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

The Mathematics (Paper I) syllabus for NDA (National Defence Academy) Exam in full: 6 chapters, 28 topics and 69 sub-topics, every one listed below, plus 53 Mathematics (Paper I) flashcards with answers.

6Chapters
28Topics
69Sub-topics
53Flashcards

Mathematics (Paper I) syllabus: all 6 chapters

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

  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. 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. 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. 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. 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. 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

NDA (National Defence Academy) Exam Mathematics (Paper I) flashcards — 53 questions with 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).

Show the other 33 Mathematics (Paper I) flashcards

  1. 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.

  2. 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.

  3. For matrix multiplication, when is the product AB defined, and what is the order of the result?

    AB is defined when the number of columns of A equals the number of rows of B; if A is m×n and B is n×p, then AB is m×p. Matrix multiplication is generally not commutative.

  4. State the property of the determinant of a product of two square matrices, and the effect of swapping two rows.

    det(AB) = det(A)·det(B). Swapping any two rows (or columns) changes the sign of the determinant.

  5. How does the determinant change when a matrix is multiplied by a scalar, and what is det(Aᵀ)?

    For an n×n matrix, det(kA) = kⁿ·det(A); also det(Aᵀ) = det(A).

  6. State the formula for the inverse of a matrix using the adjoint.

    A⁻¹ = (1/det A)·adj(A), valid when det A ≠ 0 (A is non-singular/invertible). adj(A) is the transpose of the cofactor matrix.

  7. Using Cramer's rule, how do you solve a system of linear equations, and what condition guarantees a unique solution?

    For AX = B, xᵢ = Dᵢ/D where D = det(A) and Dᵢ is D with the i-th column replaced by B. A unique solution exists iff D ≠ 0.

  8. In a system of linear equations, what do the cases D=0 with all Dᵢ=0, versus D=0 with some Dᵢ≠0, indicate?

    D=0 and all Dᵢ=0: infinitely many solutions (consistent, dependent). D=0 and at least one Dᵢ≠0: no solution (inconsistent).

  9. Convert between degrees and radians, and give the radian measure of an arc.

    π radians = 180°, so 1 radian = 180/π ≈ 57.3°. Arc length θ (radians) = arc/radius, i.e., θ = l/r, and l = rθ.

  10. State the three Pythagorean trigonometric identities.

    sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

  11. State the compound angle formulas for sin(A±B) and cos(A±B).

    sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB.

  12. State the general solutions of sinθ = 0, cosθ = 0, and tanθ = 0.

    sinθ=0 → θ = nπ; cosθ=0 → θ = (2n+1)π/2; tanθ=0 → θ = nπ, where n is any integer.

  13. State the general solution of sinθ = sinα and cosθ = cosα.

    sinθ = sinα → θ = nπ + (−1)ⁿα; cosθ = cosα → θ = 2nπ ± α, where n is any integer.

  14. State the principal value ranges of sin⁻¹x, cos⁻¹x, and tan⁻¹x.

    sin⁻¹x ∈ [−π/2, π/2]; cos⁻¹x ∈ [0, π]; tan⁻¹x ∈ (−π/2, π/2).

  15. State the sine rule and cosine rule for a triangle.

    Sine rule: a/sinA = b/sinB = c/sinC = 2R. Cosine rule: a² = b² + c² − 2bc·cosA (and cyclic).

  16. Give two formulas for the area of a triangle in terms of its sides/angles.

    Area = ½·ab·sinC (two sides and included angle); and Heron's formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2.

  17. State the distance formula and section formula in 2D coordinate geometry.

    Distance between (x₁,y₁) and (x₂,y₂) = √[(x₂−x₁)² + (y₂−y₁)²]. Section formula (ratio m:n internally): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

  18. State the various forms of the equation of a straight line: slope-intercept, point-slope, and the angle between two lines.

    Slope-intercept: y = mx + c; point-slope: y − y₁ = m(x − x₁). Angle θ between lines of slopes m₁,m₂: tanθ = |(m₁−m₂)/(1+m₁m₂)|; parallel if m₁=m₂, perpendicular if m₁m₂=−1.

