IIT-JEE 2012 Paper 2 Offline | Application of Derivatives Question 22 | Mathematics

1

IIT-JEE 2012 Paper 2 Offline

MCQ (Single Correct Answer)

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

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 $$\left( {2,\infty } \right)$$

2

IIT-JEE 2012 Paper 2 Offline

MCQ (Single Correct Answer)

+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

3

IIT-JEE 2008 Paper 1 Offline

MCQ (Single Correct Answer)

+3

-1

The total number of local maxima and local minima of the function

$$f(x) = \left\{ {\matrix{ {{{(2 + x)}^3},} & { - 3 < x \le - 1} \cr {{x^{2/3}},} & { - 1 < x < 2} \cr } } \right.$$ is

A

0

B

1

C

2

D

3

4

IIT-JEE 2007

MCQ (Single Correct Answer)

+3

-0.75

The tangent to the curve $$y = {e^x}$$ drawn at the point $$\left( {c,{e^c}} \right)$$ intersects the line joining the points $$\left( {c - 1,{e^{c - 1}}} \right)$$ and $$\left( {c + 1,{e^{c + 1}}} \right)$$

A

on the left of $$x=c$$

B

on the right of $$x=c$$

C

at no point

D

at all points