Determinants question bank (Maths)

If the system of linear equations:
$x+y+z=6$,
$x+2y+5z=10$,
$2x+3y+\lambda z=\mu$
has infinitely many solutions, then the value of $\lambda+\mu$ equals:

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