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:

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:

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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.

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Answer

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Answer

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:

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The Correct Option is D

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