Question

If \( wxyz \neq 0 \), in which quadrant of the rectangular cartesian coordinate system does \( (x, y) \) lie? Statement (I): \( (x, z) \) lies in the first quadrant.
Statement (II): \( (w, y) \) lies in the third quadrant.

• 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:

To determine the quadrant of a point, you must know the sign of both the x-coordinate and the y-coordinate. If statements give you partial information, consider if combining them completes the pair.

Answer 1

The Correct Option is C

Concept: The rectangular Cartesian coordinate system is divided into four quadrants based on the signs of the coordinates \( (x, y) \):
• Quadrant I: \( (+, +) \)
• Quadrant II: \( (-, +) \)
• Quadrant III: \( (-, -) \)
• Quadrant IV: \( (+, -) \)

Step 1: Analyze Statement (I). \( (x, z) \) lies in the first quadrant. This implies both coordinates are positive: \[ x > 0, \quad z > 0 \] This gives us the sign of \( x \), but provides no information about \( y \). Thus, Statement (I) is insufficient.

Step 2: Analyze Statement (II). \( (w, y) \) lies in the third quadrant. This implies both coordinates are negative: \[ w < 0, \quad y < 0 \] This gives us the sign of \( y \), but provides no information about \( x \). Thus, Statement (II) is insufficient.

Step 3: Combine both statements. From Statement (I), we have \( x > 0 \). From Statement (II), we have \( y < 0 \). The point \( (x, y) \) therefore has coordinates \( (+, -) \), which places it in the fourth quadrant. Combining both statements is sufficient. {(C) Both statements (I) and (II) are sufficient.}