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Overview: Modulus is one of those sneaky concepts in CAT Quant where a single sign error can cost you the entire question. In this complete guide, you'll learn every property, shortcut, and solved example you need to master CAT Modulus Questions - plus a free downloadable PDF for last-mile practice.
If you've been preparing for CAT 2026, you already know that the Quantitative Aptitude section rewards students who can think fast under sign-based traps. CAT Modulus Questions sit right at the centre of this trap zone - they look deceptively simple, but they consistently trip up even the strongest aspirants. In recent CAT slots, modulus has appeared not just as standalone questions but also blended into inequalities, quadratics, graphs, and even functions.
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This guide is built to fix that. We'll walk through what modulus actually means, every property you must memorize, the 6 most common question types asked in CAT, fully solved examples with the exact step-by-step method examiners expect, shortcut tricks, common mistakes to avoid, and a free CAT Modulus Questions you can download and revise from.
Key Highlights of This Guide
- Concept Clarity: What modulus means + 8 must-know properties.
- 6 Question Types: Every variation CAT can throw at you, mapped.
- 15+ Solved Questions: Step-by-step, CAT-level difficulty.
- Shortcut Tricks: Save 60–90 seconds per question.
- Previous Year Style Questions: Based on actual CAT slot patterns.
- Free CAT Modulus Questions PDF with answer keys.
- Common Mistakes students make and how to avoid them.
Table of Contents
What are Modulus Questions in CAT 2026?
The modulus of a real number - written as |x| - simply means the non-negative value of that number, regardless of its sign. It represents the distance from zero on the number line, and distance can never be negative.
Condition:
If x ≥ 0, then Value of |x| = x
If x < 0, then Value of |x| = −x
So |5| = 5 and |−5| = 5. Simple in theory, but in CAT it gets layered with inequalities, two or more modulus terms, and unknown variables - that's where students lose marks. CAT examiners love modulus precisely because it tests whether you can handle multiple cases without missing one.
| CAT Linear Questions PDF | |
| CAT Quadratic Questions PDF | |
| CAT Inequalities Questions PDF | |
| CAT Logarithm Questions PDF |
Important Properties of Modulus
Before you attempt any modulus CAT questions, lock these 8 properties into memory. They are the foundation of every shortcut you'll learn later.
| S.No | Property | Formula / Result |
|---|---|---|
| 1 | Non-negativity | |x| ≥ 0 for all real x |
| 2 | Identity of Indiscernibles | |x| = 0 if and only if x = 0 |
| 3 | Symmetry | |−x| = |x| |
| 4 | Product Rule | |x · y| = |x| · |y| |
| 5 | Quotient Rule | |x / y| = |x| / |y|, where y ≠ 0 |
| 6 | Triangle Inequality | |x + y| ≤ |x| + |y| |
| 7 | Reverse Triangle Inequality | | |x| − |y| | ≤ |x − y| |
| 8 | Square Property | |x|² = x² |
Types of CAT Modulus Questions
Based on the last 10 years of CAT papers, CAT Modulus Questions can be grouped into 6 dominant categories. Knowing the type tells you the method.
| Type | Example | Method |
|---|---|---|
| 1. Simple Modulus Equations | |x − 3| = 7 | Split into two cases |
| 2. Modulus Inequalities | |x − 2| < 5 | Apply inequality rules |
| 3. Multiple Modulus Expressions | |x − 1| + |x − 4| = 7 | Critical-point analysis |
| 4. Modulus in Quadratic Equations | x² − 5|x| + 6 = 0 | Substitute |x| = y |
| 5. Modulus + Functions / Graphs | Min of |x − 1| + |x − 5| + |x − 9| | Median property |
| 6. Modulus + Inequality Mix | |x² − 4| ≥ |x − 2| | Case + factor analysis |
Standard Modulus Inequality Rules
| Inequality | Solution |
|---|---|
| |x| < a | −a < x < a |
| |x| ≤ a | −a ≤ x ≤ a |
| |x| > a | x < −a or x > a |
| |x| ≥ a | x ≤ −a or x ≥ a |
CAT Modulus Questions to Practice
Let's now apply everything you've learned. Each example below is modelled on the difficulty and pattern of actual CAT slot questions.
Q1. Solve |2x − 5| = 9
Case I: 2x − 5 = 9 ⇒ 2x = 14 ⇒ x = 7
Case II: 2x − 5 = −9 ⇒ 2x = −4 ⇒ x = −2
Step 2: Verify both in the original equation. Both satisfy.
