1
JEE Main 2021 (Online) 20th July Morning Shift
+4
-1
Let a be a positive real number such that $$\int_0^a {{e^{x - [x]}}} dx = 10e - 9$$ where [ x ] is the greatest integer less than or equal to x. Then a is equal to:
A
$$10 - {\log _e}(1 + e)$$
B
$$10 + {\log _e}2$$
C
$$10 + {\log _e}3$$
D
$$10 + {\log _e}(1 + e)$$
2
JEE Main 2021 (Online) 20th July Morning Shift
+4
-1
The value of the integral $$\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx}$$ is equal to:
A
$${1 \over 2}{\log _e}2 + {\pi \over 4} - {3 \over 2}$$
B
$$2{\log _e}2 + {\pi \over 4} - 1$$
C
$${\log _e}2 + {\pi \over 2} - 1$$
D
$$2{\log _e}2 + {\pi \over 2} - {1 \over 2}$$
3
JEE Main 2021 (Online) 18th March Evening Shift
+4
-1
Let g(x) = $$\int_0^x {f(t)dt}$$, where f is continuous function in [ 0, 3 ] such that $${1 \over 3}$$ $$\le$$ f(t) $$\le$$ 1 for all t$$\in$$ [0, 1] and 0 $$\le$$ f(t) $$\le$$ $${1 \over 2}$$ for all t$$\in$$ (1, 3]. The largest possible interval in which g(3) lies is :
A
$$\left[ { - 1, - {1 \over 2}} \right]$$
B
$$\left[ { - {3 \over 2}, - 1} \right]$$
C
[1, 3]
D
$$\left[ {{1 \over 3},2} \right]$$
4
JEE Main 2021 (Online) 17th March Evening Shift
+4
-1
Let f : R $$\to$$ R be defined as f(x) = e$$-$$xsinx. If F : [0, 1] $$\to$$ R is a differentiable function with that F(x) = $$\int_0^x {f(t)dt}$$, then the value of $$\int_0^1 {(F'(x) + f(x)){e^x}dx}$$ lies in the interval
A
$$\left[ {{{331} \over {360}},{{334} \over {360}}} \right]$$
B
$$\left[ {{{330} \over {360}},{{331} \over {360}}} \right]$$
C
$$\left[ {{{335} \over {360}},{{336} \over {360}}} \right]$$
D
$$\left[ {{{327} \over {360}},{{329} \over {360}}} \right]$$
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