WB JEE 2023 | Definite Integration Question 35 | Mathematics

WB JEE 2023

MCQ (Single Correct Answer)

-0.25

If $${I_n} = \int\limits_0^{{\pi \over 2}} {{{\cos }^n}x\cos nxdx} $$, then I$$_1$$, I$$_2$$, I$$_3$$ ... are in

A.P.

G.P.

H.P.

no such relation

WB JEE 2023

MCQ (Single Correct Answer)

-0.5

$$\int\limits_0^{2\pi } {\theta {{\sin }^6}\theta \cos \theta d\theta } $$ is equal to

$${\pi \over {16}}$$

$${{3\pi } \over {16}}$$

$${{16\pi } \over 3}$$

WB JEE 2023

MCQ (Single Correct Answer)

-0.5

The average ordinate of $$y = \sin x$$ over $$[0,\pi ]$$ is :

$${2 \over \pi }$$

$${3 \over \pi }$$

$${4 \over \pi }$$

$$\pi $$

WB JEE 2022

MCQ (Single Correct Answer)

-0.25

Let f be derivable in [0, 1], then

there exists $$c \in (0,1)$$ such that $$\int\limits_0^c {f(x)dx = (1 - c)f(c)} $$

there does not exist any point $$d \in (0,1)$$ for which $$\int\limits_0^d {f(x)dx = (1 - d)f(d)} $$

$$\int\limits_0^c {f(x)dx} $$ does not exist, for any $$c \in (0,1)$$

$$\int\limits_0^c {f(x)dx} $$ is independent of $$c,c \in (0,1)$$

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