1
JEE Advanced 2014 Paper 2 Offline
+4
-1
Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt}$$ exists. Let this limit be $$g(a).$$ In addition, it is given that the function $$g(a)$$ is differentiable on $$(0,1).$$

The value of $$g'\left( {{1 \over 2}} \right)$$ is

A
$${\pi \over 2}$$
B
$$\pi$$
C
$$-{\pi \over 2}$$
D
$$0$$
2
JEE Advanced 2014 Paper 2 Offline
+4
-1
Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt}$$ exists. Let this limit be $$g(a).$$ In addition, it is given that the function $$g(a)$$ is differentiable on $$(0,1).$$

The value of $$g\left( {{1 \over 2}} \right)$$ is

A
$$\pi$$
B
$$2\pi$$
C
$${\pi \over 2}$$
D
$${\pi \over 4}$$
3
JEE Advanced 2013 Paper 1 Offline
+4
-1
The area enclosed by the curves $$y = \sin x + {\mathop{\rm cosx}\nolimits}$$ and $$y = \left| {\cos x - \sin x} \right|$$ over the interval $$\left[ {0,{\pi \over 2}} \right]$$ is
A
$$4\left( {\sqrt 2 - 1} \right)$$
B
$$2\sqrt 2 \left( {\sqrt 2 - 1} \right)$$
C
$$2\left( {\sqrt 2 + 1} \right)$$
D
$$2\sqrt 2 \left( {\sqrt 2 + 1} \right)$$
4
JEE Advanced 2013 Paper 1 Offline
+4
-1
Let $$f$$ $$:\,\,\left[ {{1 \over 2},1} \right] \to R$$ (the set of all real number) be a positive,
non-constant and differentiable function such that
$$f'\left( x \right) < 2f\left( x \right)$$ and $$f\left( {{1 \over 2}} \right) = 1.$$ Then the value of $$\int\limits_{1/2}^1 {f\left( x \right)} \,dx$$ lies in the interval
A
$$\left( {2e - 1,2e} \right)$$
B
$$\left( {e - 1,\,2e - 1} \right)$$
C
$$\left( {{{e - 1} \over 2},e - 1} \right)$$
D
$$\left( {0,{{e - 1} \over 2}} \right)$$
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