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UPSC NDA & NA Mathematics Syllabus
The Mathematics syllabus for UPSC NDA & NA in full: 7 chapters, 35 topics and 69 sub-topics, every one listed below, plus 55 Mathematics flashcards with answers.
Mathematics syllabus: all 7 chapters
Expand any chapter to see its topics and sub-topics.
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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
- Sets and Set Operations
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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
- Types of Matrices and Operations
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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
- Angles and Measurement
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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
- Rectangular Cartesian Coordinate System
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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
- Functions, Limits and Continuity
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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
- Vectors and Their Representation
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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
- Classification and Representation of Data
UPSC NDA & NA Mathematics flashcards — 55 questions with answers
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.
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).
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).
State De Morgan's laws for set complements.
(A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
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.
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.
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.
What is an equivalence relation?
A relation that is reflexive, symmetric, and transitive simultaneously.
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.
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).
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.
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.
State De Moivre's theorem.
For any integer n, (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ).
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.
State the polar (trigonometric) form of a complex number z.
z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).
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.
State the quadratic formula for the roots of ax² + bx + c = 0.
x = [−b ± √(b² − 4ac)] / (2a).
For ax² + bx + c = 0, give the sum and product of the roots.
Sum of roots = −b/a; product of roots = c/a.
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.
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.
Show the other 35 Mathematics flashcards
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.
State the nth term and sum of n terms of a geometric progression (GP).
nth term: aₙ = arⁿ⁻¹. Sum: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1.
What is the sum to infinity of a GP, and when does it exist?
S∞ = a/(1 − r), which exists only when |r| < 1.
Define the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) of two positive numbers a and b.
AM = (a + b)/2; GM = √(ab); HM = 2ab/(a + b). They satisfy AM ≥ GM ≥ HM.
Give the formulas for the sum of the first n natural numbers, their squares, and their cubes.
Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².
What is the binary number system, and which digits does it use?
A base-2 positional number system using only the digits 0 and 1. Each position represents a power of 2.
Convert the binary number 1011 to decimal.
1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11.
Describe the method to convert a decimal integer to binary.
Repeatedly divide the number by 2, recording the remainders; the binary equivalent is the remainders read from bottom (last) to top (first).
State the binary addition rules.
0+0 = 0; 0+1 = 1; 1+0 = 1; 1+1 = 10 (write 0, carry 1); 1+1+1 = 11 (write 1, carry 1).
What is the difference between a permutation and a combination?
A permutation is an arrangement where order matters; a combination is a selection where order does not matter.
State the formulas for nPr and nCr.
nPr = n!/(n−r)! and nCr = n!/[r!(n−r)!]. Also nPr = nCr × r!.
How many permutations of n objects exist when p, q, … are alike of certain kinds?
n! / (p! · q! · …), dividing by the factorials of the counts of indistinguishable objects.
State two key identities relating combinations.
nCr = nC(n−r), and Pascal's rule: nCr + nC(r−1) = (n+1)Cr.
In how many ways can n distinct objects be arranged in a circle?
(n − 1)! ways (circular permutations), since rotations are considered identical.
State the Binomial Theorem for a positive integral index n.
(a + b)ⁿ = Σ (from r=0 to n) nCr · aⁿ⁻ʳ · bʳ.
Write the general (r+1)th term in the expansion of (a + b)ⁿ.
T(r+1) = nCr · aⁿ⁻ʳ · bʳ.
How many terms are in the expansion of (a + b)ⁿ, and what is the sum of the binomial coefficients?
There are n + 1 terms. The sum of all binomial coefficients (setting a = b = 1) is 2ⁿ.
How do you find the middle term(s) in the binomial expansion of (a + b)ⁿ?
If n is even, there is one middle term: the (n/2 + 1)th term. If n is odd, there are two middle terms: the ((n+1)/2)th and ((n+3)/2)th terms.
Define a square matrix, a diagonal matrix, and a scalar matrix.
