Let f(x) = vmatrix a^2 + ^2 x x^2 & ab & ac ab & b^2 + e^x-1 x & bc ac & bc & c^2 + x x vmatrix , where x 0 . If a = 2 , b = 3 , and c = 4 , then the value of _ x 0 f(x) is equal to :

Step-by-step solution

We know the standard limits: _ x 0 ^2 x x^2 = 1 , _ x 0 e^x-1 x = 1 , and _ x 0 x x = 1 . Applying the limit x 0 to the entries of the determinant, we get: _ x 0 f(x) = vmatrix a^2+1 & ab & ac ab & b^2+1 & bc ac & bc & c^2+1 vmatrix Multiply R₁, R₂, R₃ by a, b, c respectively and divide the determinant by abc : 1 abc vmatrix a(a^2+1) & a^2b & a^2c ab^2 & b(b^2+1) & b^2c ac^2 & bc^2 & c(c^2+1) vmatrix Taking a, b, c common from C₁, C₂, C₃ respectively: vmatrix a^2+1 & a^2 & a^2 b^2 & b^2+1 & b^2 c^2 & c^2 & c^2+1 vm