and

w_{2}

in the form of

a +ib

where a and b are real

b)i)Indicate the point sets in an argand diagram corresponding to the sets of the complex numbers

A={z:\left | z \right | =3, z \epsilon \mathbb{C}}

B={z:\left | z \right | =2, z \epsilon \mathbb{C}}

ii)Shade the region corresponding to the values of z for which the inequalities

2 <\left | z \right |<3

and

30^{\circ}<arg(z)<60^{\circ}

are simultaneously satisfied.

My attempt:
a)

z^2-5-12i=0

z^2= 5+12i

z= \pm (5+12i)^(0.5)

note that

(3+2i)^2 = 5+12i

thus

z = \pm (3+2i)

w_{1}= 3+2i

w_{2}= -(3+2i)

b)

i)

A={z:\left | z \right | =3, z \epsilon \mathbb{C}}

B={z:\left | z \right | =2, z \epsilon \mathbb{C}}

ii)

2 <\left | z \right |<3

and

30^{\circ}<arg(z)<60^{\circ}

Attachment not found

I shaded the region I thought would be the right answer in purple

The part I am finding difficult is part b. Please help

(edited 5 years ago)

Original post by bigmansouf

...

Where did 33.7 come from?

Your purple region is almost fine, you just need to extend it all the way to the blue line.

Reply 2

bigmansouf

arg(z) = \frac{2}{3}

\alpha = arctan\frac{2}{3} = 33.69006753 ^{\circ} = 33.7 ^{\circ}

Thank you for your help

Original post by RDKGames

Where did 33.7 come from?

Your purple region is almost fine, you just need to extend it all the way to the blue line.

Reply 3

RDKGames

Study Forum Helper

Original post by bigmansouf

arg(z) = \frac{2}{3}

\alpha = arctan\frac{2}{3} = 33.69006753 ^{\circ} = 33.7 ^{\circ}

Thank you for your help

Part (a) has nothing to do with part (b)

Reply 4

bigmansouf

it is an old question from University of London School Examinations Department

Original post by RDKGames

Part (a) has nothing to do with part (b)

Reply 5

bigmansouf

it is an old question from University of London School Examinations Department

Original post by RDKGames

Part (a) has nothing to do with part (b)

Reply 6

bigmansouf

Original post by RDKGames

Part (a) has nothing to do with part (b)

thank you i managed to do it

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