UPSI Maths Syllabus 2026: Complete Numerical & Mental Ability Topics, Exam Pattern & Preparation Guide
The most comprehensive breakdown of the UP Police Sub Inspector Maths section — chapter-wise topics, marks distribution, weightage analysis, solved examples, shortcut tricks & a complete preparation strategy for UPPRPB SI 2026.
The UP Police Sub Inspector (UPSI) Maths Syllabus 2026 officially falls under the section titled “Numerical & Mental Ability Test” in the UPPRPB SI written examination. This section carries 100 marks (40 questions × 2.5 marks each) and is one of the four equal-weightage sections of the 400-mark written test conducted by the Uttar Pradesh Police Recruitment and Promotion Board (UPPRPB).
The Numerical & Mental Ability section is widely considered the most score-differentiating section in UPSI — candidates who master arithmetic shortcuts and mental ability techniques can complete this section in under 30 minutes with high accuracy, earning a decisive advantage over competitors who attempt all 160 questions in the 2-hour exam.
This guide provides the complete, chapter-wise maths syllabus with topic breakdowns, marks weightage, solved example questions for each topic, shortcut formulas, a weekly study plan, and expert preparation strategy — everything you need to score 80+ out of 100 in the UPSI Maths section.
UPSI Exam Pattern 2026 – All 4 Sections
The UPSI written examination is a Computer-Based Test (CBT) of 160 objective questions for a total of 400 marks. The exam is divided into 4 equal sections of 40 questions each, with each section carrying 100 marks. Each correct answer earns 2.5 marks. There is no negative marking. The total duration is 2 hours (120 minutes).
| Section | Subject | Questions | Marks | Per Question |
|---|---|---|---|---|
| Section 1 | General Hindi | 40 | 100 | 2.5 Marks |
| Section 2 | General Knowledge / Fundamental Law / Constitution | 40 | 100 | 2.5 Marks |
| Section 3 | Numerical & Mental Ability Test (MATHS) | 40 | 100 | 2.5 Marks |
| Section 4 | Mental Aptitude / Intelligence Quotient / Reasoning Ability | 40 | 100 | 2.5 Marks |
| Total | 160 | 400 | 2.5 Marks | |
Maths Section – Structure & Weightage Analysis
The UPSI Maths section (Section 3) is officially titled “Numerical & Mental Ability Test” and is split into two sub-components — Numerical Ability (pure mathematics and arithmetic) and Mental Ability (mathematical reasoning, number-based logic). Understanding this split helps allocate preparation time correctly.
| Sub-Component | Coverage | Approx. Questions | Approx. Marks | Priority |
|---|---|---|---|---|
| Numerical Ability | Arithmetic, Algebra, Geometry, Mensuration, Statistics | ~25–28 Qs | ~62–70 | 🔴 High |
| Mental Ability | Number Series, Missing Numbers, Mathematical Reasoning, Analogy | ~12–15 Qs | ~30–38 | 🟠 Medium |
| Total Section 3 | Combined | 40 Qs | 100 Marks | — |
Numerical Ability Topics (Chapter-Wise Syllabus)
The Numerical Ability sub-section covers all arithmetic, algebra, geometry, and statistics topics. Below is the complete chapter-wise breakdown with sub-topics and exam tips for each chapter:
- Natural, Whole, Rational and Irrational Numbers
- Prime and Composite Numbers
- LCM and HCF – finding and applications
- BODMAS / DMAS – order of operations
- Divisibility Rules (2, 3, 4, 5, 6, 8, 9, 10, 11)
- Fractions, Decimals – addition, subtraction, multiplication
