IIT-JEE 2012 Paper 2 Offline | Application of Derivatives Question 28 | Mathematics

IIT-JEE 2012 Paper 2 Offline

MCQ (Single Correct Answer)

-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?

$$g$$ is increasing on $$\left( {1,\infty } \right)$$

$$g$$ is decreasing on $$\left( {1,\infty } \right)$$

$$g$$ is increasing on $$(1, 2)$$ and decreasing on $$\left( {2,\infty } \right)$$

$$g$$ is decreasing on $$(1, 2)$$ and increasing on $$\left( {2,\infty } \right)$$

IIT-JEE 2008 Paper 1 Offline

MCQ (Single Correct Answer)

-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

IIT-JEE 2007 Paper 1 Offline

MCQ (Single Correct Answer)

-1

Let $$f(x)$$ be differentiable on the interval (0, $$\infty$$) such that $$f(1)=1$$, and $$\mathop {\lim }\limits_{t \to x} {{{t^2}f(x) - {x^2}f(t)} \over {t - x}} = 1$$ for each $$x > 0$$. Then $$f(x)$$ is

$${1 \over {3x}} + {{2{x^2}} \over 3}$$

$$ - {1 \over {3x}} + {{4{x^2}} \over 3}$$

$$ - {1 \over x} + {2 \over {{x^2}}}$$

$${1 \over x}$$

IIT-JEE 2005 Screening

MCQ (Single Correct Answer)

-0.5

If $$P(x)$$ is a polynomial of degree less than or equal to $$2$$ and $$S$$ is the set of all such polynomials so that $$P(0)=0$$, $$P(1)=1$$ and $$P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],$$ then

$$S = \phi $$

$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,2} \right)$$

$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,\infty } \right)$$

$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,1} \right)$$

PREVIOUS NEXT

JEE Advanced Subjects

Browse all chapters by subject