Square matrix: equal number of rows and columns. Diagonal matrix: a square matrix with all off-diagonal entries zero. Scalar matrix: a diagonal matrix with all diagonal entries equal.
What is an identity matrix and a null (zero) matrix?
Identity matrix I: a diagonal matrix with all diagonal entries 1 (acts as multiplicative identity). Null matrix: a matrix with every entry 0 (additive identity).
Define symmetric and skew-symmetric matrices.
Symmetric: A = Aᵀ (aᵢⱼ = aⱼᵢ). Skew-symmetric: A = −Aᵀ (aᵢⱼ = −aⱼᵢ), forcing all diagonal entries to be 0.
What condition must hold for matrix multiplication AB to be defined, and what is the order of the product?
The number of columns of A must equal the number of rows of B. If A is m×n and B is n×p, then AB is m×p.
State the transpose properties (AB)ᵀ and (A + B)ᵀ.
(AB)ᵀ = BᵀAᵀ and (A + B)ᵀ = Aᵀ + Bᵀ.
How is the determinant of a 2×2 matrix [[a, b], [c, d]] computed?
det = ad − bc.
State two properties of determinants regarding row interchange and a scalar multiple of a row.
Interchanging two rows (or columns) changes the sign of the determinant. Multiplying one row (or column) by k multiplies the determinant by k.
What is the value of a determinant if two rows (or columns) are identical or proportional?
The determinant equals zero.
State the property det(AB) and det(kA) for an n×n matrix.
det(AB) = det(A)·det(B), and det(kA) = kⁿ·det(A) for an n×n matrix.
Define the cofactor of an element and the adjoint (adjugate) of a matrix.
The cofactor Cᵢⱼ = (−1)^(i+j)·Mᵢⱼ, where Mᵢⱼ is the minor. The adjoint is the transpose of the cofactor matrix: adj(A) = [Cᵢⱼ]ᵀ.
State the formula for the inverse of a matrix A using its adjoint.
A⁻¹ = adj(A) / det(A), valid only when det(A) ≠ 0.
When is a matrix invertible (non-singular), and what is the key identity A·adj(A)?
A is invertible iff det(A) ≠ 0 (non-singular). The identity is A·adj(A) = adj(A)·A = det(A)·I.
State Cramer's Rule for solving a system of linear equations AX = B.
For a non-singular system, xᵢ = det(Aᵢ)/det(A), where Aᵢ is A with its ith column replaced by B. Requires det(A) ≠ 0.
How is a system of linear equations solved using the matrix inverse method?
Write the system as AX = B; if det(A) ≠ 0, the solution is X = A⁻¹B.
In a system AX = B, what do the cases det(A) ≠ 0 and det(A) = 0 indicate about solutions?
If det(A) ≠ 0: a unique solution exists. If det(A) = 0: the system has either no solution (inconsistent) or infinitely many solutions (dependent).
For a homogeneous system AX = 0, when does a non-trivial solution exist?
A non-trivial (non-zero) solution exists if and only if det(A) = 0. If det(A) ≠ 0, only the trivial solution X = 0 exists.
What is the relationship between nCr coefficients and the rows of Pascal's triangle?
The entries in the nth row of Pascal's triangle (starting from row 0) are the binomial coefficients nC0, nC1, …, nCn.
UPSC NDA & NA Mathematics FAQ
What is in the UPSC NDA & NA Mathematics syllabus?
UPSC NDA & NA Mathematics is split into 7 chapters: Algebra, Matrices and Determinants, Trigonometry, Analytical Geometry of Two and Three Dimensions, Differential and Integral Calculus, Vector Algebra and Statistics and Probability. Together they contain 35 topics and 69 sub-topics.
How many chapters are in UPSC NDA & NA Mathematics?
7 chapters, containing 35 topics and 69 sub-topics. The largest chapter is Algebra, with 8 topics.
Are there flashcards for UPSC NDA & NA Mathematics?
Yes — a 55-card Mathematics 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.