1
JEE Advanced 2021 Paper 2 Online
+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
+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 2021 Paper 1 Online
+3
-1
The area of the region

$$\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } } \right\}$$ is
A
$${{11} \over {32}}$$
B
$${{35} \over {96}}$$
C
$${{37} \over {96}}$$
D
$${{13} \over {32}}$$
4
JEE Advanced 2020 Paper 1 Offline
+3
-1
Let the functions f : R $$\to$$ R and g : R $$\to$$ R be defined by

f(x) = ex $$-$$ 1 $$-$$ e$$-$$|x $$-$$ 1|

and g(x) = $${1 \over 2}$$(ex $$-$$ 1 + e1 $$-$$ x).

The the area of the region in the first quadrant bounded by the curves y = f(x), y = g(x) and x = 0 is
A
$$(2 - \sqrt 3 ) + {1 \over 2}(e - {e^{ - 1}})$$
B
$$(2 + \sqrt 3 ) + {1 \over 2}(e - {e^{ - 1}})$$
C
$$(2 - \sqrt 3 ) + {1 \over 2}(e + {e^{ - 1}})$$
D
$$(2 + \sqrt 3 ) + {1 \over 2}(e + {e^{ - 1}})$$
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