1Q1) \begin{vmatrix} 5^{n}5+7^{n}7 & 5^{n}5^{2}+7^{n}7^{2} &5^{n}5^{3}+7^{n}7^{3} \\ 5^{n}5^{4}+7^{n}7^{4}& 5^{n}5^{5}+7^{n}7^{5} & 5^{n}5^{6}+7^{n}7^{6}\\ 5^{n}5^{7}+7^{n}7^{7}& 5^{n}5^{8}+7^{n}7^{8} & 5^{n}5^{9}+7^{n}7^{9} \end{vmatrix}
first split first column then 2nd then 3rd and take common suitable term.
columns will be identical and sum adds upto 0
2) f_{k}(x) = \frac{a_{k}x^{3}}{3}+b_{k}\frac{x^{2}}{2}+c_{k}x\Rightarrow f_{k}'(x) = a_{k}x^{2}+b_{k}x+c_{k}
follow similar way of expanding columns and taking common.
i thnk this one also adds upto to give 0
for expanding follow \begin{vmatrix} a+x & b & c\\ d+y& e& f\\ g+z&h & i \end{vmatrix} = \begin{vmatrix} a & b & c\\ d& e& f\\ g&h & i \end{vmatrix}+\begin{vmatrix} x & b & c\\ y& e& f\\ z&h & i \end{vmatrix}