1 z_(2)=a+3i Z_(1)+Z_(2),Z_(1)-2_(2),Z_(1),Z_(2),(Z_(1))/(Z_(2)) (3-z_(2))cdot z_(1)+(2z+z)/(z_(2))

$z_{2}=a+3i$<br />$Z_{1}+Z_{2},Z_{1}-2_{2},Z_{1},Z_{2},\frac {Z_{1}}{Z_{2}}$<br />$(3-z_{2})\cdot z_{1}+\frac {2z+z}{z_{2}}$<br /><br />To solve this problem, we need to perform complex number operations.<br /><br />Given:<br />$z_{2}=a+3i$<br />$Z_{1}=x+yi$<br /><br />1. $Z_{1}+Z_{2}$<br />$Z_{1}+Z_{2} = (x+yi) + (a+3i) = (x+a) + (y+3)i$<br /><br />2. $Z_{1}-2Z_{2}$<br />$Z_{1}-2Z_{2} = (x+yi) - 2(a+3i) = (x-2a) + (y-6)i$<br /><br />3. $Z_{1}$<br />$Z_{1} = x+yi$<br /><br />4. $Z_{2}$<br />$Z_{2} = a+3i$<br /><br />5. $\frac{Z_{1}}{Z_{2}}$<br />$\frac{Z_{1}}{Z_{2}} = \frac{x+yi}{a+3i} = \frac{(x+yi)(a-3i)}{(a+3i)(a-3i)} = \frac{(ax+3x)+(ay-3y)i}{a^2+9} = \frac{ax+3x}{a^2+9} + \frac{ay-3y}{a^2+9}i$<br /><br />6. $(3-Z_{2})\cdot Z_{1}+\frac{2Z_{1}+Z_{2}}{Z_{2}}$<br />$(3-Z_{2})\cdot Z_{1}+\frac{2Z_{1}+Z_{2}}{Z_{2}} = (3-(a+3i))(x+yi) + \frac{2(x+yi)+(a+3i)}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x)(x+yi) - (x+yi)(3i) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x)(x+yi) - 3i(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) + \frac{2x+2yi+a+3i}{a+3i} = (3-a-x-3i)(x+yi) +