1
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
$${\tan ^{ - 1}}\left( {{3 \over 2}} \right)$$
2
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
3
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
4
JEE Main 2021 (Online) 26th February Evening Shift
+4
-1
Let $$f(x) = {\sin ^{ - 1}}x$$ and $$g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}$$. If $$g(2) = \mathop {\lim }\limits_{x \to 2} g(x)$$, then the domain of the function fog is :
A
$$( - \infty , - 2] \cup \left[ { - {4 \over 3},\infty } \right)$$
B
$$( - \infty , - 2] \cup [ - 1,\infty )$$
C
$$( - \infty , - 2] \cup \left[ { - {3 \over 2},\infty } \right)$$
D
$$( - \infty , - 1] \cup [2,\infty )$$
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