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1
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, $$\infty$$)
2
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
3
JEE Main 2021 (Online) 27th July Morning Shift
+4
-1
Let f : R $$\to$$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of $$\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}}$$ is equal to :
A
4
B
8
C
16
D
12
4
JEE Main 2021 (Online) 25th July Morning Shift
Let $$f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$$, $$x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$$. Then, f is :
increasing in $$\left( { - {\pi \over 6},{\pi \over 2}} \right)$$
decreasing in $$\left( {0,{\pi \over 2}} \right)$$
increasing in $$\left( { - {\pi \over 6},0} \right)$$
decreasing in $$\left( { - {\pi \over 6},0} \right)$$