CBSE Class 12 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 CBSE Class 12, not a summary of it.

1. 1

   Relations and Functions

   4 topics

   • Types of relations
   • Types of functions
   • Composition of functions and invertible function
   • Binary operations
2. 2

   Algebra

   4 topics

   • Matrices
   • Determinants
   • Adjoint and inverse of a matrix
   • System of linear equations
3. 3

   Calculus

   5 topics

   • Continuity and Differentiability
   • Applications of Derivatives
   • Integrals
   • Application of the Integrals
   • Differential Equations

Mathematics flashcards for CBSE Class 12

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. What is a reflexive relation on a set A?

   A relation R on A is reflexive if (a, a) ∈ R for every a ∈ A.

2. What is a symmetric relation?

   A relation R is symmetric if (a, b) ∈ R implies (b, a) ∈ R for all a, b.

3. What is a transitive relation?

   A relation R is transitive if (a, b) ∈ R and (b, c) ∈ R together imply (a, c) ∈ R.

4. Define an equivalence relation.

   A relation that is reflexive, symmetric, and transitive simultaneously.

5. What is an equivalence class [a] for an equivalence relation R on A?

   [a] = {x ∈ A : (x, a) ∈ R}, the set of all elements related to a. Distinct classes are disjoint and partition A.

6. What is an empty relation and a universal relation on set A?

   Empty relation: R = ∅ (no element related). Universal relation: R = A × A (every element related to every element).

7. When is a function f: A → B called one-one (injective)?

   When f(x1) = f(x2) implies x1 = x2; distinct inputs give distinct outputs.

8. When is a function f: A → B called onto (surjective)?

   When every element of B is the image of at least one element of A, i.e., range = codomain B.

9. What is a bijective function?

   A function that is both one-one (injective) and onto (surjective).

10. If A and B are finite sets with n elements each, when does a one-one function f: A → B become onto?

    For finite sets of equal size, a function is one-one if and only if it is onto (hence bijective).

11. Define the composition (gof) of functions f: A → B and g: B → C.

    (gof)(x) = g(f(x)) for all x ∈ A; it is a function from A to C.

12. If f and g are both one-one, what can be said about gof?

    gof is also one-one. Likewise, if both are onto, gof is onto; if both are bijective, gof is bijective.

13. When is a function f: X → Y invertible, and what is its inverse?

    f is invertible iff it is bijective. Its inverse g: Y → X satisfies gof = I_X and fog = I_Y.

14. If f and g are invertible, what is (gof)⁻¹?

    (gof)⁻¹ = f⁻¹ o g⁻¹ (the order reverses).

15. What is a binary operation on a set A?

    A function ∗ : A × A → A; it assigns to each ordered pair (a, b) a unique element a ∗ b in A (closure).

16. What does it mean for a binary operation ∗ to have an identity element e?

    e is an identity if a ∗ e = e ∗ a = a for every a ∈ A.

17. What is the inverse of an element a under a binary operation ∗ with identity e?

    An element b such that a ∗ b = b ∗ a = e; b is the inverse of a.

18. When is a binary operation ∗ commutative and when is it associative?

    Commutative: a ∗ b = b ∗ a for all a, b. Associative: (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c.

19. What is the order of a matrix?

    Order m × n means the matrix has m rows and n columns.

20. What is a scalar matrix?

    A diagonal matrix in which all diagonal elements are equal (and off-diagonal elements are zero).

21. What conditions are required for matrix multiplication AB to be defined?

    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 is m × p.

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