AnirudhaS wrote:
Hi
IanStewart I would like to know your take on problems like this. How do we approach similar problems? For me plugging in takes so much time, its not even worth it. Please help.
I'd never just plug arbitrary numbers into a question like this. The first thing I'd want to work out is which types of numbers might matter. Often in questions like this, you can make things easy by thinking about positives and negatives. So, in a different question, if you knew x^2 > y^2, and you're wondering if y > x can be true, it's very easy to get a yes answer just by making x negative and y positive.
Unfortunately thinking about negatives doesn't help on this question, because we learn here that a is positive (we're taking its square root) and so is b (it's larger than c^2, a square). Instead what matters here are numbers larger than 1, because those increase when you square them and decrease when you square root them, and the 'fractions' between 0 and 1, because those do the opposite; they decrease when you square them and increase when you square root them:
Sajjad1994 wrote:
If \(\sqrt{a}\) > b > \(c^2\), which of the following could be true?
I. a > b > c
II. c > b > a
III. a > c > b
We know √a > b > c^2. If a, b and c are larger than 1, then a > b > c is going to be true for sure (because in that case, a > √a, and c^2 > c). So I can be true.
So the only way we'll find a situation where c > b, which is what we need in II and III, is if our letters are between 0 and 1. So I'd just take the simplest fraction possible, 1/2, and see what happens if √a = 1/2, b = 1/2, and c^2 = 1/2. Then a = 1/4, b = 1/2, and c = 1/√2 = √2/2 ~ 0.7. So in this case we find c > b > a is clearly true. Now, it can't technically be true that √a = b = c^2 = 1/2, but if √a is just negligibly larger than 1/2, and c^2 is negligibly smaller than 1/2, then √a > b > c^2 will be true, and c > b > a will be true. So II can be true, and so can III, by the same reasoning (a can be large, and b and c can be roughly 0.5).
But there are definitely other ways to look at this, and other good sets of numbers you could plug in. The important thing though is to identify that it's only the numbers between 0 and 1 that matter when you're looking at II and III.