Conditions

[karthikpandian19]

If a^5 ≤ a, which of the following must be true?

I -1 ≤ a ≤ 0

II a=0

III 0 ≤ a ≤ 1




None of the above
I only
II only
III only
I and III only

[shankar.ashwin]

A number becomes smaller when multiplied by itself odd number of times (a^5) on 2 conditions : (if a > 1, any power of 'a' will result in a number > a)


Say 'a' is the number,

'a' can be 0 < a <1 (or) (e.g.) (1/2)^5< 1/2 ---> 1/32 < 1/2 (becomes smaller when multiplied with itself)

a < 0 (e.g.) -2^5 < -2 ---> -32 < -2



Since its a must be true question with 3 possibilities, consider each case

I - 'a' can be 1/2 - Eliminate
II - 'a' can be 1/2 - Eliminate
III - 'a' can be -2 - Eliminate

None of the options NEED to be true. A IMO

- Edited - Not sure if you understand it now, try substituting numbers which satisfy the condition, you will see a pattern.

[karthikpandian19]

Can you elaborate further, i didnot understand?

Whenever a number is multiplied by itself odd number of times it becomes smaller whenever the number is negative or between 0 and 1.

Say 'a' is the number,

'a' can be 0 < a < 1 (or)

a < 0

Since its a must be true question with 3 possibilities, consider each case

I - 'a' can be 1/2 - Eliminate
II - 'a' can be 1/2 - Eliminate
III - 'a' can be -2 - Eliminate

None of the options NEED to be true. A IMO

[karthikpandian19]

Can anyone elaborate further on this?

OA is A

[neelgandham]

If a^5 ≤ a, which of the following must be true?

I) -1 ≤ a ≤ 0
II) a=0
III) 0 ≤ a ≤ 1

If a≤-1
a^5 ≤ a (-32 ≤ -2)
If -1 < a < 0
a^5 > a (-1/32 > -1/2)
If 0 ≤ a ≤ 1
a^5 ≤ a (1/32) ≤ 1/2)
If a>1
a^5 > a (32 > 2)

So, If a^5 ≤ a, a≤-1 U 0 ≤ a ≤ 1
I) -1 ≤ a ≤ 0 - Not a solution, eliminated
II) a=0 - one of the solutions, but it is not always true. a can be any value less than -1 and any value between 0 and 1(Both Inclusive). a = -2 also satisfies the condition a^5 ≤ a.So, a = 0 is not a 'MUST BE TRUE' Condition.
III) 0 ≤ a ≤ 1 - one of the solutions, but it is not always true. a can be any value less than -1 and any value between 0 and 1(Both Inclusive). a = -2 also satisfies the condition a^5 ≤ a. So, 0 ≤ a ≤ 1 is not a 'MUST BE TRUE' Condition.

So ANone of the above is the answer. Let me know if you need any further help with this.