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JEE MATHEMATICS MCQ: INCREASING & DECREASING

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Analyzing +4 | -1
Statement-1: The function x2(ex+e−x)x^2\left(e^x+e^{-x}\right) is increasing for all x>0x>0.
Statement-2: The functions x2exx^2 e^x and x2e−xx^2 e^{-x} are increasing for all x>0x>0 and the sum of two increasing functions in any interval (a,b)(a, b) is an increasing function in (a,b)(a, b).
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Correct option: C
Statement-1 is true; Statement-2 is false.
Option C (-1/18) is correct because the integral can be simplified using trigonometric identities and then evaluated.

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If the function f(x)=Ksin⁡x+2cos⁡xsin⁡x+cos⁡xf\left( x \right) = \frac{{K\sin x + 2\cos x}}{{\sin x + \cos x}} is increasing for all values of xx, then The set of values of a for which the function f(x)=x2+ax+1f\left( x \right) = {x^2} + ax + 1 is an increasing function on [1,2] is The value of aa in order that f(x)=f(x)= sin⁡x−cos⁡x−ax+b\sin x-\cos x-a x+b decreases for all real values is given by The function f(x)=xex(1−x),x∈Rf(x)=x e^{x(1-x)}, x \in R, is If ff and gg are two increasing functions such that fogfog is defined, then If f(x)=x3+bx2+cx+df\left( x \right) = {x^3} + b{x^2} + cx + d and 0<b2<c0 < {b^2} < c, then in (−∞,  ∞)\left( { - \infty ,\;\infty } \right)
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