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ICSE 12th Maths Tuition

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- To enable candidates to acquire knowledge and to develop an understanding of the terms, concepts, symbols, definitions, principles, processes and formulae of Mathematics at the Senior Secondary stage.
- To develop the ability to apply the knowledge and understanding of Mathematics to unfamiliar situations or to new problems.
- To develop an interest in Mathematics.
- To enhance ability of analytical and rational thinking in young minds.
- To develop skills of –

(a) Computation.

(b) Logical thinking.

(c) Handling abstractions.

(d) Generalizing patterns.

(e) Solving problems using multiple methods.

(f) Reading tables, charts, graphs, etc. - To develop an appreciation of the role of Mathematics in day-to-day life.
- To develop a scientific attitude through the study of Mathematics.

- Assignments every week
- 200+ MCQs
- Courseware prepared by experts
- Assessments to give you the level of improvement

There will be two papers in the subject:

Paper I : Theory (3 hours) ……80 marks

Paper II: Project Work ……20 marks

PAPER I (THEORY) – 80 Marks

The syllabus is divided into three sections A, B and C.

Section A is compulsory for all candidates. Candidates will have a choice of attempting questions from EITHER Section B OR Section C.

There will be one paper of three hours duration of 80 marks.

Section A (65 Marks): Candidates will be required to attempt all questions. Internal choice will be provided in two questions of two marks, two questions of four marks and two questions of six marks each.

Section B/ Section C (15 Marks): Candidates will be required to attempt all questions EITHER from Section B or Section C. Internal choice will be provided in one question of two marks and one question of four marks.

## Course Features

## Course Structure

S.No |
Unit |
Marks |

Section A: 65 Marks |
||

1 | Relations and Functions | 10 |

2 | Algebra | 10 |

3 | Calculus | 32 |

4 | Probability | 13 |

Section B: 15 Marks |
||

5 | Vectors | 05 |

6 | Three - Dimensional Geometry | 06 |

7 | Applications of Integrals | 04 |

OR Section C: 15 Marks |
||

8 | Application of Calculus | 05 |

9 | Linear Regression | 06 |

10 | Linear Programming | 04 |

Project Work | 20 | |

Total | 100 |

## Course Lessons

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

**(ii)** Inverse Trigonometric Functions

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

#### Matrices and Determinants

**(i)**** ****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 upto 3). 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).

**(ii) 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.

**(i)** Differentiation, 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.

**(ii) Applications of Derivatives**

Applications of derivatives: increasing/decreasing functions, tangents and normals, 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).

**(iii) 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 properties of definite integrals and evaluation of definite integrals.

**(iv) 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.

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

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.

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.

Application in finding the area bounded b y simple curves and coordinate axes. Area enclosed between two curves.

Application of Calculus in Commerce and Economics.

– Lines of regression of x on y and y on x.

– Lines of best fit.

– Regression coefficient of x on y and y on x.

– Identification of regression equations

– Estimation of the value of one variable using the value of other variable from appropriate line of regression.

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).