Question

For a real number \( x \), is \( x^3 = 125 \)? Statement (I): \( x > 4 \)
Statement (II): \( x < 6 \)

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

In Data Sufficiency questions, a statement is sufficient only when it leads to a definite YES or a definite NO. If multiple values remain possible that produce different answers, the statement is insufficient.

Answer 1

The Correct Option is D

Concept: This is a Data Sufficiency problem. We have to determine whether the given statements provide enough information to answer the question: \[ x^3 = 125 \; ? \] Since \[ 125 = 5^3, \] the equation \(x^3 = 125\) has exactly one real solution: \[ x = 5. \] Therefore, a statement will be sufficient only if it allows us to conclude with certainty that \(x = 5\) or that \(x \neq 5\).

Step 1: Analyze Statement (I). Given: \[ x > 4 \] This statement tells us only that \(x\) is greater than 4. Possible values include: \[ x=5,\quad x=6,\quad x=10,\quad x=4.5 \] If \(x=5\), \[ x^3 = 125 \] which gives a YES answer. However, if \(x=6\), \[ x^3 = 216 \neq 125 \] which gives a NO answer. Since both YES and NO are possible, Statement (I) alone is not sufficient.

Step 2: Analyze Statement (II). Given: \[ x < 6 \] This statement tells us only that \(x\) is less than 6. Possible values include: \[ x=5,\quad x=4,\quad x=0,\quad x=-2 \] If \(x=5\), \[ x^3 = 125 \] which gives a YES answer. If \(x=4\), \[ x^3 = 64 \neq 125 \] which gives a NO answer. Therefore, Statement (II) alone is also not sufficient.

Step 3: Analyze Statements (I) and (II) together. Combining the two statements gives: \[ x > 4 \] and \[ x < 6 \] Hence, \[ 4 < x < 6 \] This interval contains infinitely many real numbers: \[ 4.1,\; 4.5,\; 5,\; 5.2,\; 5.8,\ldots \] Only one of these values, namely \(x=5\), satisfies \[ x^3=125. \] Since the value of \(x\) is still not uniquely determined, we cannot answer the question definitively. Therefore, even together the statements are not sufficient. Conclusion: Neither Statement (I) nor Statement (II), nor both together, are sufficient to determine whether \[ x^3 = 125. \] \[ \boxed{\text{Neither statement is sufficient}} \]