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

The Elementary Mathematics (Paper III, for IMA/INA/AFA) syllabus for UPSC CDS (Combined Defence Services) in full: 6 chapters, 27 topics and 8 sub-topics, every one listed below, plus 50 Elementary Mathematics (Paper III, for IMA/INA/AFA) flashcards with answers.

6Chapters
27Topics
8Sub-topics
50Flashcards

Elementary Mathematics (Paper III, for IMA/INA/AFA) syllabus: all 6 chapters

Expand any chapter to see its topics and sub-topics.

  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. 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. Trigonometry

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

  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. 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. 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

UPSC CDS (Combined Defence Services) Elementary Mathematics (Paper III, for IMA/INA/AFA) flashcards — 50 questions with 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.

Show the other 30 Elementary Mathematics (Paper III, for IMA/INA/AFA) flashcards

  1. 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.

  2. 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.

  3. 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%.)

  4. State the Simple Interest formula.

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

  5. 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.

  6. How is CI calculated when interest is compounded half-yearly?

    A = P(1 + (R/2)/100)^(2T); rate is halved and time (number of periods) is doubled.

  7. For 2 years, what is the difference between CI and SI on principal P at rate R%?

    Difference = P × (R/100)². (For 1 year CI and SI are equal.)

  8. At rate R%, in how many years does a sum double under simple interest?

    T = 100 / R years (since the interest must equal the principal: P = P·R·T/100).

  9. State the basic relationship between Speed, Distance and Time.

    Speed = Distance / Time; Distance = Speed × Time; Time = Distance / Speed.

  10. How do you convert km/hr to m/s and vice versa?

    km/hr to m/s: multiply by 5/18. m/s to km/hr: multiply by 18/5.

  11. What is the relative speed for two objects moving in the same direction vs opposite directions?

    Same direction: relative speed = difference of speeds (|a − b|). Opposite directions: relative speed = sum of speeds (a + b).

  12. A train of length L crossing a pole/man covers what distance, and crossing a platform of length P?

    Crossing a pole/man: distance = L (train's own length). Crossing a platform: distance = L + P (train length plus platform length).

  13. If A does a work in x days and B in y days, how long do they take working together?

    Together they take xy/(x + y) days; their combined one-day work = 1/x + 1/y.

  14. State the average (arithmetic mean) formula.

    Average = (Sum of all observations) / (Number of observations).

  15. What is the average speed for a round trip at speeds x and y over equal distances?

    Average speed = 2xy / (x + y) (the harmonic mean), NOT the simple average of x and y.

  16. State the Rule of Alligation.

    (Quantity of cheaper)/(Quantity of dearer) = (Price of dearer − Mean price)/(Mean price − Price of cheaper).

  17. If the average of n numbers is A, what is their total sum, and what happens if each is increased by k?

    Sum = n × A. If each number increases by k, the new average becomes A + k.

  18. State the identity for (a + b)².

    (a + b)² = a² + 2ab + b².

  19. State the identity for (a − b)² and the value of (a+b)² − (a−b)².

    (a − b)² = a² − 2ab + b²; and (a + b)² − (a − b)² = 4ab.

  20. State the difference of squares identity.

    a² − b² = (a + b)(a − b).

  21. State the identities for (a + b)³ and (a − b)³.

    (a + b)³ = a³ + b³ + 3ab(a + b); (a − b)³ = a³ − b³ − 3ab(a − b).

  22. State the factorisation identities for a³ + b³ and a³ − b³.

    a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).

  23. State the identity for a³ + b³ + c³ − 3abc.

    a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). If a + b + c = 0, then a³ + b³ + c³ = 3abc.

  24. For polynomials, how are HCF and LCM related to their product?

    For two polynomials, HCF × LCM = product of the two polynomials (analogous to numbers).

  25. How do you find the HCF of two polynomials?

    Factorise each polynomial completely; the HCF is the product of all common factors taken with the lowest powers.

  26. How many solutions does a linear equation in one variable (ax + b = 0) have, and what is the solution?

    Exactly one solution: x = −b/a (provided a ≠ 0).

  27. For a pair of linear equations a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, what condition gives a unique solution?

    Unique solution when a₁/a₂ ≠ b₁/b₂ (lines intersect at one point — consistent and independent).

  28. For two linear equations, what ratio condition means no solution, and what means infinitely many?

    No solution (parallel lines): a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Infinitely many (coincident lines): a₁/a₂ = b₁/b₂ = c₁/c₂.

  29. State the quadratic formula for ax² + bx + c = 0.

    x = [−b ± √(b² − 4ac)] / (2a).

  30. What is the discriminant of a quadratic, and what do its sign cases indicate about the roots? Also give the sum and product of roots.

    Discriminant D = b² − 4ac. D > 0: two distinct real roots; D = 0: two equal real roots; D < 0: no real (complex) roots. Sum of roots = −b/a, Product of roots = c/a.

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

What is in the UPSC CDS (Combined Defence Services) Elementary Mathematics (Paper III, for IMA/INA/AFA) syllabus?

UPSC CDS (Combined Defence Services) Elementary Mathematics (Paper III, for IMA/INA/AFA) is split into 6 chapters: Arithmetic, Algebra, Trigonometry, Geometry, Mensuration and Statistics. Together they contain 27 topics and 8 sub-topics.

How many chapters are in UPSC CDS (Combined Defence Services) Elementary Mathematics (Paper III, for IMA/INA/AFA)?

6 chapters, containing 27 topics and 8 sub-topics. The largest chapter is Arithmetic, with 7 topics.

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

Yes — a 50-card Elementary Mathematics (Paper III, for IMA/INA/AFA) deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.