ISC 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 ISC Class 12, not a summary of it.

1. 1

   Relations and Functions

   4 topics

   • Types of Relations
   • Types of Functions
   • Composite Functions
   • Inverse of a Function
2. 2

   Algebra

   2 topics

   • Matrices
   • Determinants
3. 3

   Calculus

   4 topics

   • Continuity and Differentiability
   • Applications of Derivatives
   • Integrals
   • Differential Equations
4. 4

   Vectors and Three-Dimensional Geometry

   2 topics

   • Vectors
   • Three-dimensional Geometry
5. 5

   Linear Programming

   2 topics

   • Introduction to Linear Programming
   • Different Types of Linear Programming Problems
6. 6

   Probability

   4 topics

   • Conditional Probability
   • Bayes' Theorem
   • Probability Distribution
   • Binomial Distribution

Mathematics flashcards for ISC Class 12

21 of 61 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) belongs to R for every element a in A.

2. Define a symmetric relation on a set A.

   A relation R is symmetric if whenever (a, b) belongs to R, then (b, a) also belongs to R.

3. Define a transitive relation on a set A.

   A relation R is transitive if whenever (a, b) and (b, c) belong to R, then (a, c) also belongs to R.

4. What is an equivalence relation?

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

5. What is an equivalence class [a] under an equivalence relation R?

   The set of all elements related to a, i.e. [a] = {x : (a, x) belongs to R}. Distinct equivalence classes partition the set.

6. Define a one-one (injective) function.

   A function f is injective if f(x1) = f(x2) implies x1 = x2; distinct inputs give distinct outputs.

7. Define an onto (surjective) function.

   A function f: A to B is surjective if every element of B is the image of at least one element of A, i.e. range = codomain.

8. What is a bijective function?

   A function that is both one-one (injective) and onto (surjective); only bijective functions are invertible.

9. For a function from a finite set to itself, what relationship holds between injective and surjective?

   For finite sets of equal size, a function is injective if and only if it is surjective (and hence bijective).

10. How is the composite function (g o f)(x) defined?

    (g o f)(x) = g(f(x)); first apply f, then apply g. It requires range of f to be a subset of domain of g.

11. Is composition of functions commutative? State the general associativity property.

    Composition is NOT generally commutative (g o f need not equal f o g), but it IS associative: (h o g) o f = h o (g o f).

12. If f and g are both bijective, what is (g o f) inverse equal to?

    (g o f) inverse = f inverse o g inverse (note the reversed order).

13. What condition must a function satisfy to have an inverse, and what defines f inverse?

    f must be bijective. Then f inverse is the function with f inverse(y) = x if and only if f(x) = y, and f o f inverse = f inverse o f = identity.

14. What is the order of a matrix, and when are two matrices equal?

    Order is m x n (rows x columns). Two matrices are equal if they have the same order and all corresponding elements are equal.

15. Define a symmetric matrix and a skew-symmetric matrix.

    Symmetric: A transpose = A (a_ij = a_ji). Skew-symmetric: A transpose = -A (a_ij = -a_ji), so diagonal elements are 0.

16. State the rule for multiplying matrices A (m x n) and B (p x q).

    Product AB exists only if n = p; the result has order m x q, and (AB)_ij = sum over k of a_ik times b_kj.

17. What is the transpose property for a product: (AB) transpose = ?

    (AB) transpose = B transpose times A transpose.

18. Every square matrix can be expressed as the sum of which two types of matrices?

    A symmetric matrix (1/2)(A + A transpose) plus a skew-symmetric matrix (1/2)(A - A transpose).

19. How do you expand a 2x2 determinant |a b; c d|?

    The determinant equals ad - bc.

20. What is a minor M_ij and a cofactor A_ij of a determinant?

    Minor M_ij is the determinant left after deleting row i and column j. Cofactor A_ij = (-1)^(i+j) times M_ij.

21. What is the geometric meaning of a determinant of order 2 (or 3)?

    Its absolute value gives the area of the parallelogram (order 2) or volume of the parallelepiped (order 3) formed by the row/column vectors.

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