Let a, b, c be three real numbers in an Arithmetic Progression such that a If D(-12) = 0 , then the value of c is equal to

Step-by-step solution

Let the terms of the Arithmetic Progression be a = 4-d , b = 4 , and c = 4+d , since their sum is 12 . As a 0 . Expanding the determinant D(x) by applying R₁ R₁ - R₂ and R₂ R₂ - R₃ : D(x) = vmatrix a & -b & 0 0 & b & -c x & x & x+c vmatrix Expanding along the first row gives: D(x) = abc + x(ab + bc + ca) Now, express the coefficients in terms of d : abc = (4-d)(4)(4+d) = 4(16-d^2) = 64 - 4d^2 ab + bc + ca = (4-d)4 + 4(4+d) + (4-d)(4+d) = 16 - 4d + 16 + 4d + 16 - d^2 = 48 - d^2 Substitute these into D(x) : D(x) = (6