12 Days of Christmas GMAT Competition - Day 6: If a^3 > b^2 > c, which

Given ,\( a^{3} > b^{2} > c\)
The questions asks which of the following may be true. So just one instance that satisfies the conditions would be enough.

I. a > b > c
Most easiest to satisfy.
\(4>2>1\) (Taking a= 4 , b=2 , c =1)

\(4^{3} > 2^{2} > 1 => 64>4>1\)
Option 1 is true.

II. c > b > a
\(10>5>4\) (Taking a= 4 , b=5 , c =10)

\( 4^{3} > 5^{2} > 10 => 64>25>10\)
Option 2 is true.

III. a > c > b
\(10>8>3\) (Taking a= 10 , b=3 , c =8)

\( 10^{3} > 3^{2} > 8 => 1000>9>8\)
Option 3 is true.

Answer - E. I, II, and III

If a^3 > b^2 > c, which of the following may be true?

I. a > b > c
II. c > b > a
III. a > c > b

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

All three may be true
I. a > b > c

a = 100 , b = 20, c=1

II. c > b > a

a = 4, b = 5, c = 10

III. a > c > b

a= 100, c= 1, b = -2

E is the answer

I. a > b > c; a=2, b=0, c=-1 satisfies the main equation and satisfies this condition as well. Therefore, this may be true.
II. c > b > a; a=4, b=7, c=11 satisfies the main equation and satisfies this condition as well. Therefore, this may be true.
III. a > c > b a=2, b=-1, c=0 satisfies the main equation and satisfies this condition as well. Therefore, this may be true.
Since, all three options maybe true. Therefore, option E

Asked: If a^3 > b^2 > c, which of the following may be true?


If a^3= 27> b^2 = 4> c =1;
then a=3> b=2> c=1:
MAY BE TRUE


a^3 = 1.2^3 = 1.728 > b^2 = 1.3^2 = 1.69 > c = 1.5
c = 1.5 > b = 1.3 > a = 1.2
MAY BE TRUE


If a^3= 27> b^2 = 4> c =2.5;
then a=3> c=2.5 >b=2
MAY BE TRUE


A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

IMO E

All may be true
Given a^3>b^2>c

For statment one is a>b>c
Just take
a=5
b=4
c=3
For 2nd statement ( c>b>a )
Assume
a= 3
b= 4
c= 5

For 3rd statement ( a>c>b )
Assume
a= 5
c=4
b= 3

So Ans- is E

Posted from my mobile device

If a^3 > b^2 > c, which of the following may be true?

I. a > b > c If a=2, b=1, c=-1; 2>1>-1 8>1>-1 may be true
II. c > b > a If a=5, b=9, c=20; 20>9>5 125>81>20 may be true
III. a > c > b If a=4, b=-3, c=-1; 4>-1>-3 64>9>-1 may be true

Answer E

a^3 > b^2 > c

For "may be true", we need to find at least one example where the inequality holds.

I. a > b > c
Works. Eg: a = 3, b = 2, c = 1

II. c > b > a
Works
Take c = 5, b = 4, a = 3
a^3 = 27 > b^2 = 16 > c = 5

III. a > c > b
Again possible.
a = 6, c = 5, b = 4
a^3 = 216 > b^2 = 16 > c = 5

All three work. Answer: E

If a^3 > b^2 > c, which of the following may be true?

