Let \(\mathrm{z}=x+\mathrm{i} y\) be a complex number, where \(x\) and \(y\) are integers and \(\mathrm{i}=\sqrt{-1}\). Then the area of the rectangle whose vertices are the roots of the equation \(z^3+\bar{z} z^3=350\) is

\(\text {Given, } \mathrm{z}(\overline{\mathrm{z}})^3+\overline{\mathrm{z}} \mathrm{z}^3=350 \)
\( \Rightarrow \mathrm{z} \overline{\mathrm{z}}(\overline{\mathrm{z}})^2+\overline{\mathrm{z}} \mathrm{zz}^2=350\)
\(\Rightarrow|z|^2\left\{(\bar{z})^2+z^2\right\}=350 \ldots\left[\because \bar{z}=|z|^2\right] \)
\( \Rightarrow|z|^2\left[(x-\mathrm{i} y)^2+(x+\mathrm{i} y)^2\right]=350 \)
\( \Rightarrow 2\left(x^2+y^2\right)\left(x^2-y^2\right)=350 \)
\( \Rightarrow 2\left(x^4-y^4\right)=350 \)
\( \Rightarrow x^4-y^4=175 \)
\( \Rightarrow x^4=256, y^4=81 \ldots[\because x, y \text { are integers }] \)
\( \Rightarrow x^2=16, y^2=9 \)
\( \Rightarrow x= \pm 4, y= \pm 3\)
\(\therefore\) vertices of the rectangle are \((4,3) ;(-4,3)\), \((-4,-3)\) and \((4,-3)\).

\(\begin{aligned} \therefore \text {Required area} & =\text {length} \times \text {breadth} \\ & =8 \times 6 \\ & =48 \text { sq. units }\end{aligned}\)