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  4. The intervals in which the function f given by f(x)=(4 sin x-2 x-x cos x/2+ cos x) is increasing in x ∈(0,2 π) is

Q. The intervals in which the function given by is increasing in is

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Solution:






Now, or
As
or
for
and for
Thus, is increasing for

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3. The angle of intersection between the curves and where denotes the greatest integer is

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4. Let
• Column 1 contains information about zeros of and .
• Column 2 contains information about the limiting behavior of and at infinity.
• Column 3 contains information about increasing/decreasing nature of and .
Column 1 Column 2 Column 3
(I) for some (i) (P) is increasing in ()
(II) for some (ii) (Q) is decreasing in ()
(III) for some (ii) (R) is increasing in ()
(IV) for some (ii) (S) is decreasing in ()

Which of the following options is the only INCORRECT combination?

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5. Let be a positive increasing function with
Then

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