1
JEE Main 2021 (Online) 17th March Morning Shift
+4
-1
The value of
$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$$, where [ x ] denotes the greatest integer $$\le$$ x is :
A
$$\pi$$
B
$${\pi \over 4}$$
C
$${\pi \over 2}$$
D
0
2
JEE Main 2021 (Online) 16th March Evening Shift
+4
-1
Let f : S $$\to$$ S where S = (0, $$\infty$$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $$\to$$ R be defined as g(x) = loge f(x), then the value of |g''(5) $$-$$ g''(1)| is equal to :
A
1
B
$${{187} \over {144}}$$
C
$${{197} \over {144}}$$
D
$${{205} \over {144}}$$
3
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
$$\alpha$$ = $${\pi \over {\sqrt 2 }}$$
4
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
$${\tan ^{ - 1}}\left( {{3 \over 2}} \right)$$
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