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1
JEE Main 2021 (Online) 27th July Evening Shift
MCQ (Single Correct Answer)  +4  -1
Let f : (a, b) $$\to$$ R be twice differentiable function such that $$f(x) = \int_a^x {g(t)dt} $$ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :
A
twelve roots in (a, b)
B
five roots in (a, b)
C
seven roots in (a, b)
D
three roots in (a, b)
2
JEE Main 2021 (Online) 27th July Morning Shift
MCQ (Single Correct Answer)  +4  -1
The value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1) + 4n}}} $$ is equal to :
A
$$5 + {\log _e}\left( {{3 \over 2}} \right)$$
B
$$2 - {\log _e}\left( {{2 \over 3}} \right)$$
C
$$3 + 2{\log _e}\left( {{2 \over 3}} \right)$$
D
$$1 + 2{\log _e}\left( {{3 \over 2}} \right)$$
3
JEE Main 2021 (Online) 27th July Morning Shift
MCQ (Single Correct Answer)  +4  -1
The value of the definite integral

$$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $$ is equal to :
A
$$ - {\pi \over 2}$$
B
$${\pi \over {2\sqrt 2 }}$$
C
$$ - {\pi \over 4}$$
D
$${\pi \over {\sqrt 2 }}$$
4
JEE Main 2021 (Online) 27th July Morning Shift
MCQ (Single Correct Answer)  +4  -1
If the area of the bounded region
$$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$$ is ,
$$\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma $$, then the value of $${(\alpha + \beta - 2\lambda )^2}$$ is equal to :
A
8
B
2
C
4
D
1
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