1
WB JEE 2021
+1
-0.25
Let $${S_n} = {\cot ^{ - 1}}2 + {\cot ^{ - 1}}8 + {\cot ^{ - 1}}18 + {\cot ^{ - 1}}32 + ....$$ to nth term. Then $$\mathop {\lim }\limits_{n \to \infty } {S_n}$$ is
A
$${\pi \over 3}$$
B
$${\pi \over 4}$$
C
$${\pi \over 6}$$
D
$${\pi \over 8}$$
2
WB JEE 2021
+1
-0.25
Let f : D $$\to$$ R where D = [$$-$$0, 1] $$\cup$$ [2, 4] be defined by

$$f(x) = \left\{ {\matrix{ {x,} & {if} & {x \in [0,1]} \cr {4 - x,} & {if} & {x \in [2,4]} \cr } } \right.$$ Then,
A
Rolle's theorem is applicable to f in D.
B
Rolle's theorem is not applicable to f in D.
C
there exists $$\xi$$$$\in$$D for which f'($$\xi$$) = 0 but Rolle's theorem is not applicable.
D
f is not continuous in D.
3
WB JEE 2021
+2
-0.5
The $$\mathop {\lim }\limits_{x \to \infty } {\left( {{{3x - 1} \over {3x + 1}}} \right)^{4x}}$$ equals
A
1
B
0
C
e$$-$$8/3
D
e$$-$$4/9
4
WB JEE 2020
+1
-0.25
Let $$\phi (x) = f(x) + f(1 - x)$$ and $$f(x) < 0$$ in [0, 1], then
A
$$\phi$$ is monotonic increasing in $$\left[ {0,{1 \over 2}} \right]$$ and monotonic decreasing in $$\left[ {{1 \over 2}, 1} \right]$$
B
$$\phi$$ is monotonic increasing in $$\left[ {{1 \over 2}, 1} \right]$$ and monotonic decreasing in $$\left[ {0, {1 \over 2}} \right]$$
C
$$\phi$$ is neither increasing nor decreasing in any sub-interval of [0, 1]
D
$$\phi$$ is increasing in [0, 1]
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