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NDA 2 2024 Maths, question 93

Topic: Differential calculus

Passage · Q93 to Q94

Let f(t)=ln⁡(t+1+t2)f(t)=\ln \left( t+\sqrt{1+t^{2}} \right) and g(t)=tan⁡(f(t))g(t)=\tan (f(t)).

Q93

Consider the following statements :

I. f(t)f(t) is an odd function.

II. g(t)g(t) is an odd function.

Which of the statements given above is/are correct?

  1. (a) I only
  2. (b) II only
  3. (c) Both I and IICorrect answer
  4. (d) Neither I nor II

Solution.

Answer: (c) Both I and II

  1. Since (√(1+t²)−t)(√(1+t²)+t)=1, f(−t)=−f(t).
  2. Thus f is odd.
  3. Tangent is also odd, so g(−t)=tan(−f(t))=−g(t) on its symmetric domain.
  4. Both statements hold, but g has poles; oddness alone does not make every definite integral valid.