1
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
$$\left[ {{{327} \over {360}},{{329} \over {360}}} \right]$$
2
JEE Main 2021 (Online) 17th March Evening Shift
+4
-1
If the integral

$$\int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over 2}}} + \gamma$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ are integers and [x] denotes the greatest integer less than or equal to x, then the value of $$\alpha$$ + $$\beta$$ + $$\gamma$$ is equal to :
A
0
B
10
C
20
D
25
3
JEE Main 2021 (Online) 17th March Morning Shift
+4
-1
Which of the following statements is correct for the function g($$\alpha$$) for $$\alpha$$ $$\in$$ R such that

$$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx}$$
A
$$g(\alpha )$$ is a strictly increasing function
B
$$g(\alpha )$$ is an even function
C
$$g(\alpha )$$ has an inflection point at $$\alpha$$ = $$-$$$${1 \over 2}$$
D
$$g(\alpha )$$ is a strictly decreasing function
4
JEE Main 2021 (Online) 16th March Evening Shift
+4
-1
Consider the integral
$$I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}$$,
where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :
A
45 (e $$-$$ 1)
B
45 (e + 1)
C
9 (e + 1)
D
9 (e $$-$$ 1)
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