1
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
2
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)
3
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
4
JEE Main 2021 (Online) 26th February Evening Shift
+4
-1
For x > 0, if $$f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt}$$, then $$f(e) + f\left( {{1 \over e}} \right)$$ is equal to :
A
$${1 \over 2}$$
B
$$-$$1
C
0
D
1
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