1
JEE Advanced 2014 Paper 2 Offline
+4
-1
List - $$I$$
P.$$\,\,\,\,$$ The number of polynomials $$f(x)$$ with non-negative integer coefficients of degree $$\le 2$$, satisfying $$f(0)=0$$ and $$\int_0^1 {f\left( x \right)dx = 1,}$$ is
Q.$$\,\,\,\,$$ The number of points in the interval $$\left[ { - \sqrt {13} ,\sqrt {13} } \right]$$
at which $$f\left( x \right) = \sin \left( {{x^2}} \right) + \cos \left( {{x^2}} \right)$$ attains its maximum value, is
R.$$\,\,\,\,$$ $$\int\limits_{ - 2}^2 {{{3{x^2}} \over {\left( {1 + {e^x}} \right)}}dx}$$ equals
S.$$\,\,\,\,$$ $${{\left( {\int\limits_{ - {1 \over 2}}^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)} \over {\left( {\int\limits_0^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)}}$$

List $$II$$
1.$$\,\,\,\,$$ $$8$$
2.$$\,\,\,\,$$ $$2$$
3.$$\,\,\,\,$$ $$4$$
4.$$\,\,\,\,$$ $$0$$

A
$$P = 3,Q = 2,R = 4,S = 1$$
B
$$P = 2,Q = 3,R = 4,S = 1$$
C
$$P = 3,Q = 2,R = 1,S = 4$$
D
$$P = 2,Q = 3,R = 1,S = 4$$
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
$${\pi \over 2}$$
B
$$\pi$$
C
$$-{\pi \over 2}$$
D
$$0$$
3
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
$${\pi \over 4}$$
4
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
$$2\sqrt 2 \left( {\sqrt 2 + 1} \right)$$
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