CDS Elementary Mathematics Guide
How CDS Maths actually works
A 2,096-question analysis of every CDS Elementary Mathematics paper from 2016 II to 2026 II — 21 papers in all. 100 questions, 120 minutes, and a third of a mark lost for each wrong answer, so the decision on every question is whether to attempt it. We mapped the 22 chapter playbooks, the formulas, what moved over ten years, and the distractor traps.
- Past-year questions
- 2,096
- Papers (2016–2026)
- 21
- Chapters
- 26
- Playbooks
- 22
- of the bank is HARD
- 18.2%
The paper you are sitting
Elementary Mathematics is one of the three CDS papers, alongside English and General Knowledge (OTA candidates sit only those two). Everything below is the Maths paper.
- Questions
- 100
- Marks
- 100
- Duration
- 120 min
- Per question
- 1.2 min
1 mark a question
Read this before anything else
A wrong answer costs a third of a mark
That makes a blind guess among four options worth nothing on average: right one time in four for +1, wrong three times for −1/3 each. Rule out one option first and the same guess is worth something. This is the whole guessing rule:
- 4 options left: 0. A blind guess gains nothing on average and adds risk. Leave it.
- 3 options left: +1/9 of a mark. Worth it: one option ruled out makes the guess pay.
- 2 options left: +1/3 of a mark. Clearly worth it.
And the clock is tight: 120 minutes for 100 questions is 1.2 minutes a question, and 18.2% of the bank is HARD.
The difficulty split: 385 EASY (18.4%), 1,329 MODERATE (63.4%) and 382 HARD (18.2%) across the 2,096 questions.
Five chapters are nearly half the paper
Across all 21 papers the 5 cornerstone chapters average 46.1 of the 100 questions. No score is possible without them.
- Trigonometry10.81/paper · 20% HARD
- Number System10.62/paper · 16% HARD
- Mensuration 2D9.38/paper · 23% HARD
- Mensuration 3D8.14/paper · 17% HARD
- Triangles7.19/paper · 20% HARD
These are lifetime rates. Trigonometry and Number System have each risen to 13 a paper on 2024–2026 papers; see Trends.
How the 2,096 questions break down
All 26 chapters across the 21 papers, heaviest first, with the rate in the early and recent papers beside the lifetime one.
| Chapter | Q / paper | 2016–20 | 2024–26 | Questions | % HARD | Subtopics |
|---|---|---|---|---|---|---|
| Trigonometric Ratios and Identities | 10.81 | 8.78 | 13.00 | 227 | 20% | Degree, Radian and Standard Values (20 · 5% HARD), Ratios in a Right Triangle (22 · 18%), Complementary and Allied Angles (21 · 10%), Simplifying and Proving Identities (24 · 13%), Reciprocal Pairs — sec ± tan and cosec ± cot (18 · 6%), Power Identities and Given Sums (17 · 0%), Trigonometric Equations (26 · 15%), Compound and Multiple Angles (8 · 13%), Maximum, Minimum and Impossible Values (37 · 35%), Eliminating θ and Substitution Chains (34 · 47%). The largest chapter in the bank and rising: 8.8 questions a paper in 2016-2020, 13.0 in 2024-2026. Its HARD is pooled — maximum/minimum and eliminating θ hold 29 of its 46 HARD questions — so the other eight subtopics are cheap. |
| Number System | 10.62 | 9.78 | 13.00 | 223 | 16% | Division, Parity and Consecutive Integers (18 · 22% HARD), Place Value and Digit Problems (29 · 10%), Divisibility Rules and Missing Digits (9 · 22%), Unit Digit and Cyclicity (13 · 15%), Prime Numbers and Primality (22 · 0%), Factors, Divisor Counting and Trailing Zeros (20 · 20%), HCF and LCM Laws and Fractions (29 · 21%), HCF and LCM Applications and Remainder Recipes (21 · 0%), Remainders by Congruence and Cyclicity (26 · 27%), Divisibility by Factorisation (16 · 38%), Perfect Squares, Cubes and Difference of Squares (14 · 7%), Rational and Irrational Numbers (6 · 0%). Level with Trigonometry on recent papers at 13.0 a paper, up from 9.8. The HARD is spread across remainders, factorisation and HCF laws rather than pooled, and primes, HCF/LCM applications and irrationals carry none at all. |
