GATE ECE 2015 Set 3 | GA | Question: 5

Question

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

1. $\text{ln } x > \text{ln }y$
2. $e^{x}>e^{y}$
3. $y^{x}>x^{y}$
4. $\cos x> \cos y$

 

• A. $(i)$ and $(ii)$
• B. $(i)$ and $(iii)$ 
• C. $(iii)$ and $(iv)$
• D. $(ii)$ and $(iv)$

Comments

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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@anujs bro in this question for $COSX$ can i take value in degree ? or Real value is only we have to take 

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@꧁༒☬ĿọŗԀ 🆂🅷🅸🆅🅰☬༒꧂ trigonometric functions always take input from real domain and give output in real domain, it's on us how we interpret those values.

for ex: $sin(30) = \dfrac{1}{2} = 0.5; \, \, cos(30) = \dfrac{\sqrt{3}}{2} = 0.866$ (if we interpret $30$ as $30^{\circ}$)

$sin(30) = -0.988; \, \, cos(30) = 0.154$ (if we interpret $30$ as $30  \, \text{rad}$).

it a matter of choosing/defining a convention and we can go according to that convention as long as we are uniformly using it on all the elements. (here in $cos(x) > cos(y)$, $x$ and $y$ should both be of same type or both should follow the same conventions at a time).

also you can always convert from one convention to other (reference) but the relations b/w sine and cosine will not change in any convention. these relationships remain consistent no matter what specific angle values you're working with. as long as you maintain the same units (degrees or radians), the sine and cosine will have the same periodic pattern and 90 degree or π/2 radian shift b/w them.

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@anujs thnx broo got it !!

Answer 1

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

Comments

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$^{y}$.

Answer 2

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^x > e^y  holds true

So option (A) is correct