There is a question followed by two statements I and II : Question : If x and y are integers, can we find |x-y| uniquely? I. x2 + y2 = 25 II. x2 - y2 = 7 Which one of the following options is correct?
The question cannot be answered even using any of the statements
Statement I: \(x^2+y^2=25\) has integer solutions (x, y) = (0, ±5), (±5, 0), (±3, ±4), (±4, ±3). These give |x − y| values such as 5, 1 and 7 depending on which pair is taken, so the value is not unique. Statement I alone is insufficient.
Statement II: \(x^2-y^2=7\), i.e., \((x-y)(x+y)=7\). The integer factor pairs of 7 give (x, y) = (4, 3), (4, −3), (−4, 3), (−4, −3), yielding |x − y| equal to either 1 or 7 depending on the pair. Statement II alone is also insufficient.
Combining both: solving \(x^2+y^2=25\) and \(x^2-y^2=7\) together gives \(x^2=16,\ y^2=9\), so x = ±4 and y = ±3. All four sign combinations satisfy both equations simultaneously (since only the squares are fixed), giving |x − y| equal to 1 for (4, 3) or (−4, −3), and 7 for (4, −3) or (−4, 3).
Even together, the two statements do not pin down a unique value of |x − y|. Hence, the question cannot be answered even using both statements together.
Similar Questions
P, Q, R and S are four places. To travel from P to R, one travels via Q and to travel from P to S, one travels via both Q and R. Between P and Q, there are 3 routes; between Q and R, there are 4 routes; and between R and S, there are 5 routes. The time taken to travel through these routes is exhibited on the following chart in hours for a specific vehicle :
Routes Between P and Q Between Q and R Between R and S 1 2 1 2 2 3 1.5 2 3 4 2 2.5 4 - 2 3 5 - - 3.5 How many possible routes are there if the journey takes exactly 7 hours?
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