| Mensuration 2D | 9.38 | 9.44 | 9.83 | 197 | 23% | Areas of Triangles (32 · 19% HARD), Rectangles, Squares and Other Quadrilaterals (34 · 21%), Equal Perimeters and Re-bent Wires (17 · 6%), Circumference, Wheels and Rings (19 · 5%), Arcs, Sectors and Segments (25 · 4%), Inscribed and Circumscribed Figures (32 · 34%), Touching Circles (13 · 62%), Combined and Shaded Regions (25 · 40%). Steady at about 9.4 a paper. Three subtopics are nearly free (wheels, arcs and sectors, re-bent wires) and three carry the chapter's HARD: touching circles 62%, shaded regions 40%, inscribed figures 34%. |
| Mensuration 3D | 8.14 | 8.11 | 6.67 | 171 | 17% | Cubes and Cuboids (20 · 5% HARD), Diagonals and Cuboid Identities (16 · 19%), Cylinders (16 · 13%), Cones (18 · 17%), Spheres, Hemispheres and Shells (10 · 10%), Frustums and Combined Solids (22 · 9%), Melting and Recasting (18 · 11%), Water — Immersion, Flow and Rainfall (17 · 6%), Scaling and Comparing Solids (19 · 32%), Solids Inside Solids (15 · 53%). Falling: 8.1 a paper in 2016-2020, 6.7 in 2024-2026. Eight of its ten subtopics are under 20% HARD; solids inside solids (53%) and scaling (32%) are where it costs. |
| Triangles | 7.19 | 7.00 | 7.67 | 151 | 20% | Angles of a Triangle (11 · 9% HARD), Triangle Inequalities (10 · 50%), Congruence and Similarity (13 · 23%), Parallels, Midpoints and the Bisector Theorem (17 · 6%), Ratio of Areas of Triangles (15 · 13%), Pythagoras Theorem and its Converse (22 · 18%), The Altitude to the Hypotenuse (27 · 0%), Medians and Apollonius Theorem (14 · 43%), Centres of a Triangle (16 · 31%), Sine and Cosine Rules (6 · 50%). Seven a paper, steady. The altitude to the hypotenuse is 27 questions with no HARD at all; medians, triangle inequalities and the sine and cosine rules are the expensive pages. |
| Algebraic Identities and Simplification | 4.67 | 3.33 | 4.33 | 98 | 39% | Squares and Cubes of a Binomial (12 · 25% HARD), Reciprocal Sums x ± 1/x (11 · 27%), Sums and Products of Three Variables (11 · 9%), The Cube Identity and a + b + c = 0 (11 · 18%), Sums of Squares and Least Values (12 · 42%), Conditional Identities (16 · 69%), Rational Algebraic Expressions (19 · 37%), Symmetric and Cyclic Expressions (6 · 100%). The hardest chapter in the bank at 39% HARD, and it cherry-picks: conditional identities (69%) and symmetric expressions (6 of 6 HARD) are the pool, while three-variable sums and the cube identity are cheap. |
| Quadratic Equations | 4.24 | 3.78 | 3.83 | 89 | 24% | Solving and Forming Quadratic Equations (15 · 27% HARD), Symmetric Functions of the Roots (17 · 24%), Roots in a Given Relation (10 · 30%), Nature of Roots and Discriminant (14 · 14%), Perfect Squares, Signs and Location of Roots (10 · 50%), Common Roots (6 · 0%), Maximum and Minimum of Quadratic Expressions (5 · 0%), Equations Reducible to Quadratics (6 · 50%), Word Problems and Applications (6 · 0%). About four a paper at 24% HARD, spread thinly across nine subtopics — nothing to skip; common roots, maximum/minimum and word problems have never set a HARD question. |
| Surds, Indices and Simplification | 3.90 | 4.56 | 3.00 | 82 | 22% | Fractions and Decimals (11 · 0% HARD), Laws of Indices (16 · 25%), Exponential Equations (8 · 0%), Square Roots of Surds (11 · 9%), Surds and Rationalisation (11 · 27%), Equations with Surds (7 · 71%), Simplification of Expressions (6 · 50%), Continued Fractions and Nested Radicals (12 · 17%). Falling (4.6 to 3.0 a paper). Equations with surds are 71% HARD; fractions, exponential equations and square roots of surds are almost free. |
