rsarashi wrote:Is x negative?
(1) At least one of x and x^2 is greater than x^3.
(2) At least one of x^2 and x^3 is greater than x.
OAC
Hi rsarashi,
We have to determine whether x negative.
Statement 1: At least one of x and x^2 is greater than x^3.
Case a: If x > x^3 but x^2 < x^3, Statement is valid;
Case b: If x^2 > x^3 but x < x^3, Statement is valid;
Case c: If x < x^3 and x^2 < x^3, Statement is NOT valid
So, we must not choose an example such that x < x^3 and x^2 < x^3.
The following ranges are important to study.
1. x is negative: Say x = -2, then x^3 = (-2)^3 = -8, thus x > x^3, the answer is Yes, x is negative.
2. x is a positive integer: Say x = 2, then x^3 = (2)^3 = 8, thus x < x^3; again, x^2 = (2)^2 = 4, thus x^2 < x^3; this calls for Case c. It is not a valid example.
3. x is a positive fraction: Say x = 1/2, then x^3 = (1/2)^3 = 1/8, thus x > x^3, the answer is No, x is not necessarily negative.
Insufficient!
Statement 2: At least one of x^2 and x^3 is greater than x.
Case d: If x^2 > x but x^3 < x, Statement is valid;
Case e: If x^3 > x but x^2 < x, Statement is valid;
Case f: If x^3 < x and x^2 < x, Statement is NOT valid
So, we must not choose an example such that x^3 < x and x^2 < x.
Taking the same ranges to study as discussed in Statement 1,
4. x is negative: Say x = -2, then x^2 = (-2)^2 = 4, thus x^2 > x, the answer is Yes, x is negative.
5. x is a positive integer: Say x = 2, then x^3 = (2)^3 = 8, thus x < x^3, the answer is No, x is not necessarily negative.
6. x is a positive fraction: Say x = 1/2, then x^3 = (1/2)^3 = 1/8, thus x > x^3; again x^2 = (1/2)^2 = 1/4, thus x > x^2; this calls for Case f. It is not a valid example.
Insufficient!
Statement 1 & 2:
(1) At least one of x and x^2 is greater than x^3. &
(2) At least one of x^2 and x^3 is greater than x.
Keeping both the statements together only examples (1) and (4) satisfy, thus x is negative, the answer is Yes. Sufficient.
The correct answer:
C
Hope this helps!
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