1
JEE Main 2021 (Online) 16th March Evening Shift
+4
-1
Let $$\alpha$$ $$\in$$ R be such that the function $$f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$$ is continuous at x = 0, where {x} = x $$-$$ [ x ] is the greatest integer less than or equal to x. Then :
A
no such $$\alpha$$ exists
B
$$\alpha$$ = 0
C
$$\alpha$$ = $${\pi \over 4}$$
D
$$\alpha$$ = $${\pi \over {\sqrt 2 }}$$
2
JEE Main 2021 (Online) 16th March Morning Shift
+4
-1
Let $${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)}$$. Then $$\mathop {\lim }\limits_{k \to \infty } {S_k}$$ is equal to :
A
$${\cot ^{ - 1}}\left( {{3 \over 2}} \right)$$
B
$${\pi \over 2}$$
C
tan$$-$$1 (3)
D
$${\tan ^{ - 1}}\left( {{3 \over 2}} \right)$$
3
JEE Main 2021 (Online) 16th March Morning Shift
+4
-1
Let the functions f : R $$\to$$ R and g : R $$\to$$ R be defined as :

$$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right.$$ and

$$g(x) = \left\{ {\matrix{ {{x^3},} & {x < 1} \cr {3x - 2,} & {x \ge 1} \cr } } \right.$$

Then, the number of points in R where (fog) (x) is NOT differentiable is equal to :
A
0
B
3
C
1
D
2
4
JEE Main 2021 (Online) 26th February Evening Shift
+4
-1
Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4.

Then $$\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$$ equals :
A
4 $$-$$ 2a
B
2a + 4
C
a + 4
D
2a $$-$$ 4
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