1
WB JEE 2024
+1
-0.25

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a differentiable function and $$f(1)=4$$. Then the value of $$\lim _\limits{x \rightarrow 1} \int_\limits4^{f(x)} \frac{2 t}{x-1} d t$$, if $$f^{\prime}(1)=2$$ is

A
16
B
8
C
4
D
2
2
WB JEE 2024
+2
-0.5

Let $$\mathrm{I}(\mathrm{R})=\int_\limits0^{\mathrm{R}} \mathrm{e}^{-\mathrm{R} \sin x} \mathrm{~d} x, \mathrm{R}>0$$. then,

A
$$I(R)>\frac{\pi}{2 R}\left(1-e^{-R}\right)$$
B
$$I(R)<\frac{\pi}{2 R}\left(1-e^{-R}\right)$$
C
$$I(R)=\frac{\pi}{2 R}\left(1-e^{-R}\right)$$
D
$$I(R) \text { and } \frac{\pi}{2 R}(1-e^{-R}) \text { are not comparable }$$
3
WB JEE 2024
+2
-0.5

$$\lim _\limits{n \rightarrow \infty} \frac{1}{n^{k+1}}[2^k+4^k+6^k+\ldots .+(2 n)^k]=$$

A
$$\frac{2^k}{k}$$
B
$$\frac{2^{k+1}}{k+1}$$
C
$$\frac{2^k}{k+1}$$
D
$$\frac{2^{\mathrm{k}}}{\mathrm{k}-1}$$
4
WB JEE 2023
+1
-0.25

the expression $${{\int\limits_0^n {[x]dx} } \over {\int\limits_0^n {\{ x\} dx} }}$$, where $$[x]$$ and $$\{ x\}$$ are respectively integral and fractional part of $$x$$ and $$n \in N$$, is equal to

A
$${1 \over {n - 1}}$$
B
$${1 \over n}$$
C
$$n$$
D
$$n-1$$
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