1
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
2
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
3
JEE Main 2022 (Online) 26th June Evening Shift
+4
-1

$$\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}$$ is equal to :

A
$${1 \over 3}$$
B
$${1 \over 4}$$
C
$${1 \over 6}$$
D
$${1 \over 12}$$
4
JEE Main 2022 (Online) 26th June Evening Shift
+4
-1

Let f(x) = min {1, 1 + x sin x}, 0 $$\le$$ x $$\le$$ 2$$\pi$$. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

A
(2, 0)
B
(1, 0)
C
(1, 1)
D
(2, 1)
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