- Powers and Exponents
- Square Roots and Cube Roots
- Surds and Indices
- Converting fractions and decimals to percentages
- Percentage increase and decrease
- Percentage of a quantity
- Population increase/decrease problems
- Percentage change in price and quantity
- Election-based percentage problems
- Expenditure = Price × Quantity problems
- Cost Price, Selling Price, Marked Price
- Profit % and Loss % calculation
- Single and successive discounts
- Discount vs Profit – combined problems
- Dishonest dealer / false weight problems
- Selling at CP problems (x sold at cost of y)
- Break-even and no-profit-no-loss problems
- SI formula: SI = (P × R × T) / 100
- CI formula: A = P(1 + R/100)ⁿ
- Finding Principal, Rate, or Time from SI/CI
- Difference between CI and SI for 2 and 3 years
- Compounding half-yearly and quarterly
- Instalment-based CI problems
- Population growth as compound interest
- Ratio – simplification and comparison
- Direct and Inverse Proportion
- Partnership – capital × time ratio
- Alligation / Mixture Rule
- Mean price of mixture
- Replacement of mixture problems
- Ratio-based distribution of amounts
- Work done by one person, two persons
- Combined work rate problems
- Pipes and cisterns (inlet and outlet)
- Work and wages (efficiency-based wages)
- A started, B joined later problems
- Partial work completion problems
- Man-days concept (M₁D₁ = M₂D₂)
- Speed = Distance / Time basics
- Average speed (not arithmetic average)
- Relative speed – same and opposite directions
- Trains – crossing a pole, platform, another train
- Boats and Streams – upstream / downstream
- Meeting point problems
- Unit conversion – km/hr ↔ m/s (×5/18 or ÷5/18)
- Linear equations in one variable
- Linear equations in two variables
- Quadratic equations – factorisation method
- Algebraic identities: (a+b)², (a–b)², a²–b²
- Polynomials – basic operations
- Arithmetic Progressions – nth term and sum
- Problems on numbers (age-type algebra)
- Average of numbers – direct calculation
- Weighted average problems
- Average of remaining when one number is removed
- Age problems – present age, age n years ago/later
- Ratio of ages – current and future
- Average speed problems
- Lines, Angles, and Triangles – basic properties
- Pythagoras Theorem and its applications
- Area – triangle, rectangle, square, circle
- Perimeter of 2D shapes
- Surface Area – cube, cuboid, cylinder, cone, sphere
- Volume – cube, cuboid, cylinder, cone, sphere
- Area of sector, segment of circle
- Trigonometric Ratios – sin, cos, tan, cot, sec, cosec
- Standard values – 0°, 30°, 45°, 60°, 90°
- Fundamental identities: sin²θ + cos²θ = 1
- Complementary angle identities
- Heights and Distances – angle of elevation/depression
- Simple word problems using tan θ = perpendicular / base
- Mean (Arithmetic Average)
- Median – middle value of ordered data
- Mode – most frequently occurring value
- Basic Probability
- Reading data from Bar Graphs
- Reading data from Pie Charts
- Reading data from Line Graphs and Tables
Mental Ability Topics (Mathematical Reasoning)
The Mental Ability sub-component of Section 3 tests mathematical reasoning — the ability to think logically about numbers, patterns, and sequences. These are distinct from general reasoning (which appears in Section 4). Mental Ability in UPSI specifically means number-based and arithmetic-based reasoning.
