1
JEE Main 2021 (Online) 22th July Evening Shift
+4
-1
Let f : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$$

If f is continuous at x = 0, then $$\alpha$$ is equal to :
A
1
B
3
C
0
D
2
2
JEE Main 2021 (Online) 20th July Evening Shift
+4
-1
If $$f:R \to R$$ is given by $$f(x) = x + 1$$, then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \right)} \right]$$ is :
A
$${3 \over 2}$$
B
$${5 \over 2}$$
C
$${1 \over 2}$$
D
$${7 \over 2}$$
3
JEE Main 2021 (Online) 20th July Morning Shift
+4
-1
Let a function f : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if} & {0 < x < 1} \cr {2x - b} & {if} & {x \ge 1} \cr } } \right.$$

where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:
A
4
B
3
C
2
D
5
4
JEE Main 2021 (Online) 18th March Evening Shift
+4
-1
Let f : R $$\to$$ R be a function defined as

$$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr b,\,if\,x\, = 0 \hfill \cr {{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill \cr} \right.$$

If f is continuous at x = 0, then the value of a + b is equal to :
A
$$-$$3
B
$$-$$2
C
$$- {5 \over 2}$$
D
$$- {3 \over 2}$$
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