If | array lll a & a^2 & 1+a^3 b & b^2 & 1+b^3 c & c^2 & 1+c^3 array |=0 and vectors (1, a, a^2 ), (a, b, b^2 ) and (a, c, c^2 ) are non-coplanar, then the product abc equals

Step-by-step solution

| array lll a & a ^2 & 1+ a ^3 ~b & ~b ^2 & 1+ b ^3 c & c ^2 & 1+ c ^3 array |=0 | array lll a & a ^2 & 1 ~b & ~b ^2 & 1 c & c ^2 & 1 array |+ | array ccc a & a ^2 & a ^3 ~b & ~b ^2 & ~b ^3 c & c ^2 & c ^3 array |=0(a-b)(b-c)(c-a)+a b c(a-b)(b-c)(c-a)=0(a b c+1)[(a-b)(b-c)(c-a)]=0 As | array lll 1 & a & a ^2 1 & ~b & ~b ^2 1 & c & c ^2 array | 0 (given condition) abc =-1