sahil5695 wrote:

Is x<y?

1 \(\sqrt{x} < y\)

2 \(x^2 < y\)

Interesting problem! Since there are only two variables, and the math looks like it's simplified already, I'm going to start by testing cases.

Statement 1:

If \(\sqrt{x} < y\), we could have a very small x and a very large y: for instance, x = 1 and y = 100. In that case, x < y, so the answer to the question is

yes.

On the other hand, we could have a situation where \(\sqrt{x}\) is smaller than y, but x itself is actually bigger than y. You could do this by picking an x that's only a little bit bigger than y. Then, when you take the square root, it gets much smaller. For instance, x = 4 and y = 3. In that case, x > y, so the answer to the question is

no.

We got both 'yes' and 'no' answers to the question, so the statement is insufficient. Cross off A and D.

Statement 2:

Our first case will work here as well. x = 1 and y = 100. So, x < y, so the answer to the question is

yes.

Now we need to think of a situation where \(x^2\) is smaller than y, but where x is actually bigger than y. That means x would have to be bigger than \(x^2\). That only happens when x is a fraction. Let's say that x = 1/2, so \(x^2\) = 1/4. Then we need to choose a value of y that's bigger than 1/4, but smaller than 1/2. Let's say x = 1/2 and y = 1/3. That fits the statement, and x > y, so the answer is

no.

We got both 'yes' and 'no' answers to the question, so the statement is insufficient. Cross off B.

Both statements together:

The problem here is that you can't get a 'no' answer if both statements are true.

If statement 2 is true, the only way to get a 'no' answer is if x and y are fractions.

If statement 1 is true, the only way to get a 'no' answer is if x is bigger than \(\sqrt{x}\). That doesn't happen if x is a fraction.

So, there's no way to get a 'no' answer from both statements at the same time. The answer must be 'yes'. So, the right answer is

C.

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