## Karnataka class 12 commerce maths study material:

**Karnataka class 12 commerce maths study material: **Commerce is a useful stream as this discipline has great demand in different job oriented fields. Subjects like Economics, Accountancy, Business studies and others are included in this stream. After completion of their 10 standard, students can opt for this stream in class 11 and 12 and they can also take this stream in graduation level. The CBSE board is known for its well-evaluated syllabus and suitable question papers. To maintain its praiseworthy recognition, the board has prepared a comprehensive syllabus for class 11 commerce. All essential subjects and their relevant topics are incorporated in this syllabus by maintaining a proper order. Hence, students get thorough knowledge from this syllabus.

**Karnataka class 12 commerce maths study material:**

Karnataka class 12 commerce maths study material**Karnataka class 12 commerce maths study material:**

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**Karnataka class 12 commerce maths study material:**

- Time management is more important.
- Firstly do not ignore your health.
- Get proper sleep and eat a balanced nutritious diet.
- A good relaxed mind and healthy food is important for you to retain what you study.
- Besides know your strengths and weaknesses. Give more time to your weaker subjects but also take care that you do not ignore any subject altogether because a piled up syllabus in even one subject might bring down your percentage.
- Do not try to spend hours in front of books if you are not retaining what you study.Plan to study such that you do not study for long stretches and get enough break every hour or two.
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**Karnataka class 12 commerce maths study material:**

Mathematics Paper Analysis.

Part A – This part comprised one mark questions, mostly formulae and simple calculations. There were totally ten questions.

Part B – This is a little tedious section comprising 14 two marks questions with a choice to answer any ten. Each question carried two marks.

Part C – Part C is the toughest in the paper. Ten to attend of any three marks questions in this section.

Part D – Students has the option to write any six out of the ten given questions. This is the section that is a breather for students after the tricky part c.

Part E – This is the last part that carried ten marks. Only two questions were given with a choice to answer any one.

**Karnataka class 12 commerce maths study material:**

### Exam Structure

Unit | Topic | Marks |

I. | Relations and Functions | 10 |

II. | Algebra | 13 |

III. | Calculus | 44 |

IV. | Vectors and 3-D Geometry | 17 |

V. | Linear Programming | 06 |

VI. | Probability | 10 |

Total | 100 |

### Unit I: Relations and Functions

**1. Relations and Functions**

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

**2. Inverse Trigonometric Functions**

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

### Unit II: Algebra

**1. Matrices**

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Non commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2).Concept of elementary row and column operations. 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 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

### Unit III: Calculus

**1. Continuity and Differentiability**

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

**2. Applications of Derivatives**

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

**3. Integrals**

Integration as inverse process of differentiation.Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic propertiesof definite integrals and evaluation of definite integrals.

**4. Applications of the Integrals**

Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).

**5. Differential Equations**

Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

### Unit IV: Vectors and Three-Dimensional Geometry

**1. Vectors**

Vectors and scalars, magnitude and direction of a vector.Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

**2. Three – dimensional Geometry**

Direction cosines and direction ratios of a line joining two points.Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines.Cartesian and vector equation of a plane.Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.

### Unit V: Linear Programming

**1. Linear Programming**

Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

### Unit VI: Probability

**1. Probability**

Conditional probability, multiplication theorem on probability. independent events, total probability, Baye’s theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution.

**Karnataka class 12 commerce maths study material:**

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