First Principles AI

Question 1

Find the real values of $x$ and $y$ in each of the following:

(i)

$x+iy+2-3i=i(5-i)(3+4i)$

(ii)

$(x+iy)(1-i)=(2-3i)(-5+5i)-\dfrac{3}{5}i$

(iii)

$\dfrac{x}{2+i}+\dfrac{y}{3-i}=4+5i$

Solution Theory

Question 2

If $z_1=-13+24i$ and $z_2=x+yi$ , find the real values of $x$ and $y$ such that $z_1-z_2=-27+15i$ .

Solution Theory

Question 3

Find the real values of $x$ and $y$ if:

(i)

$(x+iy)^2=25+60i$

(ii)

$(x+iy)^2=64+48i$

(iii)

$(x+iy)^2=\dfrac{2i-3}{3+i}$

Solution Theory

Question 4

If $z_1=2+3i$ and $z_2=1-ai$ , find the real value of $a$ such that $\operatorname{Im}(z_1z_2)=7$ .

Solution Theory

Question 5

If $z_1=x+yi$ and $z_2=a+bi$ , find $x,y,a,b$ such that $z_1+z_2=10+4i$ and $z_1-z_2=6+2i$ .

Solution Theory

Question 6

Show that for all $z_1,z_2\in\mathbb{C}$ , $\overline{z_1z_2}=\bar{z_1}\,\bar{z_2}$ .

Solution Theory

Question 7

Find the square root of the following complex numbers:

$-7-24i$

$8-6i$

$-15-36i$

$119+120i$

Question 8

Find the square root of $13-20\sqrt{3}i$ and represent it on an Argand diagram.

Solution Theory

Question 9

Find the real values of $x$ and $y$ if $(-7+i)(x+yi)+(-1-5i)=i(11-i)$ .

Solution Theory

Question 10

Find the real values of $x$ and $y$ if $(5-2i)(x+yi)+3=i(11-i)-4i$ .

Solution Theory

Question 11

Find the real values of $u$ and $v$ if $\dfrac{u^2+v^2}{2+i}=\dfrac{v-3}{2-i}=4i$ .

Solution Theory

Question 12

If $z_1=4+5i$ and $z_2=a-2i$ , find the real values of $a$ such that $\operatorname{Re}(z_1z_2)=20$ .

Solution Theory

Questions