1
JEE Main 2021 (Online) 25th July Morning Shift
+4
-1
Let $$f:[0,\infty ) \to [0,\infty )$$ be defined as $$f(x) = \int_0^x {[y]dy}$$

where [x] is the greatest integer less than or equal to x. Which of the following is true?
A
f is continuous at every point in $$[0,\infty )$$ and differentiable except at the integer points.
B
f is both continuous and differentiable except at the integer points in $$[0,\infty )$$.
C
f is continuous everywhere except at the integer points in $$[0,\infty )$$.
D
f is differentiable at every point in $$[0,\infty )$$.
2
JEE Main 2021 (Online) 22th July Evening Shift
+4
-1
If $$\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R}$$ where [x] is the greatest integer less than or equal to x, then the value of $$\alpha$$ is :
A
200 (1 $$-$$ e$$-$$1)
B
100 (1 $$-$$ e)
C
50 (e $$-$$ 1)
D
150 (e$$-$$1 $$-$$ 1)
3
JEE Main 2021 (Online) 20th July Evening Shift
+4
-1
If [x] denotes the greatest integer less than or equal to x, then the value of the integral $$\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx}$$ is equal to :
A
$$-$$ $$\pi$$
B
$$\pi$$
C
0
D
1
4
JEE Main 2021 (Online) 20th July Evening Shift
+4
-1
If the real part of the complex number $${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$$ is $${1 \over 5}$$ for $$\theta \in (0,\pi )$$, then the value of the integral $$\int_0^\theta {\sin x} dx$$ is equal to:
A
1
B
2
C
$$-$$1
D
0
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