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Application of Derivatives question

2012 · Shift 2 · Q28

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PYQ Library
JEE Advanced
Mathematics
Application of Derivatives
2012 · Shift 2 · Q28

Application of Derivatives question

2012 · Shift 2 · Q28

JEE AdvancedMathematicsApplication of DerivativesMCQ+4 / −1

Let

f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}

for all

x \in IR

and let

g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt}

for all

x \in \left( {1,\,\infty } \right)

. Consider the statements:

P:

There exists some

x \in R

such that

f\left( x \right) + 2x = 2\left( {1 + {x^2}} \right)

Q:\,\,

There exists some

x \in R

such that

2\,f\left( x \right) + 1 = 2x\left( {1 + x} \right)

Then

A
both $P$ and $Q$ are true
B
$P$ is true and $Q$ is false
C
$P$ is false and $Q$ is true
D
both $P$ and $Q$ are false

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