2023 · Q94
Question: Are x, y, z equal, where x, y, z are real numbers?
Statement Ix² + y² + z² − xy − yz − zx = 0.
Statement IIx³ + y³ + z³ − 3xyz = 0. Mark option (a) if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone. Mark option (b) if the question can be answered by using either statement alone. Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone. Mark option (d) if the question cannot be answered even by using both the statements together.
Elementary MathematicsAlgebra
- AThe question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
- BThe question can be answered by using either statement alone
- CThe question can be answered by using both the statements together, but cannot be answered using either statement alone
- DThe question cannot be answered even by using both the statements together
Explanation
St.I: (1/2)[(x-y)^2+(y-z)^2+(z-x)^2]=0 forces x=y=z: answers alone. St.II: (x+y+z)(...)=0 also allows x+y+z=0 with unequal values (1,2,-3): cannot decide alone. One statement alone suffices. Option (a). Key disputed: verify against the official UPSC key.