Question

1) Efetue: a) (3 – 2i) + (4 – 2i) = b) (8 + 7i) – (6 – 4i) = c) 2 (7 – 3i) + 5 (4 – 2i) – 3 ( 1 + 5i) = d) 5 (3 + 7i) – 2 (8 – i) – 7 (-2 + 3i) = 2) Efetue as multiplicações: a) (7 + 2i)(-3 + 4i) = b) (5 – 3i)i = c) (5 + i)(6 + 5i)(2 – 3i) = d) (-4 + i)(3 – i)(1 + 2i)i = 3) Efetue as divisões: a) 2 – 5i b) 3 + i c) 5 – 2i d) 1 + 3i 3 + 4i 2 – 3i 3i 2 – i 4) Dado o complexo z ǂ 0, chama-se inverso de z ao número complexo z-1 tal que: z . z-1 = 1. Determine o inverso do número complexo z = 5 – 3i. 5) Determine o afixo dos números complexos e represente-os num mesmo plano complexo: a) -5 – 3i b) -7i c) 6 d) -8 + i 6) Determine o módulo dos números complexos: a) -4 + 3i b) -7i c) -4 – 3i d) 6 7) Determine o argumento dos números complexos: a) - √2 + √2 i b) - √3 + 1 i c) 1 - √3 i 2 2 2 2 2 2 8) Escreva a forma trigonométrica dos números complexos: a) 32 b) -8i c ) - 1 + √3 i d) √3 + 1 2 2 9) Determine o argumento do número complexo z = 3(cos 600 – i sen 600 ).

Answer 1

1) Efetue:

a) (3 – 2i) + (4 – 2i) =7-4i

b) (8 + 7i) – (6 – 4i) =2+11i

c) 2 (7 – 3i) + 5 (4 – 2i) – 3 ( 1 + 5i) =14-6i+20-10i-3-15i=31-31 i

d) 5 (3 + 7i) – 2 (8 – i) – 7 (-2 + 3i)

= 15+35i-16+2i+14-21i=13+16i

2) Efetue as multiplicações:

a) (7 + 2i)(-3 + 4i) = -21+28i-6i-8=-29+22i

b) (5 – 3i)i = 5i+3

c) (5 + i)(6 + 5i)(2 – 3i)

= (30+25i+6i-5)(2-3i) =(25+31i)(2-3i)

=50-75i+62i+93=143-13i

d) (-4 + i)(3 – i)(1 + 2i)i

= (-12+4i+3i+1)(i-2)

=(-11+7i)(-2+i)=22-11i-14i-7=15-25i

3) Efetue as divisões:

a) 2 – 5i b) 3 + i c) 5 – 2i d) 1 + 3i 3 + 4i 2 – 3i 3i 2 – i

(Está questão está incompleta pois não diz por quem está sendo realizada a divisão).

4) Dado o complexo z≠0, chama-se inverso de z ao número complexo z-1 tal que: z . z-1 = 1. Determine o inverso do número complexo z = 5 – 3i.

$\mathsf{z^{-1}=\dfrac{1}{(5-3i)}\cdot\dfrac{(5+3i) }{(5+3i)}}\\\mathaf{z^{-1}=\dfrac{5+3i}{25+9}}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{z^{-1}=\dfrac{5}{34}+\dfrac{3}{34}i}}}}}$

5) Determine o afixo dos números complexos e represente-os num mesmo plano complexo:

a) -5 – 3i →A(-5,-3)

b) -7i →(0,-7)

c) 6 →(6,0)

d) -8 + i→(-8,1)

6) Determine o módulo dos números complexos: a) -4 + 3i

$\mathsf{\rho=\sqrt{(-4)^2+3^2}=\sqrt{16+9}=\sqrt{25}=5}$

b) -7i

$\mathsf{\rho=\sqrt{(-7)^2}=|-7|=7}$

c) -4 – 3i

$\mathsf{\rho=\sqrt{(-4)^2+(-3)^2}=\sqrt{16+9}=5}$

d) 6

$\mathsf{\rho=\sqrt{6^2}=6}$

7) Determine o argumento dos números complexos:

a) - √2 + √2 i

$\mathsf{\rho=\sqrt{(-\sqrt{2})^2+(\sqrt{2})^2}=\sqrt{2+2}=\sqrt{4}=2}\\\mathsf{sen(\theta)=\dfrac{\sqrt{2}}{2}}\\\mathsf{cos(\theta)=-\dfrac{\sqrt{2}}{2}}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=\dfrac{3\pi}{4}}}}}}$

b) - √3 + 1 i

$\mathsf{\rho=\sqrt{(-\sqrt{3})^2+1^2}=\sqrt{3+1}=\sqrt{4}=2}\\\mathsf{sen(\theta)=\dfrac{1}{2}}\\\mathsf{cos(\theta)=\dfrac{-\sqrt{3}}{2}}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=\dfrac{5\pi}{6}}}}}}$

c) 1 - √3 i

$\mathsf{\rho=\sqrt{1^2+(-\sqrt{3})^2}=\sqrt{1+3}=\sqrt{4}=2}\\\mathsf{sen(\theta)=-\dfrac{\sqrt{3}}{2}}\\\mathsf{cos(\theta)=\dfrac{1}{2}}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=\dfrac{5\pi}{3}}}}}}$

8) Escreva a forma trigonométrica dos números complexos:

a) 32→32+0i

$\mathsf{\rho=\sqrt{32^2}=32}\\\mathsf{sen(\theta)=\dfrac{0}{32}=0}\\\mathsf{cos(\theta)=\dfrac{32}{32}=1}\\\mathsf{\theta=2\pi}\\\large[tex]\boxed{\boxed{\boxed{\boxed{\mathsf{z=32[cos(2\pi)+isen(2\pi)]}}}}}$

b) -8i

$\mathsf{\rho=\sqrt{(-8)^2}=|-8|=8}\\\mathsf{sen(\theta)=-\dfrac{8}{8}=-1}\\\mathsf{cos(\theta)=\dfrac{0}{8}=0}\\\mathsf{\theta=\dfrac{3\pi}{2}}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{z=8[cos(\dfrac{3\pi}{2})+isen(\dfrac{3\pi}{2})]}}}}}$

c ) - 1 + √3 i

$\mathsf{\rho=\sqrt{(-1)^2+(\sqrt{3})^2}=\sqrt{1+3}=\sqrt{4}=2}\\\mathsf{sen(\theta)=\dfrac{\sqrt{3}}{2}}\\\mathsf{cos(\theta)=-\dfrac{1}{2}}\\\mathsf{\theta=\dfrac{2\pi}{3}}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{z=2[cos(\dfrac{2\pi}{3})+isen(\dfrac{2\pi}{3})]}}}}}$

d) √3 + 1 2 2

$\mathsf{\rho=\sqrt{(\sqrt{3}+122)^2}=\sqrt{3}+122}} \\\mathsf{cos(\theta)=\dfrac{\sqrt{3}122}{\sqrt{3}+122}=1}\\\mathsf{\theta=2\pi}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{z=1[cos(2\pi)+isen(2\pi)]}}}}}$

9) Determine o argumento do número complexo z = 3(cos 600 – i sen 600 )

600|360

240 1

$\boxed{\boxed{\boxed{\boxed{\mathsf{\theta=240^{\circ}}}}}}$