Question: Are x, y, z <strong>EQUAL</strong>, where x, y, z are real numbers?<br>Statement-I:<br>x^{2}+y^{2}+z^{2}-xy-yz-zx=0<br>Statement-II:<br>x^{3}+y^{3}+z^{3}-3xyz=0

Consider the following for the next ten (10) items that follow :<br>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.<br>Mark option (b) if the question can be answered by using either statement alone.<br>Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone.<br>Mark option (d) if the question cannot be answered even by using both the statements together.

  1. A. The question can be answered by using one of the statements alone, but cannot be answered using the other statement alone ✓
  2. B. The question can be answered by using either statement alone
  3. C. The question can be answered by using both the statements together, but cannot be answered using either statement alone
  4. D. The question cannot be answered even by using both the statements together

Correct Answer: A. The question can be answered by using one of the statements alone, but cannot be answered using the other statement alone

Explanation

Statement-I: x^2+y^2+z^2-xy-yz-zx = 0 simplifies to \frac{1}{2}((x-y)^2+(y-z)^2+(z-x)^2) = 0. Since squares are non-negative, this strictly implies x=y=z. Sufficient. Statement-II: x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx) = 0. This implies either x=y=z OR x+y+z=0. Not sufficient alone.

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