Question

If x^1/5  > x^1/3 , then how many of the following statements are definitely true about x?

x² > x³

x^1/3 > x⁴

1 > x^1/3 > x^-3

x^-1/3 > x³

1. 1
2. 2
3. 3
4. 4

Answer 1

x^1/5  > x^1/3

Dividing by x^1/3 on both sides we get

x^{1/5 - 1/3} > 1

x^-2/15 > 1

x^2/15 < 1

So, x is less than 1. Here, x can be either < -1 or between 0 and 1.

1. x² > x³
   Is always true 
2. x^1/3 > x⁴
   Is true only for positive x. If x is < -1, this becomes false as (-1)⁴ turns out to be even.
3. 1 > x^1/3 > x^-3
   Is not true for positive x as x^-3 > x^1/3
4. x^-1/3 > x³
   Is always true

So, 1, and 4 are true. 2, 3 are false.

So, answer is 2. 2

 

Answer 2

2.2