UPSC NDA & NA Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in UPSC NDA & NA, not a summary of it.

1. 1

   Algebra

   8 topics

   • Sets and Set Operations
     • Concept of a set, subsets and power set
     • Union, intersection, complement and Venn diagrams
     • De Morgan's laws and cardinality of finite sets
   • Relations and Functions
     • Cartesian product and types of relations
     • Functions: one-one, onto and composition
   • Complex Numbers
     • Modulus, argument and conjugate
     • Cube roots of unity and Argand diagram
   • Quadratic Equations and Theory of Equations
     • Nature of roots and discriminant
     • Relation between roots and coefficients
   • Sequences and Series
     • Arithmetic, geometric and harmonic progressions
     • Sum to n terms and infinite geometric series
   • Binary Number System
     • Conversion between binary and decimal systems
     • Binary arithmetic operations
   • Permutations and Combinations
   • Binomial Theorem for Positive Integral Index
2. 2

   Matrices and Determinants

   4 topics

   • Types of Matrices and Operations
     • Addition, scalar multiplication and matrix product
     • Transpose, symmetric and skew-symmetric matrices
   • Determinants and Their Properties
     • Evaluation of second and third order determinants
     • Properties used to simplify determinants
   • Adjoint and Inverse of a Matrix
   • Solution of Linear Equations
     • Cramer's rule
     • Matrix inversion method
3. 3

   Trigonometry

   6 topics

   • Angles and Measurement
     • Degree and radian measure
     • Relation between arc, radius and angle
   • Trigonometric Ratios and Identities
     • Ratios of allied and compound angles
     • Multiple and sub-multiple angle formulae
   • Inverse Trigonometric Functions
     • Principal value branches
     • Properties and relations
   • Properties of Triangles
     • Sine and cosine rules
     • Area of a triangle
   • Heights and Distances
   • Trigonometric Equations
4. 4

   Analytical Geometry of Two and Three Dimensions

   4 topics

   • Rectangular Cartesian Coordinate System
     • Distance and section formulae
     • Slope and angle between lines
   • The Straight Line
     • Various forms of equation of a line
     • Distance of a point from a line
   • Conic Sections
     • Standard equation of a circle
     • Parabola, ellipse and hyperbola in standard form
     • Eccentricity, foci and directrix
   • Three Dimensional Geometry
     • Coordinates and distance between points in space
     • Direction cosines and direction ratios
     • Equation of a plane and a line in space
     • Angle between two lines and two planes
5. 5

   Differential and Integral Calculus

   6 topics

   • Functions, Limits and Continuity
     • Concept of limit and standard limits
     • Continuity and differentiability of functions
   • Differentiation
     • Derivatives of standard functions
     • Product, quotient and chain rules
     • Differentiation of implicit and composite functions
   • Applications of Derivatives
     • Increasing and decreasing functions
     • Maxima and minima
     • Tangents and normals
   • Indefinite Integration
     • Integration by substitution and by parts
     • Integration by partial fractions
   • Definite Integration and Area
     • Fundamental theorem of calculus
     • Area under simple curves
   • Differential Equations
     • Order and degree of a differential equation
     • Solution by variable separable method
6. 6

   Vector Algebra

   3 topics

   • Vectors and Their Representation
     • Magnitude, direction and types of vectors
     • Position vector and components
   • Operations on Vectors
     • Addition and scalar multiplication
     • Scalar (dot) product and its applications
     • Vector (cross) product and its applications
   • Applications of Vectors
     • Work done by a force
     • Area of triangle and parallelogram
7. 7

   Statistics and Probability

   4 topics

   • Classification and Representation of Data
     • Frequency distribution
     • Histograms, bar charts and pie charts
   • Measures of Central Tendency
     • Mean, median and mode
     • Cumulative frequency and ogives
   • Measures of Dispersion
     • Range, mean deviation and variance
     • Standard deviation
   • Probability
     • Random experiments, events and sample space
     • Addition and multiplication theorems
     • Conditional probability and Bayes' theorem
     • Binomial distribution

Mathematics flashcards for UPSC NDA & NA

21 of 55 cards from the Mathematics deck — real questions with worked answers.

1. What is the power set of a set A, and what is its cardinality if A has n elements?

   The power set P(A) is the set of all subsets of A (including the empty set and A itself). If A has n elements, then |P(A)| = 2^n.

2. State the formula for the number of elements in the union of two finite sets A and B.

   n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

3. State the inclusion-exclusion formula for n(A ∪ B ∪ C).

   n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C).

4. State De Morgan's laws for set complements.

   (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.

5. Define the Cartesian product A × B and give its cardinality.

   A × B = {(a, b) : a ∈ A, b ∈ B}, the set of all ordered pairs. If |A| = m and |B| = n, then |A × B| = mn.

6. What is the difference between a relation and a function from set A to set B?

   A relation is any subset of A × B. A function additionally requires that every element of A is related to exactly one element of B.

7. Define reflexive, symmetric, and transitive relations.

   Reflexive: (a,a) ∈ R for all a. Symmetric: if (a,b) ∈ R then (b,a) ∈ R. Transitive: if (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R.

8. What is an equivalence relation?

   A relation that is reflexive, symmetric, and transitive simultaneously.

9. Distinguish between one-one (injective), onto (surjective), and bijective functions.

   Injective: distinct inputs give distinct outputs. Surjective: every element of the codomain is an image of some element. Bijective: both injective and surjective.

10. How many functions and how many one-one functions exist from a set of m elements to a set of n elements?

    Total functions = n^m. One-one functions = n!/(n−m)! = nP m (defined only when n ≥ m).

11. Define the modulus and argument of a complex number z = a + bi.

    Modulus |z| = √(a² + b²); argument θ = arg(z) = tan⁻¹(b/a), the angle the vector makes with the positive real axis.

12. What is the conjugate of z = a + bi, and what is z·z̄?

    The conjugate is z̄ = a − bi. Their product z·z̄ = a² + b² = |z|².

13. State De Moivre's theorem.

    For any integer n, (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ).

14. What are the cube roots of unity, and what is their sum?

    The cube roots of unity are 1, ω, ω², where ω = (−1 + i√3)/2. Their sum is 1 + ω + ω² = 0, and ω³ = 1.

15. State the polar (trigonometric) form of a complex number z.

    z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).

16. What is the discriminant of ax² + bx + c = 0, and what does it indicate?

    D = b² − 4ac. If D > 0: two distinct real roots; D = 0: one repeated real root; D < 0: two complex conjugate roots.

17. State the quadratic formula for the roots of ax² + bx + c = 0.

    x = [−b ± √(b² − 4ac)] / (2a).

18. For ax² + bx + c = 0, give the sum and product of the roots.

    Sum of roots = −b/a; product of roots = c/a.

19. How do you form a quadratic equation given its roots α and β?

    x² − (α + β)x + αβ = 0, i.e., x² − (sum)x + (product) = 0.

20. For a polynomial equation of degree n, what does the Fundamental Theorem of Algebra state, and what is the sum of all roots of aₙxⁿ + … + a₀ = 0?

    Every degree-n polynomial has exactly n roots (counting multiplicity) in the complex numbers. Sum of all roots = −aₙ₋₁/aₙ.

21. State the nth term and sum of n terms of an arithmetic progression (AP).

    nth term: aₙ = a + (n−1)d. Sum: Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l), where l is the last term.

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