IIT-JEE 2010 Paper 1 Offline | Definite Integrals and Applications of Integrals Question 58 | Mathematics

IIT-JEE 2010 Paper 1 Offline

MCQ (More than One Correct Answer)

-1

Let $$f$$ be a real-valued function defined on the interval $$\left( {0,\infty } \right)$$
by $$\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt {1 + \sin t\,} dt.} $$ then which of the following
statement(s) is (are) true?

$$f''(x)$$ exists for all $$x \in \left( {0,\infty } \right)$$

$$f'(x)$$ exists for all $$x \in \left( {0,\infty } \right)$$ and $$f'$$ is continuous on $$\left( {0,\infty } \right)$$, but not differentiable on $$\left( {0,\infty } \right)$$

there exists $$\,\,\alpha > 1$$ such that $$\left| {f'\left( x \right)} \right| < \left| {f\left( x \right)} \right|$$ for all $$x \in \left( {\alpha ,\infty } \right)\,$$

there exists $$\beta > 0$$ such that $$\left| {f\left( x \right)} \right| + \left| {f'\left( x \right)} \right| \le \beta $$ for all $$x \in \left( {0,\infty } \right)$$

IIT-JEE 2009 Paper 2 Offline

MCQ (More than One Correct Answer)

-2

If $${I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {(1 + {\pi ^x})\sin x}}dx,n = 0,1,2,} $$ .... then

$${I_n} = {I_{n + 2}}$$

$$\sum\limits_{m = 1}^{10} {{I_{2m + 1}}} = 10\pi $$

$$\sum\limits_{m = 1}^{10} {{I_{2m}}} = 0$$

$${I_n} = {I_{n + 1}}$$

IIT-JEE 2009 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is