If xy ≠ 0, is xy = 70? (1) x > y (2) x^2 = y^2

(1) \(x > y\)

If x=35, y=2, YES
If x=2, y=1, NO

1 is not sufficient

(2) \(x^2 = y^2\)

So |x|=|y| which means x=y or x=-y

But x and y can take any value (for example x=y=√70 or x=y=1)

xy=70 may or may not be true

2 is not sufficient

(1)+(2) \(x > y\) and \(x^2 = y^2\)

Now we know that |x|=|y| and x≠y

So the only possibility is that x=-y

This means x and y have opposite signs and so xy is negative and hence surely not equal to 70

(1)+(2) is sufficient

Answer is (C)

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Statement 1: \(x > y\) Cannot tell anything about the values, so Insufficient.

A D / B C E

Statement 2: \(x^2 = y^2\) The squares are same. Either X, Y are positives or negatives, but the integer will remain the same. It means, there will be a square value such as 4, 16, 25, .. 64.. 81. We cannot get 70 from their multiplication. Hence, it is Sufficient to answer that XY ≠ 70

'B' is the winner


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What if x & y = √70
As question doesn't mentioned about x & y whether it is integer or not. So in this case it satisfies B statement.

rajatchopra1994 ... You are right I forgot the root case
If I consider √70 for both, it gives Yes and If I consider other integers, it gives No. Hence, B is also not sufficient.

Combining Statement 1 & 2: We need \(x^2\) = \(y^2\) and X > Y ... so, X has to be +ve and Y has to be -ve.

So, the answer has to be 'E'


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Since we have 2 variables (x and y) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)

Since \(x^2 = y^2\), we have \(x^2 - y^2 = (x+y)(x-y) = 0\) from the second condition, which means \(x=y\) or \(x=-y\).
However, \(x\) can't be equal to y due to the first condition, \(x = -y.\)
Since \(x\) and \(y\) have different parities, \(xy\) can't be positive number.
Thus, the answer is 'no'

Since both conditions together yield a unique solution, they are sufficient.

Since this question is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)

If \(x = 10, y = 7\), then \(xy = 70\) and the answer is 'yes'.
If \(x = 10, y = 6\), then \(xy = 60\) and the answer is 'no'.

Since condition 1) does not yield a unique solution, it is not sufficient


Condition 2)

If \(x = \sqrt{70}\) and \(y = \sqrt{70}\), then we have \(xy = 70\) and the answer is 'yes'.
If \(x = 10\) and \(y = 10\), then we have \(xy = 100\) and the answer is 'no'.

Since condition 2) does not yield a unique solution, it is not sufficient

Therefore, C is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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nehasomani33 - This one!!
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