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NDA 2 2024 Maths, question 80

Topic: Differential calculus

Q80

Let f(x)f(y)=f(xy)f(x)f(y)=f(xy) for all real x,yx,y. If f(2)=4f(2)=4, then what is the value of f(1/2)f(1/2)?

  1. (a) 1/41/4Correct answer
  2. (b) 1/21/2
  3. (c) 11
  4. (d) 44

Solution.

Answer: (a) 1/41/4

  1. Using x=2,y=1 gives 4f(1)=4, so f(1)=1.
  2. Then x=2,y=1/2 gives 4f(1/2)=1.
  3. Thus f(1/2)=1/4; no assumption that f(x)=x² everywhere is necessary.