Quantitative Aptitude – Equation – Let f(x) = x^2+1/x^2-1


XAT 2021 Exam Paper – Quantitative Aptitude – Equation – Let f(x) = x^2+1/x^2-1

Q. Let f(x) = x^2+1/x^2-1 if x ≠ 1, -1, and 1 if x = 1, -1. Let g(x) = x+1/x-1 if x ≠ 1, and 3 if x=1. What is the minimum possible value of f(x)/g(x) ?
  1. 1
  2. -1
  3. 1/4
  4. 1/3
  5. 1/2

Answer: 1/3

Solutions:
f(x)/g(x) ={((x^2+1)/(x^2-1))/((x+1)/(x-1))}=(x^2+1)/(x+1)^2 =1/(1+2x/(1+x^2 ))
Now as we can see for any values of x > 1 , 2x/(1+x^2 ) will be less than 2 thus f(x)/g(x) will be greater than 1/3 .
At x =1 , f(x) =1 and g(x) =3 thus f(x)/g(x) = 1/3
Thus minimum value of f(x)/g(x) = 1/3.

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