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1

### IIT-JEE 2012 Paper 2 Offline

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
2

### IIT-JEE 2012 Paper 2 Offline

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

Which of the following is true?

A
$$g$$ is increasing on $$\left( {1,\infty } \right)$$
B
$$g$$ is decreasing on $$\left( {1,\infty } \right)$$
C
$$g$$ is increasing on $$(1, 2)$$ and decreasing on $$\left( {2,\infty } \right)$$
D
$$g$$ is decreasing on $$(1, 2)$$ and increasing on $$\left( {2,\infty } \right)$$
3

### IIT-JEE 2008

Let the function $$g:\left( { - \infty ,\infty } \right) \to \left( { - {\pi \over 2},{\pi \over 2}} \right)$$ be given by
$$g\left( u \right) = 2{\tan ^{ - 1}}\left( {{e^u}} \right) - {\pi \over 2}.$$ Then, $$g$$ is
A
even and is strictly increasing in $$\left( {0,\infty } \right)$$
B
odd and is strictly decreasing in $$\left( { - \infty ,\infty } \right)$$
C
odd and is strictly increasing in $$\left( { - \infty ,\infty } \right)$$
D
neither even nor odd, but is strictly increasing in $$\left( { - \infty ,\infty } \right)$$
4

### IIT-JEE 2008

The total number of local maxima and local minimum of the
function $$f\left( x \right) = \left\{ {\matrix{ {{{\left( {2 + x} \right)}^3},} & { - 3 < x \le - 1} \cr {{x^{2/3}},} & { - 1 < x < 2} \cr } } \right.$$ is
A
$$0$$
B
$$1$$
C
$$2$$
D
$$3$$

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NEET

Class 12