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Definite Integration question

2008 · Shift 2 · Q30

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PYQ Library
JEE Advanced
Mathematics
Definite Integration
2008 · Shift 2 · Q30

Definite Integration question

2008 · Shift 2 · Q30

JEE AdvancedMathematicsDefinite IntegrationMCQ+4 / −1

Consider the function

f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)

defined by

f\left( x \right) = {{{x^2} - ax + 1} \over {{x^2} + ax + 1}},0 \lt a \lt 2.

Let

g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.

Which of the following is true?

A
$g'(x)$ is positive on $\left( { - \infty ,0} \right)$ and negative on $\left( {0,\infty } \right)$
B
$g'(x)$ is negative on $\left( { - \infty ,0} \right)$ and positive on $\left( {0,\infty } \right)$
C
$g'(x)$ changes sign on both $\left( { - \infty ,0} \right)$ and $\left( {0,\infty } \right)$
D
$g'(x)$ does not change sign on $\left( { - \infty ,0} \right)$

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