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CAT 2024 Slot 3 QA Question 2

Type-in-the-answer (no negative marking) · Algebra · Modulus · Try it, then check the answer and solution below.

CAT 2024 Slot 3QAAlgebra • ModulusModerate
The number of distinct integer solutions (x,y)(x, y) of the equation $|x + y| + |x - y| = 2 , is
TITA Answer:
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Answer and solution

Answer: 8

📌 Core Concept
The equation ∣x+y∣+∣x−y∣|x + y| + |x - y| can be simplified using the property that ∣a∣+∣b∣=2⋅max⁡(∣a∣,∣b∣)|a| + |b| = 2 \cdot \max(|a|, |b|). Therefore, the given equation becomes: 2⋅max⁡(∣x∣,∣y∣)=2  ⟹  max⁡(∣x∣,∣y∣)=12 \cdot \max(|x|, |y|) = 2 \implies \max(|x|, |y|) = 1
🔢 Step-by-Step Solution
1
Understanding the Equation:
The equation simplifies to max⁡(∣x∣,∣y∣)=1\max(|x|, |y|) = 1.
This means both ∣x∣|x| and ∣y∣|y| are less than or equal to 1, and at least one of them is exactly 1.
2
Possible Values:
xandandy$ can be -1, 0, or 1.
We need to count all pairs where at least one of ∣x∣|x| or ∣y∣|y| is 1.
3
Counting Solutions:
When ∣x∣=1|x| = 1:
x = 1ororx = -1$.
For each x,yx , y can be -1, 0, or 1.
Solutions: (1, 1), (1, 0), (1, -1), (-1, 1), (-1, 0), (-1, -1) → 6 solutions.
When ∣y∣=1|y| = 1 and ∣x∣=0|x| = 0:
y = 1orory = -1$.
x = 0$.
Solutions: (0, 1), (0, -1) → 2 solutions.
4
Total Solutions:
Adding both cases: 6 + 2 = 8 distinct solutions.
⚡ 30-Second Shortcut
Recognize that ∣x+y∣+∣x−y∣=2|x + y| + |x - y| = 2 simplifies to max⁡(∣x∣,∣y∣)=1\max(|x|, |y|) = 1.
Enumerate all pairs where ∣x∣|x| or ∣y∣|y| is 1, ensuring no duplicates.
🎯 Final Answer
Correct Answer: 8

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