Answer:
See below for answers and explanation
Step-by-step explanation:
Problem 1
[tex]z_1z_2=(3+3i)(-4+0i)=(3)(-4)+(3)(0i)+(3i)(-4)+(3i)(0i)=-12-12i[/tex]
So, your best answer is Q
Problem 2
A complex number, [tex]z=a+bi[/tex], has a conjugate of [tex]z=a-bi[/tex] and vice versa. Thus, your best answer is P
Problem 3
[tex]z_1-z_2=(-5+1i)-(3+5i)=-5+i-3-5i=-8-4i[/tex]
So, your best answer is R
Problem 4
To divide two complex numbers, multiply the numerator and denominator by the denominator's conjugate and simplify:
[tex]\frac{4+7i}{2-6i}\\\\\frac{4+7i}{2-6i}*\frac{2+6i}{2+6i}\\\\\frac{(4)(2)+(4)(6i)+(7i)(2)+(7i)(6i)}{(2)(2)+(2)(6i)+(-6i)(2)+(-6i)(6i)}\\\\\frac{8+24i+14i+42i^2}{4+12i-12i-36i^2}\\\\\frac{8+38i+42(-1)}{4-36(-1)}\\\\\frac{8+38i-42}{4+36}\\\\\frac{-34+38i}{40}\\\\-\frac{34}{40}+\frac{38}{40}i\\ \\-\frac{17}{20}+\frac{19}{20}i[/tex]
Thus, the correct answer is D
Problem 5
[tex]z_T=\frac{z_1z_2}{z_1+z_2}\\ \\z_T=\frac{(70+40i)(50-35i)}{(70+40i)+(50-35i)}\\\\z_T=\frac{(70)(50)+(70)(-35i)+(40i)(50)+(40i)(-35i)}{70+40i+50-35i}\\\\z_T=\frac{3500-2450i+2000i-1400i^2}{120+5i}\\\\z_T=\frac{3500-450i-1400(-1)}{120+5i}\\\\z_T=\frac{3500-450i+1400}{120+5i}\\\\z_T=\frac{4900-450i}{120+5i}[/tex]
[tex]z_T=\frac{4900-450i}{120+5i}*\frac{120-5i}{120-5i}\\\\z_T=\frac{585750-78500i}{14425}\\\\z_T=\frac{23430}{577}-\frac{3140i}{577}\\\\z_T\approx40.607-5.442i[/tex]
Therefore, the correct answer is B
