If `(x)=|a_1+x b_1+x c_1+x a_2+x b_2+x c_2+x a_3+x b_3+x c_3+x|` , show that `^(x)=0` and that`(x)=(0)+S x ,w h e r eS` denotes the sum of all the cofactors of all the element in `(0)dot`

If a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0, a_3x+b_3y+c_3z=0 and |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0 , then the given system then

|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0 then the system of equations a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0, a_3x+b_3y+c_3z=0 has (A) no solution (B) one trivial and one non trivial solutions (C) only the trivial solution (0,0,0) (D) more than two solution

If the line a_1 x + b_1 y+ c_1 = 0 and a_2 x + b_2 y + c_2 = 0 cut the coordinate axes in concyclic points, prove that : a_1 a_2 = b_1 b_2 .

Statement -1 : If Delta(x)=|{:(f_1(x),f_2(x),f_3(x)),(a_2,b_2,c_2),(a_3,b_3,c_3):}| where, f_1 , f_2,f_3 are different functions and a_2,b_2,c_2 and a_3,b_3,c_3 are constants then intDelta(x)dx=|{:(intf_1(x)dx,intf_2(x)dx,intf_3(x)dx),(a_2,b_2,c_2),(a_3,b_3,c_3):}| because Statement-2 : Integration of sum of several functions is equal to the sum of integration of individual function.

If a_1 +a_2cos 2x + a_3sin^(2)x = 0 be true for all x then a_1, a_2 and a_3 can be uniquely determined. True or False ?

Statement-1: If a, b, c are distinct real numbers, then a((x-b)(x-c))/((a-b)(a-c))+b((x-c)(x-a))/((b-c)(b-a))+c((x-a)(x-b))/((c-a)(c-b))=x for each real x. Statement-2: If a, b, c in R such that ax^(2) + bx + c = 0 for three distinct real values of x, then a = b = c = 0 i.e. ax^(2) + bx + c = 0 for all x in R .