Thread: Complex numbers question

1. Complex numbers question

Question:
a)Given that the complex numbers $\displaystyle w_{1}$ and $\displaystyle w_{2}$ are the roots of the equation $\displaystyle z^2-5-12i=0$, express $\displaystyle w_{1}$ and $\displaystyle w_{2}$ in the form of $\displaystyle a +ib$ where a and b are real

b)i)Indicate the point sets in an argand diagram corresponding to the sets of the complex numbers
$\displaystyle A= \left \{z:\left | z \right | =3, z \epsilon \mathbb{C} \right \}$
$\displaystyle B=\left \{z:\left | z \right | =2, z \epsilon \mathbb{C} \right \}$

ii)Shade the region corresponding to the values of z for which the inequalities
$\displaystyle 2 <\left | z \right |<3$ and $\displaystyle 30^{\circ}<arg(z)<60^{\circ}$

are simultaneously satisfied.

My attempt:
a)
$\displaystyle z^2-5-12i=0$
$\displaystyle z^2= 5+12i$
]$\displaystyle z= \pm (5+12i)^{(0.5)}$
note that $\displaystyle (3+2i)^2 = 5+12i$
thus $\displaystyle z = \pm (3+2i)$
$\displaystyle w_{1}= 3+2i$
$\displaystyle w_{2}= -(3+2i)$

b)i)

$\displaystyle A={z:\left | z \right | =3, z \epsilon \mathbb{C}}$
$\displaystyle B={z:\left | z \right | =2, z \epsilon \mathbb{C}}$

ii)$\displaystyle 2 <\left | z \right |<3$ and $\displaystyle 30^{\circ}<arg(z)<60^{\circ}$

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

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

2. Re: Complex numbers question

I'm not sure where you are getting the green ray from in your diagram.

The shaded area should be between the two circles and between 30 and 60 degrees. Why have you gut it off shorter at 33.7 degrees?

Also the shaded area should be bounded by dotted lines to indicate non-equality (eg a full line indicates < or = while a dotted line indicates <)

3. Re: Complex numbers question

I've just realised where you might have got the 33.7 degrees from. Is that the argument of 3 + 2i from part (a)? Part (b) is a separate question from part (a). There is no link.

4. Re: Complex numbers question

Originally Posted by Debsta
I've just realised where you might have got the 33.7 degrees from. Is that the argument of 3 + 2i from part (a)? Part (b) is a separate question from part (a). There is no link.
thank you that was the reason i did that but thanks for correcting me