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1
JEE Main 2021 (Online) 17th March Evening Shift
+4
-1
The value of $$\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$$, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :
A
r
B
$${r \over 2}$$
C
0
D
2r
2
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
3
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
$${{205} \over {144}}$$
4
JEE Main 2021 (Online) 16th March Evening Shift
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 :
no such $$\alpha$$ exists
$$\alpha$$ = 0
$$\alpha$$ = $${\pi \over 4}$$
$$\alpha$$ = $${\pi \over {\sqrt 2 }}$$