Consider the system of linear equations in x, y, z: x+2 y+t z=0, 6 x+y+5 t z=0, 3 x+t^2 y+f(t) z=0, where f: R → R is a differentiable function. If…

Consider the system of linear equations in $\displaystyle x, y, z$ : $$\begin{aligned} & x+2 y+t z=0, \\ & 6 x+y+5 t z=0, \\ & 3 x+t^2 y+f(t) z=0, \end{aligned} $$ where $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $\displaystyle t \in \mathbb{R}$, then $\displaystyle f$