1
AIEEE 2005
+4
-1
If $${I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}}dx} } }$$ and $${I_4} = \int\limits_1^2 {{2^{{x^3}}}dx}$$ then
A
$${I_2} > {I_1}$$
B
$${I_1} > {I_2}$$
C
$${I_3} = {I_4}$$
D
$${I_3} > {I_4}$$
2
AIEEE 2005
+4
-1
The area enclosed between the curve $$y = {\log _e}\left( {x + e} \right)$$ and the coordinate axes is
A
$$1$$
B
$$2$$
C
$$3$$
D
$$4$$
3
AIEEE 2005
+4
-1
The parabolas $${y^2} = 4x$$ and $${x^2} = 4y$$ divide the square region bounded by the lines $$x=4,$$ $$y=4$$ and the coordinate axes. If $${S_1},{S_2},{S_3}$$ are respectively the areas of these parts numbered from top to bottom ; then $${S_1},{S_2},{S_3}$$ is
A
$$1:2:1$$
B
$$1:2:3$$
C
$$2:1:2$$
D
$$1:1:1$$
4
AIEEE 2005
+4
-1
Let $$f(x)$$ be a non - negative continuous function such that the area bounded by the curve $$y=f(x),$$ $$x$$-axis and the ordinates $$x = {\pi \over 4}$$ and $$x = \beta > {\pi \over 4}$$ is $$\left( {\beta \sin \beta + {\pi \over 4}\cos \beta + \sqrt 2 \beta } \right).$$ Then $$f\left( {{\pi \over 2}} \right)$$ is
A
$$\left( {{\pi \over 4} + \sqrt 2 - 1} \right)$$
B
$$\left( {{\pi \over 4} - \sqrt 2 + 1} \right)$$
C
$$\left( {1 - {\pi \over 4} - \sqrt 2 } \right)$$
D
$$\left( {1 - {\pi \over 4} + \sqrt 2 } \right)$$
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