1
IIT-JEE 2010 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-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?
A
$$f''(x)$$ exists for all $$x \in \left( {0,\infty } \right)$$
B
$$f'(x)$$ exists for all $$x \in \left( {0,\infty } \right)$$ and $$f'$$ is continuous on $$\left( {0,\infty } \right)$$, but not differentiable on $$\left( {0,\infty } \right)$$
C
there exists $$\,\,\alpha > 1$$ such that $$\left| {f'\left( x \right)} \right| < \left| {f\left( x \right)} \right|$$ for all $$x \in \left( {\alpha ,\infty } \right)\,$$
D
there exists $$\beta > 0$$ such that $$\left| {f\left( x \right)} \right| + \left| {f'\left( x \right)} \right| \le \beta $$ for all $$x \in \left( {0,\infty } \right)$$
2
IIT-JEE 2009 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2

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

A
$${I_n} = {I_{n + 2}}$$
B
$$\sum\limits_{m = 1}^{10} {{I_{2m + 1}}} = 10\pi $$
C
$$\sum\limits_{m = 1}^{10} {{I_{2m}}} = 0$$
D
$${I_n} = {I_{n + 1}}$$
3
IIT-JEE 2009 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is
A
$$e-1$$
B
$$\int\limits_1^e {\ln \left( {e + 1 - y} \right)dy} $$
C
$$e - \int\limits_0^1 {{e^x}dx} $$
D
$$\int\limits_1^e {\ln y\,dy} $$
4
IIT-JEE 1999
MCQ (More than One Correct Answer)
+3
-0.75
For which of the following values of $$m$$, is the area of the region bounded by the curve $$y = x - {x^2}$$ and the line $$y=mx$$ equals $$9/2$$?
A
$$-4$$
B
$$-2$$
C
$$2$$
D
$$4$$
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