Question

Answer 1

Bài 4:

a: Số phức liên hợp của z=2+3i là: \(\overline{z}=2-3i\)

\(\left|z\right|=\sqrt{2^2+3^2}=\sqrt{13}\)

b: \(z=\left(2+3i\right)^3=2^3+3\cdot2^2\cdot3i+3\cdot2\cdot\left(3i\right)^2+\left(3i\right)^3\)

\(=8+36i+6\cdot9\cdot\left(-1\right)-27i=-46+9i\)

=>\(\overline{z}=-46-9i\)

\(\left|z\right|=\sqrt{\left(-46\right)^2+9^2}=13\sqrt{13}\)

c: \(z=\frac{2+3i}{1-2i}=\frac{\left(2+3i\right)\left(1+2i\right)}{\left(1-2i\right)\left(1+2i\right)}\)

\(=\frac{2+4i+3i+6i^2}{1-4i^2}=\frac{7i-4}{5}\)

=>\(\overline{z}=\frac{7i+4}{5}\)

\(\left|z\right|=\sqrt{\left(\frac75\right)^2+\left(\frac45\right)^2}=\frac{\sqrt{65}}{5}\)

d: \(z=\sqrt2-\frac43i\)

=>\(\overline{z}=\sqrt2+\frac43i\)

\(\left|z\right|=\sqrt{\left(\sqrt2\right)^2+\left(-\frac43\right)^2}=\sqrt{2+\frac{16}{9}}=\sqrt{\frac{34}{9}}=\frac{\sqrt{34}}{3}\)

Bài 3:

a: \(\left(3-2i\right)x+\left(5-7i\right)y=1-3i\)

=>(3x+5y)+i(-2x-7y)=1-3i

=>3x+5y=1 và -2x-7y=-3

=>3x+5y=1 và 2x+7y=3

=>6x+10y=2 và 6x+21y=9

=>6x+10y-6x-21y=2-9

=>-11y=-7

=>\(y=\frac{7}{11}\)

2x+7y=3

=>\(2x=3-7y=3-7\cdot\frac{7}{11}=3-\frac{49}{11}=\frac{33-49}{11}=-\frac{16}{11}\)

=>\(x=-\frac{8}{11}\)