FInd `x` when `|[x+a, a^2, a^3] , [x+b, b^2, b^3] , [x+c, c^2, c^3]|=0` where `a,b,c` are distinct numbers and `a!=b!=c`

Find the nature of roots of (b-x)^(2)-4(a-x)(c-x)=0 where a,b,c are distinct real numbers.

Prove [[a^3-x^3,a^2,a],[b^3-x^3,b^2,b],[c^3-x^3,c^2,c]] = (a-b)(a-c)(b-c)(abc-x^3)

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 .

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 .

If range of f (x)= ((ln x) (ln x ^(2))+ln x ^(3)+3)/(ln ^(2) x+ln x ^(2)+2) can be expressed as [ (a)/(b),(c )/(d)] where a,b,c and d are prime numbers (not nacessarily distinct) then find the value of ((a+b+c+d))/(2).

If range of f (x)= ((ln x) (ln x ^(2))+ln x ^(3)+3)/(ln ^(2) x+ln x ^(2)+2) can be expressed as [ (a)/(b),(c )/(d)] where a,b,c and d are prime numbers (not nacessarily distinct) then find the value of ((a+b+c+d))/(2).

If a ,\ b ,\ c are real numbers, then find the intervals in which f(x)=|(x+a^2,a b, a c),( a b, x+b^2,b c),( a c, b c, x+c^2)| is increasing or decreasing.

Solve the system of equations : z+ay+a^2x+a^3=0 z+by+b^2x+b^3=0 z+cy+c^2x+c^3=0] where a != b != c.