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JEE Advanced 2014Paper 2
Exam
JEE Advanced
Year
2014
Paper
Paper 2
Subject
Mathematics
Chapter
Definite Integration

JEE Advanced 2014 · Paper 2 · Mathematics · Definite Integration

Paragraph: Given that for each a ∈(0,1) , _ h arrow 0^ + ∫_ h ^ 1-h...

Paragraph: Given that for each a∈(0,1)a \in(0,1), lim⁡h→0+∫h1−ht−a(1−t)a−1dt\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t exists. Let this limit be g(a)g(a). In addition, it is given that the function g(a)g(a) is differentiable on (0,1)(0,1).
Question: The value of g^{\prime}\left(\frac{1}{2}\right) is

Options

  1. Aπ2\dfrac{\pi}{2}
  2. Bπ\pi
  3. C−π2- \dfrac{\pi}{2}
  4. D00

Correct answer

D. 00

Step-by-step solution

We have gg\left(a\right) = g(1−a)\left(1 - a\right) and gg is differentiable.
∴g′(a)=−g′(1−a)∴ {g}^{'}\left(a\right) = - {g}^{'}\left(1 - a\right)
Hence g′(12)=0{g}^{′}\left(\dfrac{1}{2}\right) = 0.
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