| Statistics | 3.81 | 3.78 | 5.00 | 80 | 8% | Data, Scales and Presentation (11 · 0% HARD), Frequency Tables and Cumulative Frequency (15 · 0%), Properties of the Arithmetic Mean (12 · 0%), Median of Ungrouped Data (15 · 13%), Mean, Median and Mode of Grouped Data (19 · 21%), Choosing a Measure of Central Tendency (8 · 0%). The cheapest chapter of any size: 8% HARD, and rising from 3.8 to 5.0 a paper. Four of its six subtopics have never set a HARD question. |
| Polynomials | 3.76 | 3.89 | 4.33 | 79 | 19% | Degree, Zeros and Coefficients (8 · 38% HARD), The Remainder Theorem (15 · 20%), The Factor Theorem (12 · 0%), Factorisation of Polynomials (19 · 16%), HCF and LCM of Polynomials (25 · 24%). About four a paper at 19% HARD. The factor theorem is free; HCF and LCM of polynomials is the largest subtopic and holds the most HARD, 6 of 15. |
| Ratio, Proportion and Variation | 3.62 | 4.00 | 3.67 | 76 | 9% | Ratio and Proportion (16 · 0% HARD), Ratios in Income, Savings and Ages (10 · 10%), Equal Ratios and Proportion Algebra (14 · 14%), Direct and Inverse Variation (18 · 0%), Partnership (5 · 40%), Mixtures and Alligation (13 · 15%). Under four a paper at 9% HARD. Ratio and proportion and direct and inverse variation have never set a HARD question. |
| Time, Speed and Distance | 3.57 | 4.00 | 3.17 | 75 | 13% | Average Speed and Speed–Time Ratios (22 · 5% HARD), Speed Changes and Equations (9 · 33%), Relative Speed: Chasing and Meeting (13 · 8%), Trains and Relative Speed (14 · 14%), Boats and Streams (7 · 0%), Races (5 · 40%), Clocks and Angles (5 · 20%). Just over three a paper at 13% HARD. Average speed is 22 questions at 5%; boats and streams has no HARD. |
| Circles | 3.29 | 3.11 | 3.33 | 69 | 26% | Chords and Perpendiculars (15 · 33% HARD), Angles in a Circle (13 · 23%), Circumcircle and Locus (9 · 44%), Tangents from an External Point (11 · 0%), Intersecting Chords and Power of a Point (5 · 40%), Common Tangents and Common Chords (6 · 17%), Touching Circles (10 · 30%). Just over three a paper at 26% HARD. Tangents from an external point is free; the circumcircle and locus and the chord pages carry the HARD. |
| Quadrilaterals | 2.57 | 2.44 | 3.67 | 54 | 20% | General Quadrilaterals and their Diagonals (11 · 36% HARD), Parallelograms (14 · 14%), Rhombus and Kite (8 · 0%), Trapeziums (8 · 25%), Cyclic Quadrilaterals (13 · 23%). Rising (2.4 to 3.7 a paper). Rhombus and kite has no HARD; the general-quadrilateral page is the expensive one at 36%. |
| Data Interpretation | 2.48 | 2.56 | 1.67 | 52 | 2% | Tables (24 · 0% HARD), Pie Charts (18 · 0%), Bar and Line Graphs (5 · 20%), Caselet Data Interpretation (5 · 0%). Falling (2.6 to 1.7 a paper), and the cheapest chapter in the bank: one HARD question in 52. The work is careful arithmetic on a table or chart. |
| Percentage, Profit and Loss | 2.38 | 3.00 | 2.50 | 50 | 6% | Percentage (14 · 7% HARD), Successive Percentage Change (12 · 0%), Profit and Loss (17 · 6%), Successive Discount and Marked Price (7 · 14%). About 2.4 a paper at 6% HARD — three HARD questions in 50. |
| Averages | 2.24 | 2.67 | 1.83 | 47 | 9% | Sum and Mean (20 · 10% HARD), Weighted and Combined Averages (19 · 5%), Averages of Consecutive Numbers (8 · 13%). About two a paper at 9% HARD. Every question is a total divided by a count, or a weighted mean. |
| Time and Work | 2.19 | 2.67 | 1.33 | 46 | 7% | Work Rates (14 · 14% HARD), Man-Days and Man-Hours (17 · 0%), Men, Women and Equivalent Workers (9 · 0%), Pipes and Cisterns (6 · 17%). Falling (2.7 to 1.3 a paper) and cheap at 7% HARD; man-days and mixed gangs have no HARD. |
| Heights and Distances | 1.95 | 2.00 | 1.33 | 41 | 34% | One Line of Sight (6 · 17% HARD), Two Points of Observation (19 · 21%), Observer Above the Ground (9 · 44%), Towers on Plane Figures and Bearings (7 · 71%). About two a paper but 34% HARD, the second-highest in the bank. Towers on plane figures (5 of 7 HARD) is the pool; one line of sight and two points of observation are cheap. |
