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1

### JEE Advanced 2014 Paper 2 Offline

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

The following integral $$\int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\left( {2\cos ec\,\,x} \right)}^{17}}dx}$$ is equal to
A
$$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {2{{\left( {{e^u} + {e^{ - u}}} \right)}^{16}}\,du}$$
B
$$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {{{\left( {{e^u} + {e^{ - u}}} \right)}^{17}}\,du}$$
C
$$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {{{\left( {{e^u} - {e^{ - u}}} \right)}^{17}}\,du}$$
D
$$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {2{{\left( {{e^u} - {e^{ - u}}} \right)}^{16}}\,du}$$
3

### JEE Advanced 2013 Paper 1 Offline

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
$$\left( {0,{{e - 1} \over 2}} \right)$$
4

### JEE Advanced 2013 Paper 1 Offline

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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