1
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
2
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
3
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)
4
JEE Main 2021 (Online) 16th March Evening Shift
+4
-1
Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that $$\int_0^1 {P(x)dx}$$ = 1 and P(x) leaves remainder 5 when it is divided by (x $$-$$ 2). Then the value of 9(b + c) is equal to :
A
9
B
11
C
7
D
15
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