12 Days of Christmas GMAT Competition - Day 12: Which of the following

Question

12 Days of Christmas GMAT Competition with Lots of Fun

Which of the following can never be correct?
 
I:  x < x^2 < 1-x

II:  x < x^2 < \frac{1}{x}

III:  x < \frac{1}{x} < x^2

IV:  x < \frac{1}{x} < x^2 < x^3

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

Bunuel · Accepted Answer

Expert's Global Official Video Explanation

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ElninoEffect · Answer

We just need to prove it right for only one case, if we find even one value that satisfies the condition that is out.

I:  x < x^2 < 1-x 
 x = -0.5 satisfies the equation so its out.

II:  x < x^2 < \frac{1}{x}  
 check for conditions:
x<0
-0.5 No.
-2 No
x>0
0.5 No
2 No
Let's keep it.

III:  x < \frac{1}{x} < x^2 
 x = -2
Satisfies the equation. Out.

IV:  x < \frac{1}{x} < x^2 < x^3 
 Putting all the values used in II we will find there are no values which satisfies the condition.
Keep it.

So our answer is II and IV.

C

gmatophobia · Answer

I: x x > -1 -1 < x < 0 When x lies between -1 and 0, 1 - x will be positive and greater than 1, while 0 < x^2 < 1 Hence this can be correct. II: x x > -1 -1 < x < 0 In this case 1 / x will be negative when x is negative, while x^2 will be positive. When x is positive and greater than 1, 1 / x will not be less greater than x^2 Hence this statement is incorrect. III: x<\frac{1}{x}

happypuppy · Answer

Nothing given for x, so x can assume any value.

For 'Can NEVER be CORRECT' , we need to find one correct value to decide:

For such questions, below is the list of values of x where values change for x, 1/x, x^2 or x^n
x = 10
x = 1
x = 0.9
x = 0.6
x = 0.5
x = 0.4
x = 0.1
x = 0
x = -0.1
x = -0.4
x = -0.5
x = -0.6
x = -0.9
x = -1
x = -10

(Actually, we need not compute for all but it helps us to judge/guess by looking the numbers)
Can never be correct OR ALWAYS incorrect. 
So, if we get JUST ONE Correct case, then the statement qualifies.

1. x < x^2 < 1 - x
Works of x = -0.1
-0.1 < 0.01 < 0.9 ------------------ Can be correct for at least one case

2. x < x^2 < 1/x
Doesn't work for any x

3. x < 1/x < x^2
Works for x=-10
-10 < -0.1 < 100 ------------------ Can be correct for at least one case

4. x < 1/x < x^2 < x^3
Doesn't work for any values of x

So, II and IV can never be correct as we got I and III correct for at least one case.

Answer: (C)

Snehaaaaa · Answer

Explanation-
In the question statement, there is no limitations define for choosing the value of x
So, x can be any value ............, 1, 10, 0.4, 0.1, 0.5, 0, -0.1, -0.4, -5, -10, .............

We will check the given conditions by substituting any values for x. If the condition holds true for even one value of x, then that condition can be correct.
In this type of question, if we get even a single value correct for a particular condition, the condition will hold true

I:  x < x^2 < 1-x 
Case 1: If x = 1,
1 < 1^2 < 1-1
The condition does not holds true

Case 2: If x= -0.1,
-0.1 < 0.01 < 1.1
The condition holds true
So, even though the condition doesn't hold true for 1 value of x, it holds true for other value of x. Therefore, this condition does not fall under 'can never be correct'

II:  x < x^2 < \frac{1}{x}  
Case 1: x= -0.1
-0.1 < 0.01 < -10
The condition does not holds true
Checked for multiple values of x, but the condition does not hold true for any value of x

III:  x < \frac{1}{x} < x^2 
Case 1: x= -10
-10 < -0.1 < 100
The condition holds true

IV:  x < \frac{1}{x} < x^2 < x^3 
Case 1: x = 0.5
0.5 < 2 < 0.25 < 0.025
The condition does not holds true

Case 2: x = -0.1
-0.1 < -10 < 0.01 < -0.001
The condition does not holds true
Checked for multiple values of x, but the condition does not hold true for any value of x

Correct answer is C

venkycs98 · Answer

I solved it in 3 steps: 1. Check if statement I is correct: x < x^2 => (i) x is either a positive value greater than 1 or (ii) -ve number x^2 < 1-x is true only when (ii) is true => x = -ve number take any number say x = -1 -1 < 1 < 1 - (-1) So clearly statement 1 can be true, so eliminate all answer choices with statement I => answer choices D and E are out Now, notice that the other answers contain statement II, so we don't have to check whether statement II is correct We only have to check if statement III or IV are correct, I chose to check if statement III is correct first IV: x < 1/x < x^2 x < 1/x => We can only have 2 types of values for x -> (i) x = +ve decimal value less than 1 (ii) -ve values lesser than -1 1/x < x^2 => Only (ii) is possible Lets say x = -5 -5 < -0.2 < 25 Fits, so eliminate all options with statement III We are left with C and A now, Check if D is correct IV: x<1/x There are only 2 types of numbers that satisfy this (i) x = +ve decimal < 1 and (ii) -ve number lesser than -1 1/x < x^2 We can eliminate (i) x^2 < x^3 Notice that this statement is not compatible with the possible types of numbers Since x^2 is always +ve and x^3 is always -ve So statement 4 can never be true We can conclude that the answer is C There's a more optimal approach: You can start with statement IV and move on to statement III, with just these 2, you can find the answer

bumpbot · Answer

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