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Paragraph: Read the following write up carefully and answer the following questions: Let f(x)=12 x^2 ₀^1 y f(y) d y+20 ₀^1 x y^2 f(y) d y+4 x Question: The number of solutions of the equation |f(|x|)|=e^ |x|

Options

  1. A0
  2. B2
  3. C4
  4. D3

Correct answer

B. 2

Step-by-step solution

aligned & Let 12 ₀^1 y f(y) d y=a and 20 ₀^1 (y^2 f(y) ) d y+4=b & f(x)=a x^2+b x f(y)=a y^2+b y & f(x)=12 x^2 ₀^1 y (a y^2+b y ) d y+20 x ₀^1 y^2 (a y^2+b y ) d y+4 x=12 x^2 ( a 4 + b 3 )+20 x [ a 5 + b 4 ]+4 x & a=3 a+4 b a=-2 b ...(1) & a n d 4 a+5 b+4=b a+b=-1 ...(2) & a=-2, b=1 & f(x)=-2 x^2+x aligned

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