Answer: x = 7 or x = −2
Q2. Solve |x − 3| < 5
Add 3 throughout: −2 < x < 8
Answer: x ∈ (−2, 8)
Q3. Find all real x satisfying |x − 1| + |x − 4| = 5 (CAT-style)
Case A (x < 1): −(x−1) − (x−4) = 5 ⇒ −2x + 5 = 5 ⇒ x = 0 ✔
Case B (1 ≤ x ≤ 4): (x−1) − (x−4) = 3 ≠ 5 ✘ (no solution)
Case C (x > 4): (x−1) + (x−4) = 5 ⇒ 2x − 5 = 5 ⇒ x = 5 ✔
Answer: x = 0 or x = 5
Q4. Solve x² − 7|x| + 12 = 0
Equation becomes: y² − 7y + 12 = 0 ⇒ (y − 3)(y − 4) = 0 ⇒ y = 3 or y = 4.
So |x| = 3 ⇒ x = ±3, and |x| = 4 ⇒ x = ±4.
Answer: x = ±3, ±4 (four solutions)
Q5. Find the minimum value of f(x) = |x − 2| + |x − 6| + |x − 10|
Median of {2, 6, 10} = 6.
f(6) = |6−2| + |6−6| + |6−10| = 4 + 0 + 4 = 8.
Answer: Minimum value = 8 (at x = 6)
Q6. (CAT 2022 Slot-style) How many integer solutions exist for |x| + |y| = 5?
On each side, integer pairs: 5 + 1 = 6 (including endpoints), but corners get counted twice.
Total integer points = 4 × 5 = 20.
Answer: 20 integer solutions
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Q7. The number of distinct integer solutions (x,y)(x, y)(x,y) of the equation ∣x+y∣+∣x−y∣=2|x+y| + |x-y| = 2∣x+y∣+∣x−y∣=2 is?
A. 8
B. 2
C. 3
D. 0
Answer: 8
Q8. The number of distinct real values of xxx, satisfying the equation max{x,2}−min{x,2}=∣x+2∣−∣x−2∣\max\{x,2\} - \min\{x,2\} = |x+2| - |x-2|max{x,2}−min{x,2}=∣x+2∣−∣x−2∣ is ______.
A. 1
B. 2
C. 3
D. 4
Answer: 2
Q9. If x and y are real numbers such that ∣x∣+x+y=15|x| + x + y = 15∣x∣+x+y=15 and x+∣y∣−y=20,x + |y| - y = 20,x+∣y∣−y=20, then (x−y)(x-y)(x−y) equals:
A. 15
B. 24
C. 32
D. 41
Answer: 15
Q10. The area of the quadrilateral bounded by the YYY-axis, the line x=5x = 5x=5, and the lines ∣x−y∣−∣x−5∣=2|x-y| - |x-5| = 2∣x−y∣−∣x−5∣=2 is ______.
A. 42
B. 76
C. 33
D. 45
Answer: 45
Q11. The largest real value of aaa for which the equation ∣x+a∣+∣x−1∣=2|x+a|+|x-1|=2∣x+a∣+∣x−1∣=2 has an infinite number of solutions for xxx is:
A. 2
B. 7
C. 3
D. 1
Answer: 1
Q12. The number of integer solutions of equation 2∣x∣(x2+1)=5x22|x|(x^2+1)=5x^22∣x∣(x2+1)=5x2 is ______.
A. 2
B. 7
C. 3
D. 4
Answer: 3
Q13. Let 0≤a≤x≤1000 \le a \le x \le 1000≤a≤x≤100 and f(x)=∣x−a∣+∣x−100∣+∣x−a−50∣.f(x)=|x-a|+|x-100|+|x-a-50|.f(x)=∣x−a∣+∣x−100∣+∣x−a−50∣. Then the maximum value of f(x)f(x)f(x) becomes 100100100 when aaa is equal to:
A. 28
B. 72
C. 50
D. 42
Answer: 50
Q14. If 3x+2∣y∣+y=73x+2|y|+y=73x+2∣y∣+y=7 and x+∣x∣+3y=1,x+|x|+3y=1,x+∣x∣+3y=1, then x+2yx+2yx+2y is:
A. 8
B. -2
C. 0
D. 2
Answer: 0
Shortcut Tricks to Solve Quickly
Speed matters in CAT. Below are the time-saving shortcuts that experienced mentors use to solve modulus CAT questions in 60 seconds or less.