| Mental Ability Topic | What Is Tested | Approx. Questions | Priority |
|---|---|---|---|
| Number Series | Identify the pattern and find the next / missing term in a numeric series | 2–3 | Very High |
| Missing Number in Matrix / Table | Find the missing number in a 3×3 or 2×3 matrix based on row/column pattern | 1–2 | High |
| Mathematical Operations | If + means ×, × means ÷ etc. — substitute and calculate | 1–2 | High |
| Number Analogy | 4 : 16 :: 5 : ? — identify the mathematical relationship | 1–2 | Medium |
| Arithmetic Reasoning | Word problems based on basic arithmetic logic | 1–2 | Medium |
| Abstract Thinking | Ability to handle abstract numbers and symbols in calculations | 1–2 | Medium |
| Data Sufficiency (Basic) | Are the given statements sufficient to answer the question? | 1 | Low |
Chapter-Wise Weightage & Priority Table
Based on previous year UPSI papers and 2026 exam trend analysis, here is the chapter-wise marks weightage and preparation priority for the Maths section:
| Chapter / Topic | Expected Questions | Expected Marks | Priority Level | Difficulty |
|---|---|---|---|---|
| Percentage | 2–3 | 5–7.5 | 🔴 Very High | Easy–Medium |
| Profit, Loss & Discount | 2–3 | 5–7.5 | 🔴 Very High | Easy–Medium |
| Time, Speed & Distance | 3–4 | 7.5–10 | 🔴 Very High | Medium |
| Time & Work | 2–3 | 5–7.5 | 🟠 High | Medium |
| Simple & Compound Interest | 2–3 | 5–7.5 | 🟠 High | Easy–Medium |
| Number Series (Mental Ability) | 2–3 | 5–7.5 | 🟠 High | Easy |
| Number System / Simplification | 2–3 | 5–7.5 | 🟠 High | Easy |
| Ratio, Proportion & Mixture | 2–3 | 5–7.5 | 🟠 High | Medium |
| Average & Ages | 2–3 | 5–7.5 | 🟡 Medium | Easy–Medium |
| Algebra | 2–3 | 5–7.5 | 🟡 Medium | Medium |
| Mensuration | 2–3 | 5–7.5 | 🟡 Medium | Medium |
| Trigonometry | 2–3 | 5–7.5 | 🟡 Medium | Medium–Hard |
| Statistics & DI | 2–3 | 5–7.5 | 🟡 Medium | Easy–Medium |
| Missing Number / Math Operations | 2–3 | 5–7.5 | 🟢 Lower | Easy |
| Total | ~40 Questions | 100 Marks | — | — |
Solved Example Questions (Topic-Wise)
Here are representative solved examples for the most frequently tested UPSI Maths question types — with solution hints to help you learn the technique, not just the answer:
Important Maths Shortcut Tricks for UPSI 2026
With only 2 hours for 160 questions (average 45 seconds per question), using shortcut methods in the Maths section is not optional — it’s essential. Here are the most powerful shortcuts:
Successive Discount / Change Shortcut
Two changes of a% and b%: Net = a + b + (ab/100). If both are decrease, use negative values.
CI–SI Difference (2 Years)
CI – SI = P × (R/100)². No need to calculate full CI and SI separately for 2-year problems.
LCM Method for Time & Work
Total work = LCM of all days given. Efficiency = LCM ÷ days. Combined work per day = sum of efficiencies.
Train Crossing Shortcut
Time to cross = (Length of Train + Length of Platform) ÷ Speed. Convert km/hr to m/s by multiplying ×5/18.
Boats & Streams Shortcut
Speed in still water = (Downstream + Upstream) / 2. Speed of stream = (Downstream – Upstream) / 2.
Alligation / Mixture Rule
Ratio of mixing = |C.P₂ – Mean Price| : |Mean Price – C.P₁|. Draw a cross and subtract diagonally.
Average Speed (Not Simple Avg)
Average Speed = 2xy / (x+y) when equal distances at speeds x and y. Never use (x+y)/2 for distance-based problems.
Area of Triangle (Heron’s Formula)
Area = √[s(s–a)(s–b)(s–c)] where s = (a+b+c)/2. Also: Area = ½ × base × height for quick calculation.
CP from Profit/Loss on SP
CP = (SP × 100) / (100 + Profit%) or (SP × 100) / (100 – Loss%). Never confuse profit% on CP vs SP.
Number Series Patterns to Know
Top 5: ×2, ×2+1, perfect squares, perfect cubes, prime numbers, +2+4+6 (increasing differences), Fibonacci.
