Solution Given z1=2+7i, z2=−5+3i.\boxed{\text{Given } z_1=2+7i,\ z_2=-5+3i.}Given z1=2+7i, z2=−5+3i. (i) ∣2z1−4z2∣|2z_1-4z_2|∣2z1−4z2∣ 2z1=2(2+7i)=4+14i4z2=4(−5+3i)=−20+12i2z1−4z2=(4+14i)−(−20+12i)=24+2i∣2z1−4z2∣=∣24+2i∣=242+22=576+4=580=2145\begin{aligned} & \boxed{2z_1=2(2+7i)=4+14i} \\ \\ & \boxed{4z_2=4(-5+3i)=-20+12i} \\ \\ & 2z_1-4z_2=(4+14i)-(-20+12i) \\ & = 24+2i \\ \\ & |2z_1-4z_2| = |24+2i| = \sqrt{24^2+2^2} \\ & = \sqrt{576+4}=\sqrt{580}=2\sqrt{145} \end{aligned}2z1=2(2+7i)=4+14i4z2=4(−5+3i)=−20+12i2z1−4z2=(4+14i)−(−20+12i)=24+2i∣2z1−4z2∣=∣24+2i∣=242+22=576+4=580=2145 (ii) ∣3z1+2z2∣|3z_1+2z_2|∣3z1+2z2∣ 3z1=3(2+7i)=6+21i2z2=2(−5+3i)=−10+6i3z1+2z2=(6+21i)+(−10+6i)=−4+27i∣3z1+2z2∣=∣−4+27i∣=(−4)2+272=16+729=745\begin{aligned} & \boxed{3z_1=3(2+7i)=6+21i} \\ \\ & \boxed{2z_2=2(-5+3i)=-10+6i} \\ \\ & 3z_1+2z_2=(6+21i)+(-10+6i) \\ & = -4+27i \\ \\ & |3z_1+2z_2| = |-4+27i| = \sqrt{(-4)^2+27^2} \\ & = \sqrt{16+729}=\sqrt{745} \end{aligned}3z1=3(2+7i)=6+21i2z2=2(−5+3i)=−10+6i3z1+2z2=(6+21i)+(−10+6i)=−4+27i∣3z1+2z2∣=∣−4+27i∣=(−4)2+272=16+729=745 (iii) ∣−7z2+2z2ˉ∣|-7z_2+2\bar{z_2}|∣−7z2+2z2ˉ∣ z2=−5+3i ⇒ z2ˉ=−5−3i−7z2=−7(−5+3i)=35−21i2z2ˉ=2(−5−3i)=−10−6i−7z2+2z2ˉ=(35−21i)+(−10−6i)=25−27i∣−7z2+2z2ˉ∣=∣25−27i∣=252+(−27)2=625+729=1354\begin{aligned} & \boxed{z_2=-5+3i\ \Rightarrow\ \bar{z_2}=-5-3i} \\ \\ & -7z_2=-7(-5+3i)=35-21i \\ & 2\bar{z_2}=2(-5-3i)=-10-6i \\ \\ & -7z_2+2\bar{z_2}=(35-21i)+(-10-6i) \\ & = 25-27i \\ \\ & |-7z_2+2\bar{z_2}| = |25-27i| = \sqrt{25^2+(-27)^2} \\ & = \sqrt{625+729}=\sqrt{1354} \end{aligned}z2=−5+3i ⇒ z2ˉ=−5−3i−7z2=−7(−5+3i)=35−21i2z2ˉ=2(−5−3i)=−10−6i−7z2+2z2ˉ=(35−21i)+(−10−6i)=25−27i∣−7z2+2z2ˉ∣=∣25−27i∣=252+(−27)2=625+729=1354 (iv) ∣(z1+z2)3∣|(z_1+z_2)^3|∣(z1+z2)3∣ z1+z2=(2+7i)+(−5+3i)=−3+10i∣(z1+z2)3∣=∣z1+z2∣3∣z1+z2∣=∣−3+10i∣=(−3)2+102=109∣(z1+z2)3∣=(109)3=109109\begin{aligned} & z_1+z_2=(2+7i)+(-5+3i)=-3+10i \\ \\ & \boxed{|(z_1+z_2)^3|=|z_1+z_2|^3} \\ \\ & |z_1+z_2|=|-3+10i|=\sqrt{(-3)^2+10^2}=\sqrt{109} \\ \\ & |(z_1+z_2)^3|=(\sqrt{109})^3=109\sqrt{109} \end{aligned}z1+z2=(2+7i)+(−5+3i)=−3+10i∣(z1+z2)3∣=∣z1+z2∣3∣z1+z2∣=∣−3+10i∣=(−3)2+102=109∣(z1+z2)3∣=(109)3=109109