In this guide
The Elementary Mathematics paper has 100 questions for 100 marks in 2 hours, with one-third of a mark deducted for each wrong answer. IMA, INA and AFA candidates sit it; OTA-only candidates do not.
For many graduates, this is the paper that decides whether they clear the written cut-off. The concepts are roughly Class 10. What makes it hard is the clock: 72 seconds a question, with no calculator, and a penalty for every careless slip. A candidate who knows the methods but has not practised under time can run out of minutes well before the last question. A candidate who has practised recognises most question types at a glance and spends the saved time on the hard ones.
This guide gives you the map of the paper, an order of study, a daily routine, and a few sample questions that show what "fast and accurate" looks like.
Starting again after years away
Many CDS candidates last did school maths three or more years ago. That is normal, and it is not a handicap for long. Class 9 and 10 concepts come back quickly once you work through them in order. Start with a Class 10 NCERT textbook for any topic that feels unfamiliar, then move straight to UPSC's previous-year questions on that topic. The textbook gives you the idea; the previous papers show you how UPSC asks it.
What the syllabus covers
The notification lists the syllabus under seven heads. In plain terms:
| Area | What it includes | Relative weight in recent papers |
|---|---|---|
| Arithmetic | Number system, percentage, profit and loss, interest, ratio, time and work, time and distance, variation | Large |
| Elementary number theory | Divisibility, primes, HCF and LCM, logarithms | Moderate |
| Algebra | Identities, factors, remainder theorem, linear and quadratic equations, inequations, sets, indices | Large |
| Trigonometry | Ratios, standard values, identities, heights and distances | Moderate |
| Geometry | Lines and angles, triangles, congruence, similarity, quadrilaterals, circles, tangents | Large |
| Mensuration | Areas of plane figures; surface area and volume of cuboid, cylinder, cone, sphere | Moderate |
| Statistics | Tables, histograms, bar and pie charts, mean, median, mode | Small to moderate |
A 12-week order of study
The order matters because topics build on each other. Percentage underpins profit and interest; identities underpin simplification and trigonometry.
| Weeks | Topics | Why here |
|---|---|---|
| 1–2 | Number system, HCF and LCM, fractions and roots, simplification | The arithmetic base for everything else |
| 3–4 | Percentage, profit and loss, interest, ratio, averages and mixtures | The largest block of word problems |
| 5 | Time and work, time and distance | Uses ratio and percentage |
| 6–7 | Identities, linear and quadratic equations, logarithms, sets | Algebra shows up inside every other area |
| 8 | Trigonometry, heights and distances | Needs algebraic ease |
| 9–10 | Lines, triangles, quadrilaterals, circles | Needs time and diagrams; do it with fresh energy |
| 11 | Mensuration, statistics | Largely formula-based once geometry is secure |
| 12 | Mixed previous papers, formula sheet revision | Converts topics into speed |
From week 4 onward, take one full maths paper every week, even while you are still learning later topics. Leave the questions you have not studied yet and note where they came from.
A daily routine
| Time | Activity |
|---|---|
| 45 min | Learn or revise one topic; write any new formula on your formula sheet |
| 45 min | 25 to 30 previous-year CDS questions on that topic, timed |
| 20 min | Mixed revision: 10 questions from earlier topics |
| 10 min | Review mistakes and note the cause of each |
The mixed revision block is easy to skip and costly to skip. Without it, percentage is sharp in week 3 and blunt by week 10.
Shortcuts: only after the method
Shortcuts save time only if you know why they work. The rule "successive changes of a% and b% give a + b + ab/100" is quick, but it comes from multiplying the two factors (1 + a/100)(1 + b/100). If you understand that, you will also know it works for decreases with negative values, and you will not misapply it to a question about addition. A shortcut used on the wrong question type is a fast route to a negative mark.
Sample questions and fast methods
These show the kind of thinking that saves time across the paper.
Example 1 (interest): A sum doubles itself at simple interest in 8 years. What is the rate?
- Doubling means the interest equals the principal: 100% in 8 years.
- Rate = 100 ÷ 8 = 12.5% a year. No formula needed.
Example 2 (algebra): If x + 1/x = 3, find x² + 1/x².
- Square both sides: x² + 2 + 1/x² = 9.
- So x² + 1/x² = 7. Never try to find x itself.
Example 3 (trigonometry): If sin θ = 3/5 and θ is acute, find tan θ.
- Think of a right triangle with opposite side 3 and hypotenuse 5. The third side is 4 (the 3-4-5 triple).
- tan θ = opposite ÷ adjacent = 3/4.
Example 4 (geometry): Each interior angle of a regular polygon is 150°. How many sides does it have?
- Work with the exterior angle: 180° − 150° = 30°.
- Exterior angles of any polygon add up to 360°, so sides = 360 ÷ 30 = 12.
Example 5 (statistics): The mean of five numbers is 20. When one number is removed, the mean of the rest is 18. Which number was removed?
- Total of five = 5 × 20 = 100. Total of four = 4 × 18 = 72.
- Removed number = 100 − 72 = 28.
Notice the pattern: in each case the fast route is to work with totals, complements or known triples rather than to solve from scratch.
Practice set
- Convert 72 km/h into metres per second.
- If x − 1/x = 2, find x² + 1/x².
- Each exterior angle of a regular polygon is 24°. How many sides does it have?
- Find the simple interest on ₹2,000 at 5% a year for 3 years.
- If tan θ = 1 and θ is acute, find sin²θ.
- Find the average of the first 10 natural numbers.
- Find the area of an equilateral triangle of side 6 cm.
- If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
Answers
- 20 m/s. Multiply by 5/18: 72 × 5 ÷ 18 = 20.
- 6. Square: x² − 2 + 1/x² = 4, so x² + 1/x² = 6.
- 15. 360 ÷ 24 = 15.
- ₹300. 2,000 × 5 × 3 ÷ 100 = 300.
- 1/2. θ = 45°, sin 45° = 1/√2, and its square is 1/2.
- 5.5. The sum is 10 × 11 ÷ 2 = 55; 55 ÷ 10 = 5.5.
- 9√3 cm². (√3/4) × 6² = (√3/4) × 36 = 9√3.
- 8 : 12 : 15. Make B equal: A : B = 8 : 12 and B : C = 12 : 15.
Exam-day tactics
- First pass (about 60 minutes): every question you can solve in about a minute. This is where most of your marks come from.
- Second pass (about 40 minutes): the longer questions you know how to solve.
- Third pass (remaining time): questions where you can eliminate two options, often by substituting values or checking units.
- Leave the rest. A blank costs nothing; a blind guess gains nothing on average.
What to do next
- Download the last five CDS maths papers and count questions per area
- Start weeks 1–2 with the number system guide
- Begin a one-page formula sheet today and add to it as you go
- Book a weekly slot for a full timed maths paper from week 4
- Read the maths speed guide once you have covered arithmetic
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Union Public Service Commission website .
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