1
IIT-JEE 2010 Paper 1 Offline
MCQ (More than One Correct Answer)
+3
-0
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 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+4
-1
The value of $$\int\limits_0^1 {{{{x^4}{{\left( {1 - x} \right)}^4}} \over {1 + {x^2}}}dx} $$ is (are)
A
$${{22} \over 7} - \pi $$
B
$${2 \over {105}}$$
C
$$0$$
D
$${{71} \over {15}} - {{3\pi } \over 2}$$
3
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+4
-1
The value of $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^3}}}\int\limits_0^x {{{t\ln \left( {1 + t} \right)} \over {{t^4} + 4}}} dt$$ is
A
$$0$$
B
$${1 \over 12}$$
C
$${1 \over 24}$$
D
$${1 \over 64}$$
4
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+4
-1
Let $$ABC$$ be a triangle such that $$\angle ACB = {\pi \over 6}$$ and let $$a, b$$ and $$c$$ denote the lengths of the sides opposite to $$A$$, $$B$$ and $$C$$ respectively. The value(s) of $$x$$ for which $$a = {x^2} + x + 1,\,\,\,b = {x^2} - 1\,\,\,$$ and $$c = 2x + 1$$ is (are)
A
$$ - \left( {2 + \sqrt 3 } \right)$$
B
$${1 + \sqrt 3 }$$
C
$${2 + \sqrt 3 }$$
D
$${4 \sqrt 3 }$$
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