Q$-1)$ Consider the equation $3x^4-2x^{2}y+y^{2}z+xy{z}^3=65$ which defines $z$ as a differentiable function of $x$ and $y.$

$($i$)$ Find the value of $z(1,2).$

Hint: $t^3+2t-33=(t^2+3t+11)(t-3).$

$($ii$)$ Calculate $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}{|}_{(1,2)}|$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}{|}_{(1,2)}|$

$($iii$)$ Write the linearization of $z(x,y)$ at the point $(x,y)=(1,2)$ in the form $L(x,y)=Ax+By+C,$ where $A,B$ and $C$ are rational numbers.

$($iv$)$ Calculate $L(\frac{3}{2},\frac{3}{2}).$

Note: The difference between $L(\frac{3}{2},\frac{3}{2})$ and $z(\frac{3}{2},\frac{3}{2})$ is $0\mathrm{.}0032$dots.