UPSC CDS (Combined Defence Services) Elementary Mathematics (Paper III, for IMA/INA/AFA) Syllabus

Elementary Mathematics (Paper III, for IMA/INA/AFA) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Elementary Mathematics (Paper III, for IMA/INA/AFA) in UPSC CDS (Combined Defence Services), not a summary of it.

1. 1

   Arithmetic

   7 topics

   • Number System and HCF/LCM
     • Natural, rational and real numbers
     • Divisibility, factors and multiples
   • Fractions, Decimals and Simplification
   • Ratio, Proportion and Variation
   • Percentage, Profit, Loss and Discount
   • Simple and Compound Interest
   • Time, Work, Speed and Distance
   • Averages, Mixtures and Alligation
2. 2

   Algebra

   6 topics

   • Basic Operations and Algebraic Identities
   • Factorisation and HCF/LCM of Polynomials
   • Linear Equations in One and Two Variables
   • Quadratic Equations and Their Roots
   • Simultaneous Equations and Word Problems
   • Set Theory and Venn Diagrams
3. 3

   Trigonometry

   3 topics

   • Trigonometric Ratios and Identities
     • Ratios of standard angles
     • Complementary angle relations
   • Trigonometric Tables and Their Use
   • Heights and Distances
4. 4

   Geometry

   5 topics

   • Lines, Angles and Their Properties
   • Triangles: Congruence and Similarity
   • Quadrilaterals, Parallelograms and Polygons
   • Circles: Chords, Tangents and Angles
   • Loci and Geometric Constructions
5. 5

   Mensuration

   3 topics

   • Perimeter and Area of Plane Figures
   • Surface Area and Volume of Solids
     • Cubes, cuboids and cylinders
     • Cones, spheres and hemispheres
   • Areas of Triangles and Quadrilaterals
6. 6

   Statistics

   3 topics

   • Collection and Tabulation of Data
   • Graphical Representation of Data
     • Bar charts, histograms and pie charts
     • Frequency polygons
   • Measures of Central Tendency: Mean, Median, Mode

Elementary Mathematics (Paper III, for IMA/INA/AFA) flashcards for UPSC CDS (Combined Defence Services)

25 of 50 cards from the Elementary Mathematics (Paper III, for IMA/INA/AFA) deck — real questions with worked answers.

1. State the divisibility rule for 8.

   A number is divisible by 8 if its last three digits form a number divisible by 8 (e.g., 17,256 → 256 ÷ 8 = 32).

2. State the divisibility rule for 11.

   A number is divisible by 11 if the difference between the sum of digits in odd positions and the sum of digits in even positions is 0 or a multiple of 11.

3. What is the relationship between the HCF and LCM of two numbers a and b?

   HCF × LCM = a × b (the product of the two numbers equals the product of their HCF and LCM).

4. How do you find the LCM of fractions?

   LCM of fractions = LCM of numerators ÷ HCF of denominators.

5. How do you find the HCF of fractions?

   HCF of fractions = HCF of numerators ÷ LCM of denominators.

6. Define a prime number and state the smallest prime.

   A prime number is a natural number greater than 1 having exactly two factors: 1 and itself. The smallest prime is 2 (and it is the only even prime).

7. Distinguish between rational and irrational numbers.

   A rational number can be expressed as p/q where p, q are integers and q ≠ 0 (terminating or recurring decimals). An irrational number cannot be so expressed (non-terminating, non-recurring decimals, e.g., √2, π).

8. What is the formula for the sum of the first n natural numbers?

   Sum = n(n + 1) / 2.

9. What is the formula for the sum of the squares of the first n natural numbers?

   Sum = n(n + 1)(2n + 1) / 6.

10. How do you convert a recurring decimal like 0.̅ (0.3 recurring) into a fraction?

    Place the repeating digit(s) over an equal number of 9s. For 0.3 recurring = 3/9 = 1/3; for 0.27 recurring = 27/99 = 3/11.

11. In a fraction, what happens to its value if a positive number is added to both numerator and denominator (numerator < denominator)?

    For a proper fraction (numerator < denominator), adding the same positive number to both increases the value, moving it closer to 1.

12. What is the BODMAS/BIDMAS order of operations?

    Brackets, Orders (powers/roots), Division and Multiplication (left to right), then Addition and Subtraction (left to right).

13. State the formula relating Ratio and Proportion: if a : b :: c : d, what is the key property?

    In a proportion a : b :: c : d, the product of means equals the product of extremes: b × c = a × d (so a/b = c/d).

14. What is the third proportional to a and b?

    The third proportional c satisfies a : b :: b : c, so c = b² / a.

15. What is the mean proportional (geometric mean) between a and b?

    Mean proportional = √(a × b).

16. Define direct and inverse variation.

    Direct variation: y = kx (y increases as x increases, y/x constant). Inverse variation: y = k/x (y decreases as x increases, xy constant).

17. If a sum is divided in the ratio a : b : c, what fraction does each part get?

    Each part is the total multiplied by its share over the sum: a/(a+b+c), b/(a+b+c), c/(a+b+c) of the total respectively.

18. What is the formula to convert a fraction to a percentage and vice versa?

    Fraction to percent: multiply by 100. Percent to fraction: divide by 100. E.g., 3/4 = 75%; 40% = 40/100 = 2/5.

19. How do you calculate Profit% and Loss%?

    Profit% = (Profit / Cost Price) × 100; Loss% = (Loss / Cost Price) × 100. Both are computed on Cost Price.

20. What is the formula for Selling Price given Cost Price and Profit%?

    SP = CP × (100 + Profit%) / 100. For a loss: SP = CP × (100 − Loss%) / 100.

21. Define Marked Price and Discount.

    Marked Price (MP) is the listed/tag price. Discount is the reduction on MP: SP = MP − Discount, and Discount% = (Discount / MP) × 100.

22. What is the formula for successive discounts of x% and y%?

    Net discount% = x + y − (xy/100). The single equivalent discount is always less than the simple sum x + y.

23. A 20% increase followed by a 20% decrease results in what net change?

    A net decrease of 4%. (Using x + y + xy/100 with x = +20, y = −20: 20 − 20 − 400/100 = −4%.)

24. State the Simple Interest formula.

    SI = (P × R × T) / 100, where P = principal, R = rate% per annum, T = time in years.

25. State the Compound Interest formula for amount compounded annually.

    A = P(1 + R/100)^T, and CI = A − P, where A is the final amount.

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