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1
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1
English
Hindi

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
2
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1
English
Hindi

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
3
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1
English
Hindi

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
4
JEE Main 2022 (Online) 28th June Morning Shift
+4
-1
English
Hindi

Let f : R $$\to$$ R be defined as

$$f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.$$

where a, b, c $$\in$$ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

A
There exists a, b, c $$\in$$ R such that f is continuous on R.
B
If f is discontinuous at exactly one point, then a + b + c = 1
C
If f is discontinuous at exactly one point, then a + b + c $$\ne$$ 1
D
f is discontinuous at at least two points, for any values of a, b and c
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