JEE Main 2021 (Online) 31st August Morning Shift | Definite Integrals and Applications of Integrals Question 113 | Mathematics

JEE Main 2021 (Online) 31st August Morning Shift

MCQ (Single Correct Answer)

-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 $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt} $$ :

equals 0

equals 1

does not exist

equals $${1 \over 2}$$

JEE Main 2021 (Online) 27th August Evening Shift

MCQ (Single Correct Answer)

-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 :

JEE Main 2021 (Online) 27th August Evening Shift

MCQ (Single Correct Answer)

-1

The value of the integral $$\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}} $$ is :

$${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 2}} \right)$$

$${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 6}} \right)$$

$${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 6}} \right)$$

$${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 2}} \right)$$

JEE Main 2021 (Online) 27th August Morning Shift

MCQ (Single Correct Answer)

-1

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 :

$${{{e^2}} \over {16}}$$

$${4 \over e}$$

$${{16} \over {{e^2}}}$$

$${4 \over {{e^2}}}$$

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