Comprehension

Direction : Consider the following for the items that follow :

Let f(t) = \(\rm \ln(t+\sqrt{1+t^2})\) and g(t) = tan(f(t)).  

Consider the following statements :

I. f(t) is an odd function.

Il. g(t) is an odd function.

Which of the statements given above is/are correct?  

This question was previously asked in
NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
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  1. I only 
  2. II only
  3. Both I and II
  4. Neither I nor II

Answer (Detailed Solution Below)

Option 3 : Both I and II
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Detailed Solution

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Explanation:

Let f(t) = \(ln (t+\sqrt{1+t^2}\)

f(-t) = \(ln (\sqrt{1+t^2} -t\)

\(ln( \frac{1}{t+ \sqrt{1+t^2}})\)

\(-ln(t+\sqrt{1+t^2})\) = -f(t)

∴  f(t) is an odd function

Now g(t) = tan f(t)

Then, g(–t) = tan f(–t)

= –tan (f(t)) = –g(t)

So g(t) is also an odd function

Hence, both statements I and II are correct.

∴ Option (c) is correct

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