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If $x > y > 1,$ which of the following must be true$?$

  1. $\ln x > \ln y$
  2. $e^{x} > e^{y} $
  3. $y^x > x^y $
  4. $\cos x > \cos y$
  1. (i) and (ii)
  2. (i) and (iii)
  3. (iii) and (iv)
  4. (ii) and (iv)
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If in the given option (i) log X > log Y will be there

then it will be incorrect too because then negative bases of log can also exist.

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2 Answers

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Both $\ln$ and $e^x$ are monotonically (continuously) increasing functions. So, (i) and (ii) are TRUE.

If we take $x = 3, y = 2;y^x = 8, x^y = 9.$ So, (iii) is false

$\cos x$ is not a monotonically increasing function. So, (iv) is false.

Correct Option: A
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sir, given that x>y but you take x<y.
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You are right @Ram_81
Take x = 3 and y =2.

y$^{x}$ = 2$^{3}$ = 8

x$^{y}$ = 3$^{2}$ = 9

So, not always y$^{x}$ is greater than x$^{y}$.

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Let x=Π & y=Π/2  then cosx=-1 & cosy=0

i.e cosx<cosy

so (iv) is incorrect & option (C) & (D) are not possible

considering (ii) for the base e to be same & the value of exponent x>y>1 ,the value e> ey  holds true

So option (A) is correct 

Answer:

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