NDA (National Defence Academy) Exam Mathematics (Paper I) Syllabus

Mathematics (Paper I) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (Paper I) in NDA (National Defence Academy) Exam, not a summary of it.

1. 1

   Algebra

   7 topics

   • Sets, Relations and Functions
     • Concept of sets, Venn diagrams, De Morgan laws
     • Cartesian product, relations, equivalence relations
     • Types of functions, domain, range, composition
   • Complex Numbers
     • Modulus, argument, polar form
     • Algebra of complex numbers, conjugate
     • Cube roots of unity and properties
   • Number Systems
     • Binary system of numbers
     • Conversion of decimal to binary and binary to decimal
   • Quadratic Equations and Inequalities
     • Roots, discriminant, nature of roots
     • Relation between roots and coefficients
     • Linear inequalities of one and two variables
   • Sequences and Series
     • Arithmetic, Geometric and Harmonic progressions
     • Sum to n terms and infinite series
   • Permutations, Combinations and Binomial Theorem
     • Fundamental principle of counting, nPr and nCr
     • Binomial theorem and its applications
   • Logarithms
     • Laws of logarithms
     • Applications and change of base
2. 2

   Matrices and Determinants

   4 topics

   • Types and Operations of Matrices
     • Square, diagonal, identity, symmetric matrices
     • Addition, multiplication, transpose
   • Determinants
     • Properties of determinants
     • Evaluation and minors, cofactors
   • Inverse of a Matrix
     • Adjoint and inverse using cofactors
     • Conditions for invertibility
   • Applications to Linear Equations
     • Cramer's rule
     • Solution by matrix method, consistency
3. 3

   Trigonometry

   4 topics

   • Angles and Measurement
     • Degree and radian measure
     • Trigonometric ratios and identities
   • Trigonometric Equations and Compound Angles
     • Sum and difference formulae
     • Multiple and sub-multiple angles
     • General solution of trigonometric equations
   • Inverse Trigonometric Functions
     • Domain, range, principal value
     • Properties and identities
   • Properties of Triangles
     • Sine rule, cosine rule, projection formulae
     • Heights and distances applications
4. 4

   Analytical Geometry of Two and Three Dimensions

   4 topics

   • Coordinate Geometry of Two Dimensions
     • Distance formula, section formula, locus
     • Equation of a straight line in various forms
     • Angle between lines, distance of point from line
   • Conic Sections
     • Standard equation of circle
     • Parabola, ellipse and hyperbola, eccentricity
   • Three Dimensional Geometry
     • Rectangular coordinate system, distance between points
     • Direction cosines and direction ratios
   • Plane and Line in Space
     • Equation of a plane and a line in space
     • Angle between two lines and two planes
     • Equation of a sphere
5. 5

   Differential and Integral Calculus

   6 topics

   • Limits and Continuity
     • Concept of limit, standard limits
     • Continuity of functions, examples
   • Differentiation
     • Derivative of a function, geometrical meaning
     • Product, quotient, chain rule
     • Derivatives of composite and implicit functions
   • Applications of Derivatives
     • Increasing and decreasing functions
     • Maxima and minima
     • Tangents and normals
   • Integration
     • Integration as inverse of differentiation
     • Substitution and by parts methods
     • Integration by partial fractions
   • Definite Integrals and Applications
     • Properties of definite integrals
     • Area under curves
   • Differential Equations
     • Order and degree of a differential equation
     • Variable separable method
     • General and particular solutions
6. 6

   Vector Algebra, Statistics and Probability

   3 topics

   • Vectors
     • Magnitude, direction, types of vectors
     • Addition, scalar (dot) product, vector (cross) product
     • Applications to work, moment and geometry
   • Statistics
     • Frequency distribution, cumulative frequency
     • Measures of central tendency: mean, median, mode
     • Variance and standard deviation, correlation, regression
   • Probability
     • Random experiment, events, sample space
     • Conditional probability, Bayes theorem
     • Binomial distribution, expectation

Mathematics (Paper I) flashcards for NDA (National Defence Academy) Exam

22 of 53 cards from the Mathematics (Paper I) deck — real questions with worked answers.

