<p>Let <code>max{|x + y|, |x - y|} = 1</code> and <code>y = x - [x]</code> be two equations which <code>x</code> and <code>y</code> satisfy, then the number of ordered pairs <code>(x, y)</code> is</p>
Step-by-Step Solution
Key Concept: The fractional part function {x} combined with the constraint max{|x+y|, |x-y|}=1 defines a square region; finding their intersection gives the number of ordered pairs.
<p><strong>Solution:</strong></p><p>We have <code>y = x - [x] = {x}</code>, the fractional part of <code>x</code>.</p><p>Also, <code>max{|x + y|, |x - y|} = 1</code></p><p>If <code>|x + y| ≥ |x - y|</code>, then <code>|x + y| = 1</code></p><p>On squaring: <code>x² + y² + 2xy = 1</code></p><p>If <code>|x + y| < |x - y|</code>, then <code>|x - y| = 1</code></p><p>This gives us a square region in the <code>xy</code>-plane.</p><p>When <code>-1 ≤ x ≤ 0</code>, both graphs coincide, giving infinite solutions.</p><p>∴ Answer is (d) ∞</p>
Correct Answer: D