NDA Examination 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 NDA Examination, not a summary of it.

1. 1

   Algebra

   8 topics

   • Sets and Venn Diagrams
     • Set operations, subsets and power sets
     • De Morgan's laws and cardinality
     • Venn diagram word problems
   • Complex Numbers
     • Modulus, argument and conjugate
     • Argand plane and polar form
     • Cube roots of unity
   • Quadratic Equations and Theory of Equations
     • Nature of roots and discriminant
     • Relation between roots and coefficients
     • Formation of equations
   • Sequences and Series
     • Arithmetic progression and AM
     • Geometric progression, GM and infinite GP
     • Harmonic progression and AM-GM-HM inequality
   • Permutations and Combinations
     • Fundamental principle of counting
     • Factorials, nPr and nCr
     • Circular permutations and restricted arrangements
   • Binomial Theorem
     • General term and middle term
     • Properties of binomial coefficients
   • Logarithms
     • Laws of logarithms
     • Change of base and applications
   • Matrices and Determinants
     • Types of matrices and operations
     • Determinant properties and minors/cofactors
     • Inverse of a matrix and Cramer's rule
2. 2

   Trigonometry

   6 topics

   • Trigonometric Ratios and Identities
     • Ratios of standard and allied angles
     • Fundamental and Pythagorean identities
   • Compound and Multiple Angles
     • Sum and difference formulae
     • Double and half angle formulae
     • Transformation of products and sums
   • Trigonometric Equations
     • General solutions of standard equations
   • Inverse Trigonometric Functions
     • Domain, range and principal values
     • Properties and identities
   • Properties of Triangles
     • Sine rule and cosine rule
     • Area of a triangle and projection formulae
   • Heights and Distances
     • Angles of elevation and depression
     • Practical applications
3. 3

   Analytic Geometry (2D and 3D)

   5 topics

   • Rectangular Cartesian Coordinate System
     • Distance and section formula
     • Locus and shift of origin
   • The Straight Line
     • Forms of equation of a line
     • Angle between lines, parallel and perpendicular conditions
     • Distance of a point from a line
   • Conic Sections
     • Circle and its equation
     • Parabola, ellipse and hyperbola in standard form
     • Eccentricity, foci and directrix
   • Three-Dimensional Geometry
     • Coordinates and distance in space
     • Direction cosines and direction ratios
     • Equation of a plane and a line in space
   • Angle Between Lines and Planes
     • Angle between two lines
     • Angle between a line and a plane
4. 4

   Differential and Integral Calculus

   6 topics

   • Functions, Limits and Continuity
     • Types of functions and composition
     • Standard limits and evaluation
     • Continuity and differentiability
   • Differentiation
     • Derivatives of standard functions
     • Product, quotient and chain rule
     • Differentiation of implicit and parametric functions
   • Applications of Derivatives
     • Increasing and decreasing functions
     • Maxima and minima
     • Tangents and normals
   • Indefinite Integration
     • Standard integrals and substitution
     • Integration by parts and partial fractions
   • Definite Integration
     • Properties of definite integrals
     • Area under curves
   • Differential Equations
     • Order and degree
     • Formation and solution by variable separable method
     • Linear differential equations of first order
5. 5

   Vector Algebra

   4 topics

   • Vectors and Scalars
     • Types of vectors and addition
     • Position vectors and components
   • Scalar (Dot) Product
     • Definition and properties
     • Projection and angle between vectors
   • Vector (Cross) Product
     • Definition and geometrical meaning
     • Area of triangle and parallelogram
   • Applications of Vectors
     • Work done by a force
     • Moment of a force
6. 6

   Statistics and Probability

   5 topics

   • Data Representation
     • Frequency distribution
     • Histograms, pie charts and frequency polygons
   • Measures of Central Tendency
     • Mean, median and mode
   • Measures of Dispersion
     • Range and mean deviation
     • Variance and standard deviation
   • Correlation and Regression
     • Correlation coefficient
     • Lines of regression
   • Probability
     • Classical and axiomatic definitions
     • Addition and multiplication theorems
     • Conditional probability and Bayes' theorem
     • Binomial distribution

Mathematics flashcards for NDA Examination

21 of 56 cards from the Mathematics deck — real questions with worked answers.

1. What is the formula for the number of subsets of a set containing n elements?

   2^n subsets (including the empty set and the set itself).

2. State the inclusion-exclusion principle for two finite sets A and B.

   n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

3. State the inclusion-exclusion principle for three finite sets 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. For a complex number z = a + ib, what is its modulus and complex conjugate?

   Modulus |z| = √(a² + b²); conjugate z̄ = a − ib. Also z·z̄ = |z|².

5. State De Moivre's theorem for (cos θ + i sin θ)^n where n is an integer.

   (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ).

6. What are the cube roots of unity and a key property they satisfy?

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

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

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

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

   D > 0: real distinct roots; D = 0: real equal roots; D < 0: complex conjugate roots.

9. What is the nth term of an arithmetic progression with first term a and common difference d?

   aₙ = a + (n − 1)d.

10. Give the formula for the sum of the first n terms of an AP.

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

11. What is the sum of an infinite geometric series with first term a and ratio r (|r| < 1)?

    S∞ = a / (1 − r).

12. Give the formulas for the sum of 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.

13. State the formulas for permutations nPr and combinations nCr.

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

14. How many ways can n distinct objects be arranged in a circle?

    (n − 1)! arrangements (or (n−1)!/2 if reflections are considered identical).

15. State the general (r+1)th term in the binomial expansion of (a + b)^n.

    T_(r+1) = nCr · a^(n−r) · b^r.

16. What is the sum of the binomial coefficients in the expansion of (1 + x)^n at x = 1?

    nC0 + nC1 + ... + nCn = 2^n.

17. State the change of base formula for logarithms.

    log_b(a) = log_c(a) / log_c(b) for any valid base c.

18. Give the product, quotient, and power rules for logarithms.

    log(mn) = log m + log n; log(m/n) = log m − log n; log(m^k) = k·log m.

19. How is the determinant of a 2×2 matrix [[a, b],[c, d]] computed?

    det = ad − bc.

20. What is the formula for the inverse of an invertible matrix A in terms of its adjoint?

    A⁻¹ = adj(A) / det(A), valid when det(A) ≠ 0.

21. What condition on the determinant determines whether a square matrix is singular?

    A matrix is singular (non-invertible) if det(A) = 0; non-singular (invertible) if det(A) ≠ 0.

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