Question

Find out if t $<$ 0.
1. \(|t| > t\)
2. \(t^2 > 0\)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

Memorize the key properties of absolute values and squares for inequalities: - \(|x| = x\) if \(x \geq 0\). - \(|x| = -x\) if \(x < 0\). - The inequality \(|x| > x\) is true only for \(x < 0\). - \(x^2 > 0\) is true for all real \(x \neq 0\). - \(x^2 \geq 0\) is true for all real \(x\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question is a 'Yes/No' data sufficiency problem. We need to determine if the variable \( t \) is negative. A statement is sufficient if it allows us to conclude with certainty that \( t < 0 \) (a 'Yes' answer) or that \( t \geq 0 \) (a 'No' answer).

Step 2: Detailed Explanation:
Analyzing Statement (1): \(|t| > t\)
Let's analyze the properties of the absolute value function \( |t| \).
- If \( t > 0 \), then \( |t| = t \). In this case, the inequality becomes \( t > t \), which is false.
- If \( t = 0 \), then \( |t| = 0 \). In this case, the inequality becomes \( 0 > 0 \), which is false.
- If \( t < 0 \), then \( |t| = -t \). The inequality becomes \( -t > t \). Since \( t \) is negative, \( -t \) is positive, so a positive number is always greater than a negative number. This is true. For example, if \(t = -5\), then \(|-5| > -5\), which means \(5 > -5\), which is true.
The only case where \(|t| > t\) holds is when \( t \) is negative. Therefore, this statement guarantees that \( t < 0 \). The answer to the question "is t < 0?" is a definitive 'Yes'.
Thus, statement (1) is sufficient.
Analyzing Statement (2): \(t^2 > 0\)
The square of any real number is non-negative. \( t^2 \) will be greater than 0 for any real number \( t \) except for \( t = 0 \).
So, this statement tells us that \( t \neq 0 \).
- \( t \) could be positive (e.g., \( t = 2 \), \( 2^2 = 4 > 0 \)). In this case, the answer to "is t < 0?" is 'No'.
- \( t \) could be negative (e.g., \( t = -2 \), \( (-2)^2 = 4 > 0 \)). In this case, the answer to "is t < 0?" is 'Yes'.
Since the answer could be 'Yes' or 'No', this statement is not sufficient.

Step 3: Final Answer:
Statement (1) alone is sufficient, but statement (2) alone is not.