1. Determine whether the following indeterminate equations have non-zero solutions. If solvable, find a set of solutions according to the method of Theorem 1:
(i) $3 x^{2}+7 y^{2}-5 z^{2}=0$;
(ii) $5 x^{2}-17 y^{2}+3 z^{2}=0$;
(iii) $19 x^{2}-7 y^{2}-11 z^{2}=0$;
(iv) $7 x^{2}-43 y^{2}-3 z^{2}=0$;
(v) $4 x^{2}+3 y^{2}-5 z^{2}=0$;
(vi) $41 x^{2}+3 y^{2}-11 z^{2}=0$.
1. (i) no solution; (ii) has solution, $x=1, y=1, z=2$; (iii) no solution; (iv) has solution, $x=5, y=$ $2, z=1$; (v) no solution; (vi) has solution, $x=y=1, z=2$.