*Karnataka Class 12 Commerce Maths Weightage to Content*

*Karnataka Class 12 Commerce Maths Weightage to Content*

*Karnataka Class 12 Commerce Maths Weightage to Content : The Karnataka Secondary Education examination board is a state level examination which is conducted in the month of March-April. Also known as the Pre-University Examination, it offers Science, Arts and commerce streams. The question paper is usually of three hours duration and as per the Science stream, it is divided into practical and written. Written examination is purely of subjective category.*

**Eligibility**: All candidates who are presuming to take the Pre-university examination are required to clear the SSLC examination successfully. The results declared are purely based on the merit criteria.

*Here you will find all the necessary details under Karnataka Class 12 commonly called as PUC for Commerce Students on Maths subject syllabus, Marks weightage topicwise etc*

*Karnataka Class 12 Commerce Maths Weightage to Content*

**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}( 2x/1-x^{2}) = sin^{-1}( 2x/1+x^{2}) = cos^{-1}(1-x^{2}/1+x^{2}) without proof.*Conversion of one inverse trigonometric function to another w.r.t to right angled triangle.**Problems.*

*Karnataka Class 12 Commerce Maths Weightage to Content*

**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 2x*^{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.*

*Karnataka Class 12 Commerce Maths Weightage to Content*

**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***∫ dx/(a**and problems .^{2}+/-x^{2}), ∫ dx/√ (x^{2}+/- a^{2 }),∫ dx/√ (a^{2}– x^{2}^{ })*Problems on Integrals of functions like***∫[(px+q)/(ax**^{2}+ bx + c)] dx , ∫[(px+q)/√ (ax^{2}+ bx + c)] dx*Integration by parts : Problems , Integrals of type***∫e**and related simple problems.^{x}[f (x) + f ‘(x)] dx*Evaluation of Integrals of some more types like***√ (x**and problems^{2}+/- a^{2 }),√ (a^{2}– x^{2}^{ })*Definite integrals: Definition, Definite Integral as a limit of a sum to evaluate integrals of the form ∫*^{a}_{o }(f(x) dx) 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***dy/dx+ py = q**where p and q are functions of x or constant**dx/dy+ py = 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)

*Karnataka Class 12 Commerce Maths Weightage to Content*

**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.*

*Karnataka Class 12 Commerce Maths Weightage to Content*

**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).*

*Karnataka Class 12 Commerce Maths Weightage to Content*

**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.**Note: Unsolved miscellaneous problems given in the prescribed text book need not be considered.*

*Karnataka Class 12 Commerce Maths Weightage to Content*

**Design of the Question Paper **

**MATHEMATICS CLASS : II PUC **

**Time: 3 hours 15 minute; Max. Mark:100**

*The weightage of the distribution of marks over different dimensions of the question paper shall be as follows:*

**I. Weightage to Objectives: **

Objective | Weightage | Marks |

Knowledge | 40% | 60/150 |

Understanding | 30% | 45/150 |

Application | 20% | 30/150 |

Skill | 10% | 15/150 |

**II. Weightage to level of difficulty: **

Level | Weightage | Marks |

Easy | 35% | 53/150 |

Average | 55% | 82/150 |

Difficult | 10% | 15/150 |

**III. Weightage to content:**

Chapter No. | Chapter | No. of teaching Hours | Marks |

1. | RELATIONS AND FUNCTIONS | 11 | 11 |

2. | INVERSE TRIGONOMETRIC FUNCTIONS | 8 | 8 |

3. | MATRICES | 8 | 9 |

4. | DETERMINANTS | 13 | 12 |

5. | CONTINUITY AND DIFFERENTIABILITY | 19 | 20 |

6. | APPLICATION OF DERIVATIVES | 11 | 10 |

7. | INTEGRALS | 21 | 22 |

8. | APPLICATION OF INTEGRALS | 8 | 8 |

9. | DIFFERENTIAL EQUATIONS | 9 | 10 |

10. | VECTOR ALGEBRA | 11 | 11 |

11. | THREE DIMENSIONAL GEOMETRY | 12 | 11 |

12. | LINEAR PROGRAMMING | 7 | 7 |

13. | PROBABILITY | 12 | 11 |

| Total | 150 | 150 |

**IV. Pattern of the question paper:**

PART | Type of questions | Number of questions to be set | Number of questions to be answered | Remarks |

