1
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
2
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
3
JEE Main 2022 (Online) 28th June Morning Shift
+4
-1

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
4
JEE Main 2022 (Online) 27th June Morning Shift
+4
-1

Let a be an integer such that $$\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}$$ exists, where [t] is greatest integer $$\le$$ t. Then a is equal to :

A
$$-$$6
B
$$-$$2
C
2
D
6
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