Question

Let A = (x, y) \(\in\) {R \(\times\) {R} : |x + y| \(\geq\) 3} and} B = (x, y) \(\in\) {R \(\times\) {R} : |x| + |y| \(\leq 3\)}. If C = (x, y) \(\in\) A \(\cap\) B : x = 0 \(\text{ or y = 0}\), then \(\sum_{(x, y) \in C} |x| + |y|\) is:

• A. 15
• B. 18
• C. 24
• D. 12

Hint:

To solve problems involving intersections of sets defined by absolute values, carefully analyze the constraints and identify the points that satisfy both conditions.

Answer 1

The Correct Option is D

The set \( A \) includes all points \( (x, y) \) such that \( |x + y| \geq 3 \).
The set \( B \) includes all points \( (x, y) \) such that \( |x| + |y| \leq 3 \).
The set \( C \) is the intersection of \( A \) and \( B \) where either \( x = 0 \) or \( y = 0 \).
The points in \( C \) where \( x = 0 \) are on the line \( |y| \geq 3 \), but within the bounds of \( |x| + |y| \leq 3 \). These points are \( (0, 3) \) and \( (0, -3) \), contributing a total of \( |x| + |y| = 3 + 3 = 6 \).
The points in \( C \) where \( y = 0 \) are on the line \( |x| \geq 3 \), within the bounds of \( |x| + |y| \leq 3 \).
These points are \( (3, 0) \) and \( (-3, 0) \), contributing a total of \( |x| + |y| = 3 + 3 = 6 \). \[ \text{Total sum} = 6 + 6 = 12. \]

So, The correct answer is (4), as the sum is 12.