A | 1 mark questions | 10 | 10 | Compulsory part |

B | 2 mark questions | 14 | 10 | —————— |

C | 3 mark questions | 14 | 10 | —————— |

D | 5 mark questions | 10 | 6 | Questions must be asked from the specific set of topics as mentioned below, under section V |

E | 10 mark questions
| 2 | 1 |

*Karnataka Class 12 Commerce Maths Weightage to Content*

**GUIDELINES TO THE QUESTION PAPER SETTER **

*1. The question paper must be prepared based on the individual blue print without changing the weightage of marks fixed for each chapter.*

*2. The question paper pattern provided should be adhered to.*

*Part A : 10 compulsory questions each carrying 1 mark;**Part B : 10 questions to be answered out of 14 questions each carrying 2 mark ;**Part C : 10 questions to be answered out of 14 questions each carrying 3 mark;**Part D : 6 questions to be answered out of 10 questions each carrying 5 mark**Part E : 1 question to be answered out of 2 questions each carrying 10 mark with subdivisions (a) and (b) of 6 mark and 4 mark respectively. (The questions for PART D and PART E should be taken from the content areas as explained under section V in the design of the question paper)*

*3. There is nothing like a single blue print for all the question papers to be set. The paper setter should prepare a blue print of his own and set the paper accordingly without changing the weightage of marks given for each chapter.*

*4. Position of the questions from a particular topic is immaterial.*

*5. In case of the problems, only the problems based on the concepts and exercises discussed in the text book (prescribed by the Department of Pre-university education) can be asked. Concepts and exercises different from text book given in Exemplar text book should not be taken. Question paper must be within the frame work of prescribed text book and should be adhered to weightage to different topics and guidelines.*

*6. No question should be asked from the historical notes and appendices given in the text book.*

*7. Supplementary material given in the text book is also a part of the syllabus.*

*8. Questions should not be split into subdivisions. No provision for internal choice question in any part of the question paper.*

*9. Questions should be clear, unambiguous and free from grammatical errors. All unwanted data in the questions should be avoided.*

*10. Instruction to use the graph sheet for the question on LINEAR PROGRAMMING in PART E should be given in the question paper.*

*11. Repetition of the same concept, law, fact etc., which generate the same answer in different parts of the question paper should be avoided.*

*Karnataka Class 12 Commerce Maths Weightage to Content*

**Weightage to content:**

Chapter No. | Chapter | No. of teaching Hours | PART A
| PART B
| PART C
| PART D
| PART E
| PART E
| Marks |

1. | RELATIONS AND FUNCTIONS | 11 | 1 | 1 | 1 | 1 | | | 11 |

2. | INVERSE TRIGONOMETRIC FUNCTIONS | 8 | 1 | 2 | 1 | | | | 8 |

3. | MATRICES | 8 | 1 | | 1 | 1 | | | 9 |

4. | DETERMINANTS | 13 | 1 | 1 | | 1 | | 1 | 12 |

5. | CONTINUITY AND DIFFERENTIABILITY | 19 | 1 | 2 | 2 | 1 | | 1 | 20 |

6. | APPLICATION OF DERIVATIVES | 11 | | 1 | 1 | 1 | | | 10 |

7. | INTEGRALS | 21 | 1 | 2 | 2 | 1 | 1 | | 22 |

8. | APPLICATION OF INTEGRALS | 8 | | | 1 | 1 | | | 8 |

9. | DIFFERENTIAL EQUATIONS | 9 | | 1 | 1 | 1 | | | 10 |

10. | VECTOR ALGEBRA | 11 | 1 | 2 | 2 | | | | 11 |

11. | THREE DIMENSIONAL GEOMETRY | 12 | 1 | 1 | 1 | 1 | | | 11 |

12. | LINEAR PROGRAMMING | 7 | 1 | | | | 1 | | 7 |

13. | PROBABILITY | 12 | 1 | 1 | 1 | 1 | | | 11 |

| Total | 150 | | | | | | | 150 |

**To download***Karnataka Class 12 Commerce Maths syllabus copy check here Syllabus*

**To download***Karnataka Class 12 Commerce Maths Model papers check here Model papers *

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