**Karnataka class 12 commerce maths syllabus**

**Karnataka class 12 commerce maths syllabus :- **

Pre-University College (PUC) is the most popular option after Class 10. Before deciding on which stream to choose in PUC, a student must know the following facts:

- PUC is a 2-year course with 1 Major Board Exam at the end of the second year PUC/12th.
- Every student must study 4 core subjects in each branch along with 1st language—English & 2nd language (Kannada / Sanskrit / Hindi / Urdu / Tamil / Telugu / Malayalam / Marathi / French / German / Arabic ).
- Options in Arts/Humanities—History, Geography, Sociology, Political Science, Psychology, Education, Logic, Home Science, Optional Kannada, Karnataka Music & Hindustani Music, etc.
- Options in Commerce—History, Economics, Business Studies, Accountancy, Computer Science, Geography, Statistics & Basic Maths.
- Options in Science—Physics, Chemistry, Mathematics, Biology, Electronics, Computer Science, Home Science & Geology

**Flexible Combinations**

- If you don’t like to study Biology, you can pursue PCME (Electronics) or PCMCs ( Computer Science) or PCMG (Geology). This makes you eligible for career in Engineering Stream.
- If you don’t like to study Mathematics, you can pursue PCBH ( Home Science ) or PCBS ( Statistics).
- This is apt for students focused on medicine and who find it difficult to cope up with mathematics.

### Karnataka class 12 commerce maths syllabus

**Main subjects in class 11 & 12**

In India, Commerce is possibly the most popular academic stream for students.

The main subjects that are covered in the Commerce stream in Class 11 and 12 are:

- Accountancy
- Economics
- Business Studies
- Mathematics
- Informatics Practices
- English

**Karnataka class 12 commerce maths syllabus**

Mathematics is not just a subject that is restricted to the four walls of a classroom. Its philosophy and applications are to be looked for in the daily course of our life. The knowledge of mathematics is essential for us, to explore and practice in a variety of fields like business administration, banking, stock exchange and in science and engineering.

**Karnataka class 12 commerce maths syllabus**

**Contents of Syllabus**

**UNIT I: RELATIONS AND FUNCTIONS **

**1. Relations and Functions**

Types of relations: Reflexive, symmetric, transitive, empty, universal and equivalence relations. Examples and problems. Types of functions: One to one and onto functions, inverse of a function composite functions, mentioning their properties only , examples and problems. Binary operations: associative, commutative, identity, inverse with examples

**2. Inverse Trigonometric Functions**

Definition, range, domain, principal value branches. Mentioning domain and range of trigonometric and inverse trigonometric functions. Graphs of inverse trigonometric functions. Properties and proofs of inverse trigonometric functions given in NCERT prescribed text book, mentioning formulae for sin-1 x sin-1 y, cos-1 x cos-1 y, 2 tan-1 x = tan-1 ( ) = sin-1 ( ) = cos -1 ( ) without proof. Conversion of one inverse trigonometric function to another w.r.t to right angled triangle. Problems.

**UNIT II: ALGEBRA**

**1. Matrices**

Concept, notation, order, Types of matrices: column matrix, row matrix, rectangular matrix, square matrix, zero matrix, diagonal matrix, scalar matrix and unit matrix. Algebra of matrices: Equality of matrices, Addition, multiplication, scalar multiplication of matrices, Transpose of a matrix. Mentioning properties with respect to addition, multiplication, scalar multiplication and transpose of matrices. Symmetric and skew symmetric matrices: Definitions, properties of symmetric and skew symmetric matrices: proofs of i) If A is any square matrix A+A′ is symmetric and A-A′ is skew symmetric ii) Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix. Concept of elementary row and column operations and finding inverse of a matrix restricted to 2×2 matrices only. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

**2. Determinants**

Determinant of a square matrix (up to 3 × 3 matrices): Definition, expansion, properties of determinants, minors , cofactors and problems. Applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix, definition of singular and non-singular matrices, mentioning their properties: a)If A and B are nonsingular matrices of same order, then AB and BA are nonsingular matrices of same order b)A square matrix A is invertible if and only if A is non-singular matrix Consistency, inconsistency and number of solutions of system of linear equations by examples, Solving system of linear equations in two and three variables (having unique solution) using inverse of a matrix.

