if equations 2x-y+z=0 ,x-2y+z=0 and tx-y+2z=0 have non trivial solutions and f(x)be continuous functions such that f(5+x)+f(x)=2 then integrate f(x) dx (limit 0 to -2t)
Here are some hints to get you started.
First find the determinant of this matrix and set it equal to zero and solve for $\displaystyle t$
$\displaystyle \begin{vmatrix}2 & -1 & 1 \\ 1 & -2 & 1 \\ t & -1 & 2 \end{vmatrix}=0$
For the 2nd part use the linearlity of the integral to break it up into two parts
$\displaystyle \displaystyle \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b} f(x)dx$
Then make a u-substitution.
Good luck and post your workings if you get stuck.