1
IIT-JEE 2012 Paper 2 Offline
+4
-1
The value of the integral $$\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\cos xdx}$$ is
A
$$0$$
B
$${{{\pi ^2}} \over 2} - 4$$
C
$${{{\pi ^2}} \over 2} + 4$$
D
$${{{\pi ^2}} \over 2}$$
2
IIT-JEE 2011 Paper 1 Offline
+4
-1
The value of $$\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx}$$ is
A
$${1 \over 4}\,\ell n{3 \over 2}$$
B
$$\,{1 \over 2}\,\ell n{3 \over 2}$$
C
$$\ell n{3 \over 2}$$
D
$$\,\,{1 \over 6}\,\ell n{3 \over 2}$$
3
IIT-JEE 2011 Paper 1 Offline
+4
-1
Let the straight line $$x=b$$ divide the area enclosed by
$$y = {\left( {1 - x} \right)^2},y = 0,$$ and $$x=0$$ into two parts $${R_1}\left( {0 \le x \le b} \right)$$ and
$${R_2}\left( {b \le x \le 1} \right)$$ such that $${R_1} - {R_2} = {1 \over 4}.$$ Then $$b$$ equals
A
$${3 \over 4}$$
B
$${ 1\over 2}$$
C
$${1 \over 3}$$
D
$${1 \over 4}$$
4
IIT-JEE 2011 Paper 2 Offline
+4
-1
Let f $$:$$$$\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$$ be a continuous function such that
$$f\left( x \right) = f\left( {1 - x} \right)$$ for all $$x \in \left[ { - 1,2} \right]$$

Let $${R_1} = \int\limits_{ - 1}^2 {xf\left( x \right)dx,}$$ and $${R_2}$$ be the area of the region bounded by $$y=f(x),$$ $$x=-1,$$ $$x=2,$$ and the $$x$$-axis. Then

A
$${R_1} = 2{R_2}$$
B
$${R_1} = 3{R_2}$$
C
$${2R_1} = {R_2}$$
D
$${3R_1} = {R_2}$$
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