Correct Answer: Neither I nor II

Solution : Statement I. If $x=12, y=-2$ and $z=-10$, then $x^3+y^3+z^3=360$.
$x^3 = (12)^3 = 1728$
$y^3 = (-2)^3 = -8$
$z^3 = (-10)^3 = -1000$
$\therefore$ $x^3 + y^3 + z^3 = 1728 + (-8) + (-1000) $
⇒ $x^3 + y^3 + z^3 = 720$
$\therefore$ Statement I is false.
Statement II. If $x+y=48$ and $4 x y=128$, then $4 x^2+4 y^2=4480$
$x+y = 48$
$4xy = 128$
⇒ $2xy = 64$
squaring both sides,
$x^2 + y^2 + 2xy = 2304$
$x^2 + y^2 + 64 = 230$
⇒ $x^2 + y^2 = 166$
Multiplying by 4 on both sides,
$4x^2 + 4y^2 = 4\times 166$
⇒ $4x^2 + 4y^2 = 664$
$\therefore$ Statement II is also incorrect.
Hence, the correct answer is neither I nor II.