I. a > b > c
Take a=2, b=1, c=0. (true)

II. c > b > a
Take a=3, b=4, c=5 (true)

III. a > c > b
Take a=2, b=-1,c=0 (true)

SO E)

1. Take a=3,b=2,c=1 Yes (any positive value here will work)
2. Take a=3,b=4,c=5 Yes (try to take value of a,b,c large enough such that the squares and cubes be significant enough to make the question stem inequality hold true)
3. Take a=4,b=2, c=3 Yes (try to take a value of b <c such that b^2>c)

Ans E

I. a>b>c
let a= 4, b=3, c=1
a^3 = 64, b=9, c=1
a^3>b^2>c holds true

II.
c>b>a

let a= 2, b=sqrt(6), c= 4
a^3= 8, b^2 = 6 , c= 4
a^3>b^2>c holds true

III.

a>c>b
a= 3, c= 1.2, b= 1.1
a^3 = 27, b^2 = 1.21, c= 1.2
a^3>b^2>c holds true

Correct Answer E

I. a > b > c True for a=10, b=5, c=1

II. c > b > a True for a=3, b=4, c=6

III. a > c > b True for a=3, b=-4, c=1

Hence IMO E.
_________________

Solution:

Given, a^3 > b^2 > c.
Question asks us which may be true and not must be true. So we need to find just 1 case to justify the given statements.

I. a > b > c
For a=5, b=4, c=2, both a > b > c and a^3 > b^2 > c hold true. POSSIBLE.

II. c > b > a
For a=5, b=6, c=7, both c > b > a and a^3 > b^2 > c hold true. POSSIBLE.

III. a > c > b
For a=5, b=2, c=3, both a > c > b and a^3 > b^2 > c hold true. POSSIBLE.

ANSWER E.

a=3, b=2, c=1
I may be true

a=3, b=4, c=5
II may be true

a=2, b=-1, c=0
III may be true

Hence, answer is E
_________________

a^3 > b^2 > c

The options I, II and III must satisfy atleast one example to prove that that specific option may be true.


b can be negative.


But, b^2 will be always positive.

Then, a must be always positive as, if a is negative, a^3 will be negative.

b and c can be positive or negative, but a must be always positive.


I. a > b > c

Assume a=5, b=4, c = 2.

This satisfies, both a^3 > b^2 > c and a > b > c.

Hence keep choices that always include I. Eliminate choice B.


II. c > b > a

In this case c must be positive as we know a must be positive.
So, all a,b,c must be positive.

Let b= 2, c=3, a=1.6

a^3 = 4.096
b^2 = 4
c = 3
Then a^3 > b^2 > c is satisfied.

Also, c > b > a is satisfied.


So keep choices that include II. Eliminate choices D.


III. a > c > b

In this case a is positive, but b and c can be negative.

Let a = 5, c= -2. b= -4.

Then a^3= 125, b^2 = 16, and c = -2.

Then a^3 > b^2 > c is satisfied.

but you will see, c > b here.

Also it satisfies, a > c > b.

Hence, keep options that include III. That is, I, II and III all satisfies.

Hence the best answer choice is E.

This can very easily be done by plugging in values.

First, let a=3, b=2, c=1. Here, a^3>b^2>c and a>b>c. So, I is true.

Now, keep rest of the values same and make b=-2. Then, a>c>b will also become true. Hence, III is true.

Finally, let c=24, b=5, a=4. Here, a^3>b^2>c and c>b>a. So, II is also true.

Hence, IMO I,II and III are true. Option E.

Answer is E
If a^3 > b^2 > c, which of the following may be true?
we know that a and c is bigger than zero. b can both be positive and be negative.
I. a > b > c May be true
II. c > b > a may be true
III. a > c > bthis is true when b is negative.

Let a,b,c=5,6,7 respectively. c>b>a
Hence a^3>b^2>c
Let a,b,c=2,-1,0 respectively. a>c>b
Hence a^3>b^2>c
Since only option having I and II correct is option E, thats the answer.

I)
This clearly may be true. a=10, b=2, c=1

II) Similar situation.
c=5, b=4, a=3

III) Case of a=10, c=3, b=2.

All are possible, therefore choice E is the best.

If a^3 > b^2 > c, which of the following may be true?

I. a > b > c
II. c > b > a
III. a > c > b

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

-----------------------------------------

IMO E

I. It is possible. a= 10, b=2 , c=1

II. It is possible. a= 3, b=4 , c=5

III. It is possible. a= 5, b=-2 , c=3