1
JEE Main 2021 (Online) 31st August Morning Shift
+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 $${1 \over 2}$$
2
JEE Main 2021 (Online) 27th August Evening Shift
+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
$${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 2}} \right)$$
3
JEE Main 2021 (Online) 27th August Morning Shift
+4
-1
Out of Syllabus
If $${U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n$$, then $$\mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}}$$ is equal to :
A
$${{{e^2}} \over {16}}$$
B
$${4 \over e}$$
C
$${{16} \over {{e^2}}}$$
D
$${4 \over {{e^2}}}$$
4
JEE Main 2021 (Online) 27th August Morning Shift
+4
-1
$$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx}$$ is equal to :
A
6
B
8
C
5
D
10
JEE Main Subjects
Physics
Mechanics
Electricity
Optics
Modern Physics
Chemistry
Physical Chemistry
Inorganic Chemistry
Organic Chemistry
Mathematics
Algebra
Trigonometry
Coordinate Geometry
Calculus
EXAM MAP
Joint Entrance Examination