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Most Important Selected Qs for JEE AdvancedMathematicsIndefinite Integration

If f(x) is a polynomial function such that f(x)+f^ (x)+f^ (x)+f^ (x)=x^3 , g(x)= f(x) x^3 d x and g(1)=1 , then: [Note: Where [k] denotes greatest integer function less than or equal to k .]

Options

  1. Ag(x) is strictly increasing function in (3, )
  2. Bthe value of _ x 1 (g(x))^ 1 x-1 is equal to e^2
  3. Cnumber of solution of the equation g(x)=0 is 2
  4. Dthe value of [g(e)] is equal to -1

Correct answer

D. the value of [g(e)] is equal to -1

Step-by-step solution

Let, f(x)=x^3+a x^2+b x+c , then aligned & f(x)=x^3+a x^2+b x+c & f^ (x)=3 x^2+2 a x+b & f^ (x)=6 x+2 & f^ (x)=6 aligned Equating the coefficient of f(x)+f^ (x)+f^ (x)+f^ (x)=x^3 We have 3+a=0 a=-3b+2 a+6=0 b=0 and c+b+2 a+6=0 c=0 Therefore, f(x)=x^3-3 x^2 and g(x)= f(x) x^3 d x= x^3-3 x^2 x^3 d x= (1- 3 x ) d x=x-3 x+C Where, C is the constant of integration g(1)=1-3 1+C=1 C=0 g(x)=x-3 x

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