Under which of the following conditions does the determinant | array lll a & b & c b & c & a c & a & b array | vanish? 1. a+b+c=0 2. a^3+b^3+c^3=3 a b c 3. a^2+b^2+c^2-a b-b c-c a=0 Select the correct answer using the code given below :

Step-by-step solution

Let = | array lll a & b & c b & c & a c & a & b array |=0 Applying R₁ R₁+R₂+R₃= | array ccc a+b+c & a+b+c & a+b+c b & c & a c & a & b array |=0=(a+b+c) | array ccc 1 & 1 & 1 b & c & a c & a & b array |=0 Applying c₂ c₂-c₁ and c₃ c₃-c₁=(a+b+c) | array ccc 1 & 0 & 0 b & c-b & a-b c & a-c & b-c array |=0 aligned & =(a+b+c) [-a^2-b^2-c^2+a b+b c+c a ]=0 & =-(a+b+1) [a^2+b^2+c^2-a b-b c-c a ]=0 & =- (a^3+b^3+c^3-3 a b c )=0 & a+b+ c =0 or a^2+b^2+c^2-a b-b c-c a=0 & or a^3+b^3+c^3=3 a b c aligned Hence all statement 1,2