In this guide
This sheet is for the last two or three weeks before the exam, when you are no longer learning topics but keeping them fresh. It covers the whole CDS Elementary Mathematics syllabus in tables, one section per area. Read it once a day, and when a line doesn't make sense any more, go back to the full guide for that topic.
A formula sheet only helps if you know where each formula applies and where it fails. So each section adds a short note on the mistake the paper is designed to catch. At the end there are six worked questions and a mixed self-test.
Number system and arithmetic
| Topic | Result |
|---|---|
| Divisibility | 3 and 9: digit sum. 4: last two digits. 8: last three digits. 11: difference of alternate digit sums is 0 or a multiple of 11 |
| Number of factors | For N = pᵃ × qᵇ × rᶜ, the count is (a + 1)(b + 1)(c + 1) |
| Unit digits | Powers repeat in cycles of 4 (2: 2, 4, 8, 6; 3: 3, 9, 7, 1; 7: 7, 9, 3, 1; 8: 8, 4, 2, 6). 0, 1, 5 and 6 never change |
| HCF and LCM | HCF × LCM = product, for two numbers only |
| Fractions | HCF = HCF of numerators ÷ LCM of denominators. LCM = LCM of numerators ÷ HCF of denominators |
Watch out: HCF × LCM = product does not hold for three numbers.
Percentage, profit and interest
| Topic | Result |
|---|---|
| Successive change | a% then b% gives a + b + ab/100 (use negative signs for decreases) |
| Keeping spending fixed | If price rises r%, cut consumption by r/(100 + r) × 100% |
| Profit and loss | Profit % is on cost price. Discount % is on marked price |
| Simple interest | SI = PRT/100 |
| Compound interest | Amount = P(1 + R/100)ⁿ |
| CI − SI, 2 years | P(R/100)² |
| CI − SI, 3 years | P(R/100)²(3 + R/100) |
| Doubling at CI | If a sum doubles in n years, it becomes 2ᵏ times in kn years |
Watch out: a 20% rise followed by a 20% fall is a 4% loss, not zero.
Ratio, averages, work and speed
| Topic | Result |
|---|---|
| Alligation | Cheaper : dearer = (dearer − mean) : (mean − cheaper) |
| Combined average | (n₁a₁ + n₂a₂) ÷ (n₁ + n₂) |
| Work together | A in a days, B in b days: ab/(a + b) days |
| Unit conversion | km/h × 5/18 = m/s |
| Average speed, equal distances | 2ab/(a + b) |
| Trains | Distance = train length + whatever it crosses; relative speed adds if opposite, subtracts if same direction |
| Boats | Still water = (down + up)/2; stream = (down − up)/2 |
Watch out: average speed is never the plain average of the speeds unless the times are equal.
Algebra
| Topic | Result |
|---|---|
| Squares | (a ± b)² = a² ± 2ab + b²; a² − b² = (a + b)(a − b) |
| Cubes | (a + b)³ = a³ + b³ + 3ab(a + b); a³ + b³ = (a + b)(a² − ab + b²) |
| Reciprocal pairs | If x + 1/x = k: x² + 1/x² = k² − 2, and x³ + 1/x³ = k³ − 3k |
| Three variables | a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca); so if a + b + c = 0, a³ + b³ + c³ = 3abc |
| Linear pair a₁x + b₁y = c₁, a₂x + b₂y = c₂ | Unique if a₁/a₂ ≠ b₁/b₂. None if a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Infinitely many if all three ratios are equal |
| Quadratic ax² + bx + c = 0 | Sum of roots −b/a; product c/a; D = b² − 4ac (D > 0 two real, D = 0 equal, D < 0 no real roots) |
| Logarithms | log(mn) = log m + log n; log(m/n) = log m − log n; log mⁿ = n log m; log of a to base b = log a ÷ log b |
Watch out: log(m + n) is not log m + log n.
Trigonometry
| Topic | Result |
|---|---|
| Values | sin: 0, 1/2, 1/√2, √3/2, 1 for 0°, 30°, 45°, 60°, 90°. cos is the same list reversed. tan = sin/cos |
| Identities | sin²θ + cos²θ = 1; sec²θ − tan²θ = 1; cosec²θ − cot²θ = 1 |
| Complementary | sin(90° − θ) = cos θ; tan(90° − θ) = cot θ; sec(90° − θ) = cosec θ |
| Pairs | If sec θ + tan θ = p, then sec θ − tan θ = 1/p |
| Heights | tan for height and distance; sin or cos when a slant length is given |
| Special triangles | 30-60-90: 1 : √3 : 2. 45-45-90: 1 : 1 : √2 |
Watch out: the angle of depression is measured from the horizontal, not from the vertical.