**UNIT III: CALCULUS**

**1. Continuity and Differentiability**

**Continuity**: Definition, continuity of a function at a point and on a domain. Examples and problems, Algebra of continuous functions, problems , continuity of composite function and problems

** Differentiability**: Definition, Theorem connecting differentiability and continuity with a counter example. Defining logarithm and mentioning its properties , Concepts of exponential, logarithmic functions, Derivative of ex , log x from first principles, Derivative of composite functions using chain rule, problems. Derivatives of inverse trigonometric functions, problems. Derivative of implicit function and problems. Logarithmic differentiation and problems . Derivative of functions expressed in parametric forms and problems. Second order derivatives and problems Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric Interpretations and problems

**2. Applications of Derivatives**

Tangents and normal: Equations of tangent and normal to the curves at a point and problems Derivative as a Rate of change: derivative as a rate measure and problems Increasing/decreasing functions and problems Maxima and minima : introduction of extrema and extreme values, maxima and minima in a closed interval, first derivative test, second derivative test. Simple problems restricted to 2 dimensional figures only Approximation and problems.

**3. Integrals**

Integration as inverse process of differentiation: List of all the results immediately follows from knowledge of differentiation. Geometrical Interpretation of indefinite integral, mentioning elementary properties and problems. Methods of Integration: Integration by substitution, examples. Integration using trigonometric identities, examples, Integration by partial fractions: problems related to reducible factors in denominators only. Integrals of some particular functions : Evaluation of integrals of ∫ , ∫ √ ∫ √ and problems . Problems on Integrals of functions like ∫ , ∫ √ Integration by parts : Problems , Integrals of type ∫ [ ( ) ( )] and related simple problems. Evaluation of Integrals of some more types like √ , √ and problems Definite integrals: Definition, Definite Integral as a limit of a sum to evaluate integrals of the form ∫ ( ) only. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

**4. Applications of the Integrals:**

Area under the curve : area under simple curves, especially lines, arcs of circles/parabolas/ellipses (in standard form only), Area bounded by two above said curves: problems

**5. Differential Equations**

Definition-differential equation, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution containing at most two arbitrary constants is given. Solution of differential equations by method of separation of variables, Homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type – + py = q where p and q are functions of x or constant + px = q where p and q are functions of y or constant (Equation reducible to variable separable , homogeneous and linear differential equation need not be considered)

**UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY **

**1. Vectors**

Definition of Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors: direction angles, direction cosines, direction ratios, relation between direction ratio and direction cosines. Problems. Types of vectors :Equal, unit, zero, parallel and collinear vectors, coplanar vector position vector of a point, negative of a vector. Components of a vector, Algebra of vectors: multiplication of a vector by a scalar addition of vectors: triangle law, parallelogram law, properties of addition of vectors, position vector of a point dividing a line segment in a given ratio(section formula). Scalar (dot) product of vectors: definition, properties, problems projection of a vector on a line. Vector (cross) product of vectors: definition, properties and problems Scalar triple product: definition, properties and problems.

**2. Three-dimensional Geometry:**

Direction cosines/ratios of a line joining two points. Straight lines in space: Cartesian and vector equation of a line passing through given point and parallel to given vector, Cartesian and vector equation of aline passing through two given points, coplanar and skew lines, distance between two skew lines(Cartesian and vector approach), distance between two parallel lines (vector approach). Angle between two lines. Problems related to above concepts. Plane: Cartesian and vector equation of a plane in normal form, equation of a plane passing through the given point and perpendicular to given vector, equation of a plane passing through three non- collinear points, Intercept form of equation of a plane, angle between two planes, equation of plane passing through the intersection of two given planes, angle between line and plane, condition for the coplanarity of two lines, distance of a point from a plane (vector approach) ,Problems related to above concepts.

**Unit V: Linear Programming**

Introduction of L.P.P. definition of constraints, objective function, optimization, constraint equations, non- negativity restrictions, feasible and infeasible region, feasible solutions, Mathematical formulation-mathematical formulation of L.P.P. Different types of L.P.P: problems namely manufacturing, diet and allocation problems with bounded feasible regions only, graphical solutions for problem in two variables, optimum feasible solution(up to three non-trivial constraints).

**Unit VI: Probability**

Conditional probability – definition, properties, problems. Multiplication theorem, independent events, Baye’s theorem, theorem of total probability and problems. Probability distribution of a random variable-definition of a random variable, probability distribution of random variable, Mean , variance of a random variable and problems. Bernoulli trials and Binomial distribution: Definition of Bernoulli trial, binomial distribution, conditions for Binomial distribution, and simple problems.

**Karnataka class 12 commerce maths syllabus**

Students who completed class 11 and entering into class 12 they can download complete karnataka class 12 commerce maths syllabus by following the steps below:

1. Go to website http://www.kar.nic.in/pue/home.asp

2. Click on Syllabus & Text book Tab

3. You will to directed to a new page showing puc 1 and puc 2

4. Select PUC 2 Syllabus and the subject which you want to know the syllabus

**Karnataka class 12 commerce maths syllabus**

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