| Linear Equations | 1.90 | 2.67 | 0.17 | 40 | 20% | Solving Linear Systems (10 · 10% HARD), Word Problems and Applications (13 · 15%), Age Problems (12 · 17%), Integral Solutions and Diophantine Equations (5 · 60%). Fading fast: 2.7 a paper in 2016-2020, one question in the six 2024-2026 papers. The integral-solutions page is the only HARD pool. |
| Simple and Compound Interest | 1.62 | 1.44 | 2.00 | 34 | 15% | Simple Interest (13 · 8% HARD), Compound Interest (13 · 23%), Instalments and Difference of SI and CI (8 · 13%). About two a paper on recent papers at 15% HARD; compound interest holds most of it. |
| Logarithms | 1.57 | 1.89 | 1.33 | 33 | 24% | Logarithm Identities and Change of Base (13 · 15% HARD), Number of Digits and Characteristic (10 · 10%), Solving Logarithmic Equations (10 · 50%). About 1.6 a paper at 24% HARD, and most of that (5 of 8) is the equations page; the laws and digit-counting pages are cheap. |
| Sets | 1.33 | 1.89 | 0.83 | 28 | 14% | Sets and Set Operations (8 · 13% HARD), Venn Diagrams and Inclusion-Exclusion (20 · 15%). Below the playbook line (1.3 a paper, falling). Mostly Venn-diagram counting at 15% HARD. |
| Lines, Angles and Polygons | 1.05 | 1.78 | 0.83 | 22 | 0% | Lines, Angles and Parallels (12 · 0% HARD), Interior and Exterior Angles of Polygons (10 · 0%). Below the playbook line (1.05 a paper, falling from 1.8) and the only chapter with no HARD question at all. |
| Sequence and Series | 0.90 | 0.89 | 1.00 | 19 | 11% | Progressions and Special Sums (10 · 20% HARD), Arithmetic, Geometric and Harmonic Means (9 · 0%). Below the playbook line: 19 questions in 21 papers at 11% HARD. |
| Inequalities | 0.62 | 0.33 | 0.67 | 13 | 15% | Solving Linear and Quadratic Inequalities (5 · 0% HARD), Signs, Powers and Comparisons (8 · 25%). The thinnest chapter: 13 questions in 21 papers, mostly data-sufficiency items on signs. |
“Q / paper” is over all 21 papers; 2016–20 is nine papers and 2024–26 six, so a small move in the recent column is noise. 22 chapters ship a playbook; the four smallest are covered on Strategy.
What’s inside
Strategy
Five chapters carry half the paper. With a one-third penalty, the decision is which questions to attempt — and when a guess is worth making.
Playbooks
22 playbooks — one per chapter at 1.5 questions a paper or more. The subtopic split, where the HARD sits, and whether the chapter cherry-picks.
Formulas
One page of the formulas and results CDS Maths actually tests, by chapter. At 1.2 minutes a question, recall has to be instant.
Trends
Trigonometry and Number System have risen to 13 questions a paper each; Linear Equations and Time and Work have faded. What moved between 2016-2020 and 2024-2026.
Traps
The distractors CDS reuses: the average of two speeds, the percentage on the wrong base, the chord on the other side of the centre, and the blind guess that is worth nothing.
The formula sheet holds 65 formulas across 22 chapters.
Want the teaching, not just the strategy?
This guide tells you what the paper asks and in what order to take it. The notes teach each chapter: foundations, worked examples, self-checks, drills and a mastery check for every subtopic. All 22 playbook chapters have notes, and so do the four small ones.
All CDS Maths notesWhere the numbers come from
Every question from the 21 papers is in the bank, tagged by chapter, subtopic and difficulty. Click any “Drill” link and you see the exact questions a claim is about.
UPSC published no answer key for CDS Maths before 2026 II, so the answers to earlier papers were worked out by our team; 2026 II uses UPSC’s provisional key.
Data snapshot: 29 September 2026.