Chapter-Wise Study Plan for UPSI Maths
Use this prioritised 4-phase study plan to cover the complete UPSI Maths syllabus systematically:
Master the Basics
- Number System & BODMAS
- Percentage (all types)
- Profit, Loss & Discount
- LCM & HCF applications
- Ratio & Proportion
- Learn all formulas (flash cards)
Build Speed
- Simple & Compound Interest
- Time & Work (LCM method)
- Time, Speed & Distance
- Trains & Boats problems
- Average & Ages
- Mixture & Alligation
Cover Remaining Topics
- Algebra & Equations
- Geometry & Mensuration
- Trigonometry (trig table)
- Statistics & DI
- Number Series patterns
- Mathematical Operations
Test & Refine
- Full mock tests (complete paper)
- Maths section timed practice
- Analyse weak areas daily
- Revise shortcut formulas
- Previous year papers
- Final formula revision week
Preparation Tips & Strategy for UPSI Maths 2026
Prioritise the 5 Core Topics First
Percentage, Profit-Loss, SI/CI, Time-Work, and Time-Speed-Distance collectively account for 50–60% of the Maths section. Master these 5 before touching geometry or trigonometry.
Memorize All Formulas (One Sheet)
Create a single A4 formula sheet with every key formula. Review it every morning. In the 2-hour exam, you cannot afford to derive formulas from scratch — they must be instant recall.
Attempt All Questions — No Negative Marking
With zero penalty for wrong answers, never leave a Maths question blank. If you have eliminated 2 options, you have a 50% chance of being correct. A random guess gives 25% chance. Always attempt.
Use the LCM Method for Time & Work
For Time & Work problems, always use the LCM approach (assign total work = LCM of all given days). This eliminates fractions and dramatically speeds up calculation. Practice it until it’s automatic.
Solve 30 Maths Questions Daily
Speed and accuracy only come through repetition. Solve at least 30 maths questions every day across different topics. Rotate through the priority topics — don’t spend 5 days on one chapter.
Learn Number Series Patterns
Number Series questions in Mental Ability are the fastest marks in Section 3. Learn the 10 most common patterns (×2, ×2+1, squares, cubes, primes, Fibonacci, alternating series) — each question takes under 25 seconds once you recognise the pattern.
Trig Table is Non-Negotiable
Memorise the trigonometry values table (sin, cos, tan for 0°, 30°, 45°, 60°, 90°) completely. Heights and Distances questions always use tan θ = opposite/adjacent. Without this table memorised, trig questions cost valuable time.
Meet the 35% Sectional Cutoff First
The UPSI sectional qualifying mark is 35% (35 marks = 14 correct questions out of 40). With no negative marking, answering 14 questions correctly is the minimum safety net. Focus on easy chapters first to guarantee passing the section before targeting higher scores.
Best Books for UPSI Maths 2026
| Book | Best For | Why Recommended |
|---|---|---|
| Quantitative Aptitude for Competitive Examinations – R.S. Aggarwal | All Topics (Numerical Ability) | The gold standard for competitive maths. Covers every UPSI topic with 1,000+ practice questions, shortcut methods, and worked examples. The primary preparation book. |
| Fast Track Objective Arithmetic – Rajesh Verma (Arihant) | Speed & Shortcuts | Excellent for quick shortcut techniques and time-saving methods. Ideal for candidates who have basic knowledge but need to improve exam speed. |
| Magical Book on Quicker Maths – M. Tyra (BSC Publication) | Shortcut Methods | The most popular Vedic Maths and shortcut method book for competitive exams. Particularly useful for percentage, SI/CI, and time-work shortcuts. |
| UPSI Previous Year Papers (Various Publishers) | Real Exam Practice | The single most accurate preparation resource. Shows actual question difficulty, topic frequency, and question types from UPPRPB. Solve at least 5 previous year papers. |
| NCERT Mathematics Class 9 & 10 | Geometry, Algebra, Trigonometry | Free resource. The best reference for building concept clarity in Geometry, Algebra, and Trigonometry before moving to competitive level problem-solving. |