Bunuel wrote:
What is the sum of the squares of x^2 and y^2?
(1) x^8 – y^8 = 6
(2) x^4 – y^4 = 2
This is a piece of the puzzle type of question, where we need to get the exact value
Question: To find \(x^2 + y^2\)
Statement 1: \(x^8 – y^8 = 6\)
\(x^8 – y^8 = (x^4)^2 - (y^4)^2 = (x^4 + y^4)(x^4 - y^4) = (x^4 + y^4)(x^2 + y^2)(x^2 - y^2) = 6\)
To find \(x^2 + y^2\), we need the value of \(x^2 - y^2\) and \(x^4 + y^4\)
Therefore Statement 1 Alone is Insufficient. Answer options could be B, C or E
Statement 2: \(x^4 – y^4 = 2\)
\(x^4 – y^4 = (x^2 + y^2)(x^2 - y^2) = 2\)
This data too is not enough to get the value of \(x^2 + y^2\).
Therefore Statement 2 Alone is Insufficient. Answer Options could be C or E
Combining Both Statements:\((x^4 + y^4)(x^2 + y^2)(x^2 - y^2) = 6\) and \((x^2 + y^2)(x^2 - y^2) = 2\)
With this, we can see that the value of \(x^4 + y^4 = 3\)
Adding \(x^4 + y^4 = 3\) and \(x^4 - y^4 = 2\), we get \(2x^4 = 5\) or \(x^4 = \frac{5}{2}\).
Therefore \(x^2 = \sqrt{\frac{5}{2}}\)
With this we can find the value of \(y^2\)
Therefore Both Statements together are Sufficient.
Option CArun Kumar