Question

What is the value of xyz?
Statement 1: x_a = y^b = z^c and ab + bc + ca 0 where a, b and c are non-zero integers
Statement 2: a^x = b, b^y = c, c^z = a where a, b and c are non-zero integers

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is D

To solve the question of finding the value of \(xyz\), we need to examine both of the provided statements individually. Our goal is to determine if either statement alone is sufficient to obtain a result. 

Statement 1:

The equation given is \(x^a = y^b = z^c\) and \(ab + bc + ca = 0\), where \(a\), \(b\), and \(c\) are non-zero integers.

From \(x^a = y^b = z^c = k\), where \(k\) is a constant, we can write:

• \(x = k^{1/a}\)
• \(y = k^{1/b}\)
• \(z = k^{1/c}\)

Now, calculate \(xyz\):

\(xyz = k^{(1/a + 1/b + 1/c)}\)

We know from the equation \(ab + bc + ca = 0\) that:

\(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0\)

Thus, \(xyz = k^0 = 1\). Therefore, statement 1 alone is sufficient to answer the question.

Statement 2:

Here, the statement is \(a^x = b, b^y = c, c^z = a\) for non-zero integers \(a\), \(b\), and \(c\).

From the given relations, calculate the product \(xyz\):

• Take logarithm of both sides:
  • \(x \log a = \log b\)
  • \(y \log b = \log c\)
  • \(z \log c = \log a\)
• Multiply all three equations:
• \((xyz)(\log a)(\log b)(\log c) = (\log a)(\log b)(\log c)\)

Hence, \(xyz = 1\). Statement 2 alone is also sufficient to answer the question.

Conclusion:

Since either statement alone is sufficient to determine that \(xyz = 1\), the correct answer to the question is: Either statement (1) alone or statement (2) alone is sufficient to answer the question.