Suppose that the Wronskian of two functions $f_1(t)$ and $f_2(t)$ is given by

$W(t) = t^2-4 = \det$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $f_1(t)$ $f_2(t)$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $f_1'(t)$ $f_2'(t)$

Even though you don't know the functions $f_1$ and $f_2$ you can determine whether the following questions are true or false.

? True False  1. The equations $$\begin{array}{cc} a f_1(2) + b f_2(2) &= 0 \cr a f_1'(2) + b f_2'(2) &= 0 \end{array}$$ have more than one solution.

? True False  2. The equations $$\begin{array}{cc} a f_1(2) + b f_2(2) &= c \cr a f_1'(2) + b f_2'(2) &= d \end{array}$$ have a unique solution for any $c$ and $d$

? True False  3. The vectors $(f_1(0) ,f_1'(0) )$ and $(f_2(0) ,f_2'(0) )$ are linearly independent

? True False  4. The vectors $(f_1(4) ,f_1'(4) )$ and $(f_2(4) ,f_2'(4) )$ are linearly independent

? True False  5. The vectors $(f_1(-2) ,f_1'(-2) )$ and $(f_2(-2) ,f_2'(-2) )$ are linearly independent