| Trick | When to Use | How It Helps |
|---|---|---|
| Median Trick | Min of |x − a₁| + |x − a₂| + ... + |x − aₙ| | Min occurs at median (odd n) or between two middle values (even n) |
| Geometric Interpretation | Sum of modulus expressions equals a constant | Think distance on number line instead of algebra |
| Squaring Both Sides | |A| = |B| form | Eliminates modulus, gives clean polynomial |
| |x|² = x² Substitution | Whenever |x|² appears | Removes modulus instantly |
| Plot, Don't Solve | Complex multi-modulus expressions | Visual intersections faster than algebra |
Exam Tip: In CAT, if a modulus question has more than 3 cases, attempt it last. The time cost is rarely worth it unless you're sure of the method.
Common Mistakes to Avoid in CAT 2026
Even strong students lose easy marks in CAT Modulus Questions because of these recurring errors. Don't be one of them.
| Mistake | What Goes Wrong | How to Avoid It |
|---|---|---|
| Forgetting the negative case | Solving |x − 4| = 6 and writing only x = 10, missing x = −2 | Always write both cases before solving |
| Not verifying solutions | Some case-split solutions don't satisfy the original equation | Plug every answer back into the original |
| Missing critical points | In |x − 1| + |x − 3| + |x − 7|, skipping one critical point | List all critical points before dividing the number line |
| Squaring carelessly | Squaring |x − 2| = x without checking that x ≥ 0 | Always check the domain after squaring |
| Confusing < and > rules | Treating |x| > a like |x| < a or vice versa | Memorize: "less than" = bounded; "greater than" = two rays |
CAT Modulus Questions PDF - Free Download
Want to revise on the go? We've curated a comprehensive CAT Modulus Questions PDF containing all the question types covered above, plus 50+ extra problems with detailed solutions, previous year CAT-style sets, and a quick-revision sheet of properties and formulas.
CAT Modulus Questions PDF (2026 Edition)
50+ solved questions • Property revision sheet • Previous year patterns • Answer key included
What's inside the PDF: Concept summary, 8 properties at a glance, 6 question types with examples, 50 practice problems sorted by difficulty, solutions with shortcut tricks, and a CAT-pattern mini mock.
Practice Questions with Answers
Time to test yourself. Set a timer for 12 minutes and try these 6 questions. Answers are provided below.
| Q.No | Question | Answer |
|---|---|---|
| 1 | Solve |3x − 4| = 11 | x = 5 or x = −7/3 |
| 2 | Solve |x + 2| ≤ 7 | x ∈ [−9, 5] |
| 3 | Find min of f(x) = |x − 1| + |x − 4| + |x − 7| + |x − 10| | 12 (any x in [4, 7]) |
| 4 | Solve x² − 8|x| + 15 = 0 | x = ±3, ±5 |
| 5 | How many integer solutions of |x| + |y| ≤ 3? | 25 |
| 6 | Solve |x − 1| + |x − 3| = 6 | x = −1 or x = 5 |
CAT Modulus Preparation Tips
Concept is one thing; exam-ready mastery is another. Here's how toppers approach this topic:
- Build property fluency first. Don't attempt practice questions until the 8 properties are second nature.
- Practice case-splitting daily. Even 5 questions a day for 10 days will transform your speed.
- Always sketch the number line for multi-modulus expressions. Visual = fewer mistakes.
- Mix modulus with inequalities. CAT rarely tests them in isolation, so practice combined sets.
- Time yourself. Target: 90 seconds for simple, 2 minutes for moderate, 2.5 minutes for advanced.
- Solve previous year CAT and XAT papers. Modulus appears in 2–3 questions almost every year.
- Maintain a mistake log. Note every modulus question you got wrong and revisit weekly.
Conclusion
CAT Modulus Questions are not hard - they are unforgiving. One missed case, one sign flipped, and the question is gone. But the inverse is also true: students who master modulus rarely lose marks on it, and they often crack 2–3 "free" questions per slot while others struggle.
The roadmap is clear. Lock in the 8 properties, drill case-splitting on at least 40 questions, use the median trick and geometric interpretation as your speed weapons, and revise from the CAT Modulus Questions PDF regularly. Combine this with consistent mock practice, and modulus will quickly shift from a weak area to a guaranteed scoring zone in your CAT 2026 attempt.
Bookmark this guide, download the PDF, and revisit it before every mock. All the best for your CAT journey.
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