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

To solve the given problem, we need to understand and process the sets \(A\), \(B\), and \(C\).

Define: 

• Set \(A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\}\).
• Set \(B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\}\).
• Set \(C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}\).

Step 1: Identify points satisfying both conditions.

For points where \(x = 0\) or \(y = 0\) in set \(A\), the conditions are:

• If \(x = 0\), then \(|y| \geq 3\).
• If \(y = 0\), then \(|x| \geq 3\).

In set \(B\), the points satisfy \(|x| + |y| \leq 3\), and we need to find intersections with the conditions above.

Step 2: Calculate intersection points in \(C\).

Possible boundary values are:

• For \(x = 0\): \((0, 3)\) and \((0, -3)\).
• For \(y = 0\): \((3, 0)\) and \((-3, 0)\).

These points are inside set \(B\) since \(|0| + |3| = 3\), and similarly for others.

Step 3: Calculate \(\sum_{(x, y) \in C} |x| + |y|\).

• For \((0, 3)\): \(|0| + |3| = 3\).
• For \((0, -3)\): \(|0| + |-3| = 3\).
• For \((3, 0)\): \(|3| + |0| = 3\).
• For \((-3, 0)\): \(|-3| + |0| = 3\).

Summing these values gives \(3 + 3 + 3 + 3 = 12\).

The correct answer is: 12

Answer 2

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.