stonecold wrote:
If x=a+1 and b=x+2 then which of the following must be false?
(I)a>b
(II)b>a
(III)a=b
(IV)a=b^2
(V)b=a^2
A) I and III
B) I, III, V
C) I, III, IV
D) II, III, V
E) I, III, IV, V
There are many issues with this question - GMAT Roman Numeral questions always have exactly three items (not the five you see here), there is no reason to even look at item III because it's in every answer choice, and it's bizarrely convoluted to ask which items must be false instead of which could be true.
We know a = x-1 and b = x + 2. Since x+2 is exactly 3 greater than x-1, certainly b > a. So I and III must be false.
We know b is precisely 3 greater than a, so b = a + 3. If we want to see if IV can be true, we can then ask if it's possible for a = (a + 3)^2. Notice this won't be true if a+3 is a 'fraction' between 0 and 1, because then a is negative, so could never equal a square, which is positive. But if a+3 is not between 0 and 1 (inclusive), then (a+3)^2 is always greater than a+3, which is always greater than a. So (a+3)^2 is always larger than a, and b^2 can never equal a, so IV must be false.
On the other hand, it's definitely going to be possible for a^2 = a + 3 to be true, for two values of a. If you picture y = x^2 and y = x + 3 in the coordinate plane, you can see immediately that the line y = x + 3 will meet the parabola y = x^2 in two points. Or you can notice that when a = 2, a^2 < a + 3, but when a = 3, a^2 > a + 3, so for some value of a between 2 and 3 they're equal.
But this is not a realistic GMAT question at all.
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