If the system of equations ax+y+z=0, x+by+z=0,x+y+cz=0,a,b,c≠1  has a non trivial solution ( non-zero solution) , then 11−a+11−b+11−c=

Given system of equations has a nontrivial solution   ⇒ coefficient matrix is singular⇒a111b111c=⇒abc−1−1c−1+11−b=0⇒abc−a−c+1+1−b=0⇒abc=a+b+c−211−a+11−b+11−c=(1−b)(1−c)+(1−a)(1−c)+(1−a)(1−b)(1−a)(1−b)(1−c)⇒3−2(a+b+c)+bc+ac+ab1−(a+b+c)+ab+bc+ca−abc=3−2(a+b+c)+bc+ca+ab1−(a+b+c)+ab+bc+ca−(a+b+c−2)=1