JEE Advanced 2016 Paper 2 Offline | Definite Integration Question 22 | Mathematics

1

JEE Advanced 2016 Paper 2 Offline

MCQ (Single Correct Answer)

+3

-1

The value of $$\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx} $$ is equal to

A

$${{{\pi ^2}} \over 4} - 2$$

B

$${{{\pi ^2}} \over 4} + 2$$

C

$${\pi ^2} - {e^{{\pi \over 2}}}$$

D

$${\pi ^2} + {e^{{\pi \over 2}}}$$

2

JEE Advanced 2015 Paper 2 Offline

MCQ (Single Correct Answer)

+4

-1

Let $$f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}$$ for all $$x \in R\,\,$$ with $$f\left( {{1 \over 2}} \right) = 0$$.
If $$m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,} $$ then the possible values of $$m$$ and $$M$$ are

A

$$m=13,$$ $$M=24$$

B

$$\,m = {1 \over 4},M = {1 \over 2}$$

C

$$m=-11,$$ $$M=0$$

D

$$m=1,$$ $$M=12$$

3

JEE Advanced 2014 Paper 2 Offline

MCQ (Single Correct Answer)

+3

-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

$$0$$

4

JEE Advanced 2014 Paper 2 Offline

MCQ (Single Correct Answer)

+3

-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

$${\pi \over 4}$$