1
WB JEE 2022
+1
-0.25

The values of a, b, c for which the function $$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin x} \over x},x < 0 \hfill \cr c,x = 0 \hfill \cr {{{{(x + b{x^2})}^{{1 \over 2}}} - {x^{{1 \over 2}}}} \over {b{x^{{1 \over 2}}}}},x > 0 \hfill \cr} \right.$$ is continuous at x = 0, are

A
$$a = {3 \over 2},b = - {3 \over 2},c = {1 \over 2}$$
B
$$a = - {3 \over 2},c = {3 \over 2},b$$ is arbitrary non-zero real number.
C
$$a = - {5 \over 2},b = - {3 \over 2},c = {3 \over 2}$$
D
$$a = - 2,b \in R - \{ 0\} ,c = 0$$
2
WB JEE 2022
+1
-0.25

Let $$f(x) = {a_0} + {a_1}|x| + {a_2}|x{|^2} + {a_3}|x{|^3}$$, where $${a_0},{a_1},{a_2},{a_3}$$ are real constants. Then f(x) is differentiable at x = 0

A
whatever be $${a_0},{a_1},{a_2},{a_3}$$.
B
for no values of $${a_0},{a_1},{a_2},{a_3}$$.
C
only if $${a_1} = 0$$
D
only if $${a_1} = 0,{a_3} = 0$$
3
WB JEE 2022
+1
-0.25

$$\mathop {\lim }\limits_{x \to 0} \left( {{1 \over x}\ln \sqrt {{{1 + x} \over {1 - x}}} } \right)$$ is

A
$${1 \over 2}$$
B
0
C
1
D
does not exist
4
WB JEE 2022
+1
-0.25

Let f : [a, b] $$\to$$ R be continuous in [a, b], differentiable in (a, b) and f(a) = 0 = f(b). Then

A
there exists at least one point $$c \in (a,b)$$ for which $$f'(c) = f(c)$$
B
$$f'(x) = f(x)$$ does not hold at any point of (a, b)
C
at every point of $$(a,b),f'(x) > f(x)$$
D
at every point of $$(a,b),f'(x) < f(x)$$
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