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Questions
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1
JEE Main 2021 (Online) 31st August Evening Shift
MCQ (Single Correct Answer)
+4
-1
If [x] is the greatest integer $$\le$$ x, then

$${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx$$ is equal to :
A
2($$\pi$$ $$-$$ 1)
B
4($$\pi$$ $$-$$ 1)
C
4($$\pi$$ + 1)
D
2($$\pi$$ + 1)
2
JEE Main 2021 (Online) 31st August Morning Shift
MCQ (Single Correct Answer)
+4
-1
Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $$\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} } $$, $$0 \le x \le 1$$ and f(0) = 0, then $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt} $$ :
A
equals 0
B
equals 1
C
does not exist
D
equals $${1 \over 2}$$
3
JEE Main 2021 (Online) 27th August Evening Shift
MCQ (Single Correct Answer)
+4
-1
The area of the region bounded by the parabola (y $$-$$ 2)2 = (x $$-$$ 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :
A
9
B
10
C
4
D
6
4
JEE Main 2021 (Online) 27th August Evening Shift
MCQ (Single Correct Answer)
+4
-1
The value of the integral $$\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}} $$ is :
A
$${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 2}} \right)$$
B
$${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 6}} \right)$$
C
$${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 6}} \right)$$
D
$${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 2}} \right)$$
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