UPSC NDA 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 UPSC NDA, not a summary of it.

1. 1

   Algebra

   11 topics

   • Concept of Sets
   • Operations on Sets
   • Venn Diagrams
   • De Morgan Laws
   • Cartesian Product
   • Relation and Functions
   • Quadratic Equations
   • Linear Inequations
   • Permutations and Combinations
   • Binomial Theorem
   • Logarithms
2. 2

   Matrices and Determinants

   6 topics

   • Types of Matrices
   • Operations on Matrices
   • Determinants
   • Properties of Determinants
   • Inverse of a Matrix
   • Applications of Determinants and Matrices
3. 3

   Trigonometry

   7 topics

   • Angles and Their Measures
   • Trigonometric Ratios
   • Trigonometric Identities
   • Trigonometric Equations
   • Inverse Trigonometric Functions
   • Properties of Triangles
   • Heights and Distances
4. 4

   Analytical Geometry

   9 topics

   • Cartesian Coordinate System
   • Distance Formula
   • Section Formula
   • Locus
   • Straight Lines
   • Circles
   • Parabola
   • Ellipse
   • Hyperbola
5. 5

   Differential Calculus

   5 topics

   • Concept of a Real-Valued Function
   • Limits and Continuity
   • Derivatives
   • Application of Derivatives
   • Maxima and Minima
6. 6

   Integral Calculus and Differential Equations

   6 topics

   • Integration as Inverse of Differentiation
   • Definite Integrals
   • Application of Integrals
   • Differential Equations
   • Formation of Differential Equations
   • General and Particular Solutions
7. 7

   Vector Algebra

   6 topics

   • Vectors and Scalars
   • Addition of Vectors
   • Scalar Multiplication
   • Dot Product
   • Cross Product
   • Applications of Vectors
8. 8

   Statistics and Probability

   7 topics

   • Measures of Central Tendency
   • Measures of Dispersion
   • Correlation and Regression
   • Probability
   • Random Variables
   • Binomial Distribution
   • Normal Distribution

Mathematics flashcards for UPSC NDA

21 of 60 cards from the Mathematics deck — real questions with worked answers.

1. What is the difference between a subset and a proper subset of a set A?

   B is a subset of A (B ⊆ A) if every element of B is in A, allowing B = A. B is a proper subset (B ⊂ A) if B ⊆ A and B ≠ A, i.e. A has at least one element not in B.

2. For finite sets A and B, state the formula for the cardinality of their union.

   n(A ∪ B) = n(A) + n(B) − n(A ∩ B). For three sets: n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C).

3. In a Venn diagram, which region represents A − B (the difference of sets)?

   The part of circle A that does NOT overlap with circle B — i.e. elements in A but not in B. Formally A − B = {x : x ∈ A and x ∉ B} = A ∩ B'.

4. State De Morgan's Laws for two sets A and B.

   (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'. The complement of a union is the intersection of complements, and the complement of an intersection is the union of complements.

5. 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 n(A) = m and n(B) = n, then n(A × B) = m × n.

6. What conditions must a relation from A to B satisfy to be a function?

   Every element of the domain A must be related to exactly one element of B — no element of A is left unmapped, and no element of A maps to two different values.

7. For the quadratic ax² + bx + c = 0, what does the discriminant tell you about the roots?

   D = b² − 4ac. If D > 0: two distinct real roots; if D = 0: two equal real roots; if D < 0: two complex conjugate roots.

8. For roots α and β of ax² + bx + c = 0, give the sum and product of the roots.

   Sum: α + β = −b/a. Product: αβ = c/a.

9. When solving a linear inequation, when must you reverse the inequality sign?

   When you multiply or divide both sides by a negative number, the direction of the inequality sign reverses (e.g. −x < 2 becomes x > −2).

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

    ⁿPr = n!/(n−r)! (arrangements, order matters); ⁿCr = n!/[r!(n−r)!] (selections, order does not matter). Relation: ⁿPr = ⁿCr × r!.

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

    T_{r+1} = ⁿCr · a^{n−r} · b^{r}, for r = 0, 1, ..., n. The expansion has (n + 1) terms.

12. State the three core laws of logarithms for products, quotients, and powers.

    log(mn) = log m + log n; log(m/n) = log m − log n; log(m^p) = p·log m. Also change of base: log_b a = log a / log b.

13. Define a scalar matrix and how it relates to a diagonal and identity matrix.

    A scalar matrix is a diagonal matrix whose diagonal entries are all equal. If that common value is 1 it becomes the identity matrix; all off-diagonal entries are 0.

14. What is the condition for two matrices A and B to be multipliable, 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 exists and has order m×p.

15. State the rule for expanding a 2×2 determinant.

    For [[a, b],[c, d]], the determinant = ad − bc.

16. State two key properties of determinants regarding row/column operations.

    (1) Interchanging two rows (or columns) changes the determinant's sign. (2) If two rows (or columns) are identical or proportional, the determinant is 0. Also, multiplying a row by k multiplies the determinant by k.

17. Give the formula for the inverse of a non-singular matrix A.

    A⁻¹ = (1/|A|) · adj(A), valid when |A| ≠ 0, where adj(A) is the transpose of the cofactor matrix.

18. How is a system of linear equations AX = B solved using matrices when A is invertible?

    Multiply both sides by A⁻¹: X = A⁻¹B. A unique solution exists when |A| ≠ 0 (Cramer's Rule equivalently gives x_i = D_i/D).

19. How many degrees are there in one radian, and what is the relation between radians and degrees?

    π radians = 180°, so 1 radian ≈ 57.2958°. To convert: degrees = radians × (180/π); radians = degrees × (π/180).

20. Give the exact values of sin, cos, and tan at 30°, 45°, and 60°.

    sin: 1/2, 1/√2, √3/2. cos: √3/2, 1/√2, 1/2. tan: 1/√3, 1, √3 (for 30°, 45°, 60° respectively).

21. State the three Pythagorean trigonometric identities.

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

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