  19. Give the standard equations and eccentricities of the parabola, ellipse, and hyperbola.

    Parabola: y²=4ax, e=1. Ellipse: x²/a²+y²/b²=1, e<1 (e=√(1−b²/a²)). Hyperbola: x²/a²−y²/b²=1, e>1 (e=√(1+b²/a²)). Circle is a special ellipse with e=0.

  20. State the distance formula between two points in three-dimensional space.

    Distance between (x₁,y₁,z₁) and (x₂,y₂,z₂) = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].

  21. What are direction cosines of a line, and what relation do they satisfy?

    Direction cosines (l,m,n) are the cosines of the angles the line makes with the x, y, z axes; they satisfy l² + m² + n² = 1.

  22. State the equation of a plane in normal form and the angle between a line and a plane.

    Plane: lx + my + nz = p (l,m,n direction cosines of normal, p = perpendicular distance from origin). The angle φ between a line and a plane satisfies sinφ = |line direction · plane normal| / (|d||n|).

  23. What are the necessary conditions for a function f(x) to be continuous at x = a?

    f(a) is defined, lim_{x→a} f(x) exists (left limit = right limit), and lim_{x→a} f(x) = f(a).

  24. State the standard limit results: lim_{x→0} (sin x)/x and lim_{x→0} (e^x − 1)/x.

    lim_{x→0} (sin x)/x = 1 and lim_{x→0} (e^x − 1)/x = 1 (with x in radians).

  25. State the product rule and quotient rule of differentiation.

    Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

  26. State the chain rule and give d/dx of sin x, ln x, and aˣ.

    Chain rule: dy/dx = (dy/du)(du/dx). d/dx(sin x)=cos x; d/dx(ln x)=1/x; d/dx(aˣ)=aˣ ln a.

  27. How are increasing/decreasing functions and points of local maxima/minima determined using derivatives?

    f'(x)>0 → increasing, f'(x)<0 → decreasing. At a critical point (f'(x)=0): if f''(x)<0 it is a local maximum, if f''(x)>0 it is a local minimum (second-derivative test).

  28. Give the basic integration results for ∫xⁿ dx, ∫(1/x) dx, and ∫eˣ dx.

    ∫xⁿ dx = x^{n+1}/(n+1) + C (n≠−1); ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C.

  29. State the Fundamental Theorem of Calculus for definite integrals.

    If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

  30. How do you find the order and degree of a differential equation, and how is a variable-separable equation solved?

    Order = highest order derivative present; degree = power of the highest-order derivative (when free of radicals/fractions). Variable-separable: write as f(y)dy = g(x)dx, then integrate both sides.

  31. State the scalar (dot) product and vector (cross) product of two vectors in terms of the angle between them.

    a·b = |a||b|cosθ (scalar; zero if perpendicular). a×b = |a||b|sinθ·n̂ (vector perpendicular to both; magnitude = area of parallelogram; zero if parallel).

  32. In statistics, define mean, median, and mode, and state the empirical relation between them.

    Mean = sum/number of observations; median = middle value when ordered; mode = most frequent value. Empirical relation: Mode = 3·Median − 2·Mean.

  33. State the addition theorem of probability and the formula for conditional probability.

    P(A∪B) = P(A)+P(B) − P(A∩B) (for mutually exclusive events, = P(A)+P(B)). Conditional: P(A|B) = P(A∩B)/P(B), P(B)≠0; for independent events P(A∩B)=P(A)·P(B).

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

What is in the NDA (National Defence Academy) Exam Mathematics (Paper I) syllabus?

NDA (National Defence Academy) Exam Mathematics (Paper I) is split into 6 chapters: Algebra, Matrices and Determinants, Trigonometry, Analytical Geometry of Two and Three Dimensions, Differential and Integral Calculus and Vector Algebra, Statistics and Probability. Together they contain 28 topics and 69 sub-topics.

How many chapters are in NDA (National Defence Academy) Exam Mathematics (Paper I)?

6 chapters, containing 28 topics and 69 sub-topics. The largest chapter is Algebra, with 7 topics.

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

Yes — a 53-card Mathematics (Paper I) 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.