Question

Directions: Each of the following questions is followed by two statements. As the answer, mark (1) if the question can be answered with the help of statement I alone, mark (2) if the question can be answered with the help of statement II alone, mark (3) if both statement I and statement II together are needed to answer the question, and mark (4) if the question cannot be answered even with the help of both the statements.

Is line MN perpendicular to the X-axis?
I. M = (5, 12)
II. N lies on the X-axis at a distance of 5 units from the origin.

• A. The question can be answered with the help of statement I alone.
• B. The question can be answered with the help of statement II alone.
• C. Both statement I and statement II together are needed to answer the question.
• D. The question cannot be answered even with the help of both the statements.

Hint:

Being 5 units from the origin on the X-axis fixes only the distance, not the sign, so N could be at (5,0) or (-5,0), check whether that changes the answer.

Answer 1

The Correct Option is D

Step 1: Understand the Concept.
A line is perpendicular to the X-axis exactly when it is a vertical line, and a vertical line is one where every point on it has the same x-coordinate. So MN is perpendicular to the X-axis if and only if M and N share the same x-coordinate.

Step 2: Use Statement I alone.
Statement I gives \( M = (5, 12) \), so M's x-coordinate is 5. It tells us nothing about N's coordinates at all. We cannot check whether N shares that x-coordinate, so statement I alone is not enough.

Step 3: Use Statement II alone.
Statement II says N lies on the X-axis at a distance of 5 units from the origin. A point on the X-axis has the form \( (x, 0) \), and being 5 units from the origin means \( |x| = 5 \), so \( x = 5 \) or \( x = -5 \). This gives two possible points for N, \( (5, 0) \) or \( (-5, 0) \), and statement II alone does not even mention M. So statement II alone is not enough either.

Step 4: Combine both statements.
Take M = (5, 12) from statement I, and N = (5, 0) or N = (-5, 0) from statement II.
If N = (5, 0): M and N both have x-coordinate 5, so line MN is a vertical line, perpendicular to the X-axis.
If N = (-5, 0): M has x-coordinate 5 and N has x-coordinate -5, so line MN is slanted, not perpendicular to the X-axis.
The two allowed positions for N give two different answers to the question, one yes and one no. Since even combining both statements leaves the question unresolved, the data given is not enough.

Final Answer:
The question cannot be answered even with the help of both the statements. \[ \boxed{\text{Option 4}} \]