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NDA Examination Mathematics Syllabus
The Mathematics syllabus for NDA Examination in full: 6 chapters, 34 topics and 82 sub-topics, every one listed below, plus 56 Mathematics flashcards with answers.
Mathematics syllabus: all 6 chapters
Expand any chapter to see its topics and sub-topics.
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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
- Sets and Venn Diagrams
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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
- Trigonometric Ratios and Identities
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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
- Rectangular Cartesian Coordinate System
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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
- Functions, Limits and Continuity
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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
- Vectors and Scalars
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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
- Data Representation
NDA Examination Mathematics flashcards — 56 questions with answers
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).
State the inclusion-exclusion principle for two finite sets A and B.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
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).
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.
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).
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.
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.
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.
What is the nth term of an arithmetic progression with first term a and common difference d?
aₙ = a + (n − 1)d.
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.
What is the sum of an infinite geometric series with first term a and ratio r (|r| < 1)?
S∞ = a / (1 − r).
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.
State the formulas for permutations nPr and combinations nCr.
nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!]. Also nPr = nCr · r!.
How many ways can n distinct objects be arranged in a circle?
(n − 1)! arrangements (or (n−1)!/2 if reflections are considered identical).
State the general (r+1)th term in the binomial expansion of (a + b)^n.
T_(r+1) = nCr · a^(n−r) · b^r.
What is the sum of the binomial coefficients in the expansion of (1 + x)^n at x = 1?
nC0 + nC1 + ... + nCn = 2^n.
State the change of base formula for logarithms.
log_b(a) = log_c(a) / log_c(b) for any valid base c.
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.
How is the determinant of a 2×2 matrix [[a, b],[c, d]] computed?
det = ad − bc.
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.
Show the other 36 Mathematics flashcards
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.
State the Pythagorean trigonometric identities.
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
Give the sine and cosine addition formulas for (A + B).
sin(A+B) = sin A cos B + cos A sin B; cos(A+B) = cos A cos B − sin A sin B.
State the double angle formulas for sin 2θ and cos 2θ.
sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
What is the formula for tan 2θ in terms of tan θ?
tan 2θ = 2 tan θ / (1 − tan²θ).
Give the general solution of the trigonometric equation sin θ = sin α.
θ = nπ + (−1)^n α, where n is an integer.
Give the general solutions of cos θ = cos α and tan θ = tan α.
cos θ = cos α: θ = 2nπ ± α; tan θ = tan α: θ = nπ + α, n ∈ ℤ.
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).
State the sine rule for a triangle with sides a, b, c and opposite angles A, B, C.
a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.
State the cosine rule for side a of a triangle.
a² = b² + c² − 2bc cos A (so cos A = (b² + c² − a²)/(2bc)).
In heights and distances, define the angle of elevation and angle of depression.
Angle of elevation: angle the line of sight makes above the horizontal to an object above the observer. Angle of depression: angle below the horizontal to an object below the observer.
Give the distance formula between two points (x₁, y₁) and (x₂, y₂).
d = √[(x₂ − x₁)² + (y₂ − y₁)²].
Give the section formula for a point dividing the segment joining (x₁,y₁) and (x₂,y₂) in ratio m:n internally.
((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).
What is the slope of a line and the condition for two lines to be perpendicular?
Slope m = tan θ = (y₂ − y₁)/(x₂ − x₁). Two lines are perpendicular if m₁·m₂ = −1; parallel if m₁ = m₂.
Give the perpendicular distance from point (x₁, y₁) to the line ax + by + c = 0.
d = |ax₁ + by₁ + c| / √(a² + b²).
Give the standard equations of a circle, parabola, ellipse, and hyperbola (centered/vertex at origin).
Circle: x² + y² = r²; Parabola: y² = 4ax; Ellipse: x²/a² + y²/b² = 1; Hyperbola: x²/a² − y²/b² = 1.
What is the eccentricity range for an ellipse, parabola, and hyperbola?
Ellipse: 0 < e < 1; Parabola: e = 1; Hyperbola: e > 1; Circle: e = 0.
Give the distance formula between two points in 3D space.
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].
How is the angle between two lines with direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂) found?
cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)].
How is the angle between a line and a plane related to the angle between the line and the plane's normal?
If φ is the angle between the line and the normal, the angle between the line and the plane is θ = 90° − φ, so sin θ = cos φ.
What three conditions must hold for a function f(x) to be continuous at x = a?
(1) f(a) is defined; (2) lim(x→a) f(x) exists; (3) lim(x→a) f(x) = f(a).
State the two standard limits: lim(x→0) sin x / x and lim(x→0) (eˣ − 1)/x.
Both equal 1: lim(x→0) sin x / x = 1 and lim(x→0) (eˣ − 1)/x = 1.
State the product rule and quotient rule for differentiation.
Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².
What does the first derivative test say about increasing/decreasing functions and maxima/minima?
f'(x) > 0 ⇒ increasing; f'(x) < 0 ⇒ decreasing. At a critical point, f' changing + to − gives a local maximum; − to + gives a local minimum.
Give the integrals ∫xⁿ dx (n ≠ −1) and ∫(1/x) dx.
∫xⁿ dx = x^(n+1)/(n+1) + C; ∫(1/x) dx = ln|x| + C.
State the fundamental theorem of calculus for a definite integral.
∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative of f (F' = f).
What is the order and degree of a differential equation?
Order = highest derivative present; Degree = power of the highest-order derivative (after the equation is made polynomial/free of radicals in derivatives).
Define scalar (dot) product of vectors a and b and state when they are perpendicular.
a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. They are perpendicular when a·b = 0.
Define the vector (cross) product of a and b, including its magnitude and direction.
a × b has magnitude |a||b| sin θ and direction perpendicular to both a and b (right-hand rule). It is a vector; a × b = −(b × a). It is zero for parallel vectors.
What is the scalar triple product [a b c] and what does it represent geometrically?
[a b c] = a·(b × c); it equals the volume of the parallelepiped with edges a, b, c. If it is 0, the vectors are coplanar.
Give the formulas for mean, median, and mode (define each measure of central tendency).
Mean = (Σx)/n (arithmetic average); Median = middle value of ordered data; Mode = most frequently occurring value.
Define variance and standard deviation for a data set.
Variance σ² = Σ(xᵢ − x̄)²/n (mean of squared deviations); Standard deviation σ = √(variance).
State the classical (theoretical) definition of probability of an event E.
P(E) = (number of favourable outcomes) / (total number of equally likely outcomes), with 0 ≤ P(E) ≤ 1.
State the addition theorem of probability for events A and B.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0.
State Bayes' theorem and the conditional probability formula.
Conditional: P(A|B) = P(A ∩ B)/P(B). Bayes: P(Aᵢ|B) = [P(Aᵢ)P(B|Aᵢ)] / Σ P(Aⱼ)P(B|Aⱼ).
What does the Karl Pearson correlation coefficient r measure and what is its range?
It measures the strength and direction of a linear relationship between two variables; −1 ≤ r ≤ 1 (−1 perfect negative, +1 perfect positive, 0 no linear correlation).
NDA Examination Mathematics FAQ
What is in the NDA Examination Mathematics syllabus?
NDA Examination Mathematics is split into 6 chapters: Algebra, Trigonometry, Analytic Geometry (2D and 3D), Differential and Integral Calculus, Vector Algebra and Statistics and Probability. Together they contain 34 topics and 82 sub-topics.
How many chapters are in NDA Examination Mathematics?
6 chapters, containing 34 topics and 82 sub-topics. The largest chapter is Algebra, with 8 topics.
Are there flashcards for NDA Examination Mathematics?
Yes — a 56-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.