1
JEE Main 2021 (Online) 26th August Evening Shift
+4
-1
$$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$$ is equal to :
A
$${9 \over {44}}$$
B
$${5 \over {24}}$$
C
$${1 \over 5}$$
D
$${7 \over {36}}$$
2
JEE Main 2021 (Online) 27th July Evening Shift
+4
-1
The value of

$$\mathop {\lim }\limits_{x \to 0} \left( {{x \over {\root 8 \of {1 - \sin x} - \root 8 \of {1 + \sin x} }}} \right)$$ is equal to :
A
0
B
4
C
$$-$$4
D
$$-$$1
3
JEE Main 2021 (Online) 27th July Evening Shift
+4
-1
Let $$f:[0,\infty ) \to [0,3]$$ be a function defined by

$$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.$$

Then which of the following is true?
A
f is continuous everywhere but not differentiable exactly at one point in (0, $$\infty$$)
B
f is differentiable everywhere in (0, $$\infty$$)
C
f is not continuous exactly at two points in (0, $$\infty$$)
D
f is continuous everywhere but not differentiable exactly at two points in (0, $$\infty$$)
4
JEE Main 2021 (Online) 27th July Morning Shift
+4
-1
Let $$f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R$$ be defined as $$f(x) = \left\{ {\matrix{ {{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr b & , & {x = 0} \cr {{e^{\cot 4x/\cot 2x}}} & , & {0 < x < {\pi \over 4}} \cr } } \right.$$

If f is continuous at x = 0, then the value of 6a + b2 is equal to :
A
1 $$-$$ e
B
e $$-$$ 1
C
1 + e
D
e
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