1. What is the formula for the number of subsets and proper subsets of a set with n elements?

   Total subsets = 2^n; proper subsets = 2^n − 1 (excluding the set itself); number of non-empty proper subsets = 2^n − 2.

2. State De Morgan's laws for two sets A and B.

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

3. For finite sets, what is 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(C∩A) + n(A∩B∩C).

4. Define a function as a special type of relation, and state the condition for a function to be a bijection.

   A function from A to B is a relation in which every element of A has exactly one image in B. It is a bijection if it is both one-one (injective) and onto (surjective).

5. What is the value of i^n for the four residue cases of n mod 4?

   i^0=1, i^1=i, i^2=−1, i^3=−i; in general i^n depends only on n mod 4 (n≡0→1, 1→i, 2→−1, 3→−i).

6. For a complex number z = a + bi, state its modulus and conjugate, and the relation between them.

   |z| = √(a²+b²), conjugate z̄ = a − bi, and z·z̄ = |z|² = a²+b².

7. State the polar (Euler) form of a complex number and De Moivre's theorem.

   z = r(cosθ + i sinθ) = r e^{iθ}, where r=|z|, θ=arg z. De Moivre: (cosθ + i sinθ)^n = cos nθ + i sin nθ.

8. What are the cube roots of unity and their key properties?

   1, ω, ω² where ω = (−1 + i√3)/2. Properties: 1 + ω + ω² = 0 and ω³ = 1.

9. Classify the real number system: how do rational and irrational numbers differ in decimal form?

   Rationals (p/q, q≠0) have terminating or recurring (repeating) decimals; irrationals have non-terminating, non-repeating decimals (e.g., √2, π).

10. 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.

11. For the quadratic ax² + bx + c = 0, state the sum and product of the roots.

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

12. What does the discriminant D = b² − 4ac tell you about the roots of a quadratic equation?

    D>0: two distinct real roots; D=0: two equal real roots; D<0: two complex conjugate roots. (A perfect-square D>0 with rational coefficients gives rational roots.)

13. For a quadratic expression ax² + bx + c with a>0, when is it positive for all real x?

    When a>0 and the discriminant b² − 4ac < 0 (no real roots, so the parabola stays entirely above the x-axis).

14. State the nth term and sum of the first 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.

15. State the nth term and sum formulas for a geometric progression (GP), including the infinite sum.

    aₙ = a·r^{n−1}; Sₙ = a(r^n − 1)/(r − 1), r≠1; infinite sum S∞ = a/(1 − r) for |r| < 1.

16. State the relationship between the arithmetic mean (AM) and geometric mean (GM) of two positive numbers.

    AM ≥ GM, i.e., (a+b)/2 ≥ √(ab), with equality if and only if a = b.

17. 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.

18. State the formulas for permutations (nPr) and combinations (nCr).

    nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!]; relation: nPr = nCr · r!.

19. State the general term in the binomial expansion of (x + y)^n.

    The (r+1)th term: T_{r+1} = nCr · x^{n−r} · y^r, for r = 0,1,...,n.

20. What is the change-of-base formula for logarithms?

    log_b a = (log_c a)/(log_c b) for any valid base c; in particular log_b a = (ln a)/(ln b) = (log a)/(log b).

21. State the three core logarithm laws.

    log(mn) = log m + log n; log(m/n) = log m − log n; log(m^p) = p·log m. Also log_b b = 1 and log_b 1 = 0.

22. Define symmetric, skew-symmetric, and identity matrices.

    Symmetric: Aᵀ = A. Skew-symmetric: Aᵀ = −A (diagonal entries all zero). Identity (I): square matrix with 1s on the main diagonal and 0s elsewhere.

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