JEE Advanced 2021 Paper 2 Online | Definite Integration Question 10 | Mathematics

1

JEE Advanced 2021 Paper 2 Online

MCQ (Single Correct Answer)

+3

-1

Let $${\psi _1}:[0,\infty ) \to R$$, $${\psi _2}:[0,\infty ) \to R$$, f : (0, $$\infty$$) $$\to$$ R and g : [0, $$\infty$$) $$\to$$ R be functions such that f(0) = g(0) = 0,

$${\psi _1}(x) = {e^{ - x}} + x,x \ge 0$$,

$${\psi _2}(x) = {x^2} - 2x - 2{e^{ - x}} + 2,x \ge 0$$,

$$f(x) = \int_{ - x}^x {(|t| - {t^2}){e^{ - {t^2}}}dt,x > 0} $$ and

$$g(x) = \int_0^{{x^2}} {\sqrt t {e^{ - t}}dt,x > 0} $$.

Which of the following statements is TRUE?

A

$$f(\sqrt {\ln 3} ) + g(\sqrt {\ln 3} ) = {1 \over 3}$$

B

For every x > 1, there exists an $$\alpha$$ $$\in$$ (1, x) such that $${\psi _1}(x) = 1 + \alpha x$$

C

For every x > 0, there exists a $$\beta$$ $$\in$$ (0, x) such that $${\psi _2}(x) = 2x({\psi _1}(\beta ) - 1)$$

D

f is an increasing function on the interval $$\left[ {0,{3 \over 2}} \right]$$

2

JEE Advanced 2021 Paper 2 Online

MCQ (Single Correct Answer)

+3

-1

Let $${\psi _1}:[0,\infty ) \to R$$, $${\psi _2}:[0,\infty ) \to R$$, f : (0, $$\infty$$) $$\to$$ R and g : [0, $$\infty$$) $$\to$$ R be functions such that f(0) = g(0) = 0,

$${\psi _1}(x) = {e^{ - x}} + x,x \ge 0$$,

$${\psi _2}(x) = {x^2} - 2x - 2{e^{ - x}} + 2,x \ge 0$$,

$$f(x) = \int_{ - x}^x {(|t| - {t^2}){e^{ - {t^2}}}dt,x > 0} $$ and

$$g(x) = \int_0^{{x^2}} {\sqrt t {e^{ - t}}dt,x > 0} $$.

Which of the following statements is TRUE?

A

$${\psi _1}(x) \le 1$$, for all x > 0

B

$${\psi _2}(x) \le 0$$, for all x > 0

C

$$f(x) \ge 1 - {e^{ - {x^2}}} - {2 \over 3}{x^3} + {2 \over 5}{x^5}$$, for all $$x \in \left( {0,{1 \over 2}} \right)$$

D

$$g(x) \le {2 \over 3}{x^3} - {2 \over 5}{x^5} + {1 \over 7}{x^7}$$, for all $$x \in \left( {0,{1 \over 2}} \right)$$

3

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

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

4

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

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