Question

Are the two lines L1 and L2 parallel?
1. Both lines lie in the first, second and fourth quadrants.
2. The y intercepts of the lines L1 and L2 are 8 and 4 respectively.

• 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:

For a "Yes/No" Data Sufficiency question, a statement is sufficient only if it always leads to the same answer (always "Yes" or always "No"). If you can find scenarios that satisfy the conditions but give different answers, the information is not sufficient.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The question asks whether two lines, L1 and L2, are parallel. For two distinct lines to be parallel, they must have the same slope and different y-intercepts.
Step 2: Key Formula or Approach:
Let the equations of the lines be \( y = m_1x + c_1 \) for L1 and \( y = m_2x + c_2 \) for L2.
For L1 and L2 to be parallel:
1. Slopes must be equal: \( m_1 = m_2 \)
2. Y-intercepts must be different: \( c_1 \neq c_2 \)
Step 3: Detailed Explanation:
Analyze Statement (1): "Both lines lie in the first, second and fourth quadrants."
A line that passes through Quadrant I, II, and IV must cross the y-axis at a positive value (to be in I and II) and have a negative slope (to go from II down to I and then IV). So, for both lines, \( m<0 \) and \( c>0 \). This tells us that both slopes are negative, but not that they are equal. For example, L1 could be \( y = -2x + 5 \) and L2 could be \( y = -3x + 6 \). Both satisfy the condition but are not parallel. Therefore, Statement (1) is not sufficient.
Analyze Statement (2): "The y intercepts of the lines L1 and L2 are 8 and 4 respectively."
This tells us that \( c_1 = 8 \) and \( c_2 = 4 \). We know the y-intercepts are different (\( c_1 \neq c_2 \)), which is a necessary condition for two distinct lines to be parallel. However, we have no information about their slopes (\( m_1 \) and \( m_2 \)). The slopes could be equal or unequal. Therefore, Statement (2) is not sufficient.
Analyze Statements (1) and (2) Together:
From (1), we know both lines have negative slopes (\( m_1<0, m_2<0 \)).
From (2), we know their y-intercepts are 8 and 4.
Combining these, we know that L1 and L2 are two distinct lines with positive y-intercepts and negative slopes. However, we still do not know if their slopes are equal. For example:
Case A (Parallel): L1 is \( y = -2x + 8 \) and L2 is \( y = -2x + 4 \). Both satisfy the conditions.
Case B (Not Parallel): L1 is \( y = -3x + 8 \) and L2 is \( y = -2x + 4 \). Both also satisfy the conditions.
Since we cannot definitively determine if the lines are parallel, the combined information is not sufficient.
Step 4: Final Answer:
Because even with both statements, we cannot be certain whether the slopes are equal, the information is not sufficient. The correct option is (E).