1
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1

If $$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0$$, then $$8(\alpha+\beta)$$ is equal to :

A
4
B
$$-$$8
C
$$-$$4
D
8
2
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1

The value of $$\mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}$$ is equal to:

A
$${{{\pi ^2}} \over 6}$$
B
$${{{\pi ^2}} \over 3}$$
C
$${{{\pi ^2}} \over 2}$$
D
$$\pi$$2
3
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1

Let f, g : R $$\to$$ R be functions defined by

$$f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.$$ and $$g(x) = \left\{ {\matrix{ {{e^x} - x} & , & {x < 0} \cr {{{(x - 1)}^2} - 1} & , & {x \ge 0} \cr } } \right.$$ where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :

A
one point
B
two points
C
three points
D
four points
4
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1

The value of

$$\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right)} } \right\}$$ is equal to :

A
1
B
2
C
3
D
6
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