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NDA 2 2025 Maths, question 77

Topic: Differential calculus

Passage · Q76 to Q77

Consider the function f(x)=x1−xf(x)=\frac{x}{1-x} (x>0,x≠1)(x>0,x\ne 1)

Q77

What is (1−x)f(x)+xf(x+1)(1-x)f(\sqrt{x})+xf(\sqrt{x}+1) equal to?

  1. (a) −f(x)-f(x)
  2. (b) f(x)f(x)
  3. (c) xx
  4. (d) 00Correct answer

Solution.

Answer: (d) 00

  1. Let t=√x.
  2. Then (1−x)f(√x)=(1−t²)t/(1−t)=t(1+t), while xf(√x+1)=t²[−(t+1)/t]=−t(t+1).
  3. The terms cancel, giving 0.