Geometry
| Topic | Result |
|---|---|
| Parallel lines | Alternate and corresponding angles equal; co-interior angles add to 180° |
| Triangle | Angle sum 180°; exterior angle = sum of the two opposite interior angles; any two sides exceed the third |
| Similar triangles | Area ratio = (side ratio)² |
| Bisector theorem | BD : DC = AB : AC |
| Centres | Centroid 2 : 1 on the median; ∠BIC = 90° + A/2; ∠BOC = 2A; ∠BHC = 180° − A |
| Circle angles | Centre = 2 × circumference; semicircle 90°; same segment equal; cyclic quadrilateral opposite angles add to 180° |
| Tangents | Perpendicular to the radius; equal from an external point; PT² = PA × PB |
| Chords | d² + (c/2)² = r²; intersecting chords AP × PB = CP × PD |
| Polygons | Interior sum (n − 2) × 180°; each exterior 360°/n; diagonals n(n − 3)/2 |
Watch out: ∠BOC = 2A needs A to be acute, and the angle-at-centre rule depends on which arc the point lies on.
Mensuration
| Figure | Area or volume | Surface or perimeter |
|---|---|---|
| Square | a², or d²/2 | 4a |
| Equilateral triangle | (√3/4)a² | 3a |
| Any triangle | √(s(s − a)(s − b)(s − c)) | a + b + c |
| Trapezium | ½(a + b)h | sum of sides |
| Circle | πr² | 2πr |
| Sector, angle θ | (θ/360)πr² | 2r + (θ/360)2πr |
| Cylinder | πr²h | curved 2πrh; total 2πr(r + h) |
| Cone | ⅓πr²h | curved πrl, l = √(r² + h²) |
| Sphere | ⁴⁄₃πr³ | 4πr² |
| Hemisphere | ⅔πr³ | curved 2πr²; total 3πr² |
Watch out: scaling lengths by k scales area by k² and volume by k³.
Statistics
| Topic | Result |
|---|---|
| Mean | Σfx ÷ Σf |
| Median, grouped | L + ((N/2 − cf) ÷ f) × h |
| Mode, grouped | L + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h |
| Empirical | Mode = 3 median − 2 mean |
| Pie chart | Angle = (value ÷ total) × 360° |
Watch out: a combined mean is weighted by group sizes.
Six worked questions
Question 1: The price of sugar rises by 25%. By how much must a household cut its consumption to keep spending the same?
- 25/125 × 100 = 20%. The tempting wrong option is 25%.
Question 2: Find CI − SI on ₹10,000 at 10% a year for 2 years.
- 10,000 × (1/10)² = ₹100.
Question 3: A car goes to a town at 40 km/h and returns at 60 km/h. Find the average speed.
- 2 × 40 × 60 ÷ 100 = 48 km/h, not 50.
Question 4: A shop gives successive discounts of 20% and 10%. What single discount is this equal to?
- −20 − 10 + 200/100 = −28, so 28%, not 30%.
Question 5: A sum doubles in 5 years at compound interest. In how many years will it become 8 times?
- 8 = 2³, so 3 × 5 = 15 years, not 20.
Question 6: How many factors does 360 have?
- 360 = 2³ × 3² × 5, so (3 + 1)(2 + 1)(1 + 1) = 24.
Mixed self-test
- If x + 1/x = 3, find x³ + 1/x³.
- If a = 1, b = 2 and c = −3, find a³ + b³ + c³ without cubing.
- Find the sum and product of the roots of 2x² − 7x + 3 = 0.
- Find log₂ 64.
- Find tan 30° × tan 60°.
- Each exterior angle of a regular polygon is 45°. How many diagonals does it have?
- A cone has radius 3 cm and height 4 cm. Find its curved surface area.
- The mean of a data set is 22 and the median is 20. Find the mode.
Answers
- 18. 27 − 9.
- −18. a + b + c = 0, so the sum is 3abc = 3 × 1 × 2 × (−3).
- Sum 7/2, product 3/2.
- 6. 2⁶ = 64.
- 1. Complementary angles.
- 20. It is an octagon, and 8 × 5 ÷ 2 = 20.
- 15π cm². The slant is 5 cm.
- 16. 3 × 20 − 2 × 22.
How to use this sheet
- Read one section a day aloud, and cover the right-hand column to test yourself
- Mark any line you can't explain, and reread the full guide: for example trigonometry or circles
- Take a full mock with the three-pass plan the same week
- On the day before the exam, read only the "Watch out" lines and the self-test
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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