JEE Advanced 2014 · Paper 2 · Mathematics Q53

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MathematicsSingle correct+3 / -1Question 53 of 60

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Paragraph For Question 53 and 54

Given that for each $a\in(0,1)$ ,
$\lim_{h\to0^+}\int_h^{1-h}t^{-a}(1-t)^{a-1}\,dt$
exists. Let this limit be $g(a)$ . In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$ .

The value of $g\left(\frac12\right)$ is

\pi

2\pi

\dfrac\pi2

\dfrac\pi4

Answer and worked solution

Official answer: A

At $a=\frac12$ .
$g\left(\tfrac12\right)=\int_0^1\frac{dt}{\sqrt{t(1-t)}}.$
Put $t=\sin^2\theta$ , $dt=2\sin\theta\cos\theta\,d\theta$ :
$\int_0^{\pi/2}\frac{2\sin\theta\cos\theta}{\sin\theta\cos\theta}d\theta=\pi.$
(In general $g(a)=B(1-a,a)=\dfrac{\pi}{\sin\pi a}$ .)

(A)

Topics:Properties of definite integrals Substitution

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JEE Advanced 2014 · Paper 2, Mathematics question 53