Curriculum Mathematics Senior Secondary NIOS

SENIOR SECONDARY COURSE IN MATHEMATICS

Rationale

The curriculum in Mathematics has been designed to cater to the specific needs of NIOS learners. The thrust is on the applicational aspects of mathematics and relating learning to the daily life and work situation of the learners. The course is modular in nature with – eight compulsory modules forming the core curriculum and four optional modules out of which the learner is to choose one optional module. An attempt has been made to reduce rigour and abstractness.

Objectives

The course aims at enabling learners to :

become precise, exact and logical.
acquire knowledge of mathematical terms, symbols, facts and formulae.
develop an understanding of mathematical concepts.
develop problem solving ability.
acquire skills in applying the learning to situation including reading charts, tables, graphs etc.
apply the above skills in solving problems related to Science, Commerce and daily life.
develop a positive attitude towards Mathematics and its application.

COURSE STRUCTURE

The compulsory modules are :

Complex Numbers and Quadratic Equations
Determinants and Matrices
Permutations and Combinations
Sequences and Series
Trigonometry
Coordinate Geometry
Differential Calculus
Integral Calculus

The optional modules are :

Statistics and Probability
Vectors and Analytical Solid Geometry
Linear Programming

MODULE WISE DISTRIBUTION OF STUDY HOURS AND MARK

Module No.	Compulsory Modules	Study Hours	Marks
1.	Complex Numbers & Quadratic Equations	15	10
2.	Determinants & Matrices	15	10
3.	Permutations & Combinations	20	8
4.	Sequences & Series	20	8
5.	Trigonometry	30	10
6.	Coordinate Geometry	30	10
7.	Differential Calculus	45	17
8.	Integral Calculus	45	17

Optional Modules	Study Hours	Marks
Statistics & Probability	20	10
Vectors & Analytical Solid Geometry	20	10
Linear Programming	20	10
TOTAL	240	100

Module 1: Complex Numbers and Quadratic Equations

The evaluation system for this subject would consist of internal evaluation through Tutor Marked Assignment (TMA) and external examination. The awards of TMA will be reflected separately on the Mark Sheet. The external examination will be conducted twice a year. Besides these, there will be certain inbuilt components of self evaluation such as Intext Questions, Terminal exercise, etc.

Study Time: 15 hrs. Max. Marks: 10

Pre-requisites: Real numbers and quadratic equations with real coefficients.

Content and Extent of Coverage

Complex Numbers

Definition in the form x + iy
Real and imaginary parts of a complex number.
Modulus and argument of a complex number
Conjugate of a complex number

Algebra of Complex number

Equality of complex numbers
Operations on complex numbers (addition, subtraction, multiplication and division)
Properties of operations (closure, commutativity, associativity, identity, inverse, distributivity)
Elementary properties of modulus namely

(i) |z| = 0 ⇔ z = 0 and |z₁| = |z₂| ⇔ z₁ = z₂

(ii) |z₁ + z₂| ≤ |z₁| + |z₂|

(iii) |z₁ / z₂| = |z₁| / |z₂| (z₂ ≠ 0)

Argand Diagram

Representation of a complex number by a point in a plane.

Quadratic Equations

Solution of quadratic equation with real coefficients using the quadratic formula
Square root of a complex number
Cube roots of unity

Extended Learning

Polar representation of a complex number
Quadratic equations with complex coefficients

NOTE :

“Division by zero is not allowed in complex numbers” to be stressed.
Lack of order in complex numbers to be highlighted.
The fact that complex roots of a quadratic equation with real coefficients occur in conjugate pairs but the same may not be true if the coefficients are complex numbers is to be verified using different examples.

Module 2: Determinants and Matrices

Study Time: 15 hrs. Max. Marks: 10

Pre-requisites : Knowledge of number systems; solution of system of linear equations.

Content and Extent of Coverage

Determinants and their Properties – Minors and Cofactors

Expansion of a determinant
Properties of determinants

Matrices

Introduction as a rectangular array of numbers
Matrices upto order 3 × 4

Types of matrices

Square and rectangular matrices
Unit matrix, zero matrix, diagonal, row and column matrices
Symmetric and skew symmetric matrices

Algebra of matrices

Multiplication of a matrix by a number
Sum and difference of matrices
Multiplication of matrices

Inverse of a square matrix

Minor and cofactors of a matrix
Adjoint of a matrix
Inverse of a matrix

Solution of a system of linear equations

Solution by Cramer’s Rule
Solution by matrix method

NOTE :

If any two rows or columns of a determinant are interchanged, then the sign of the determinant is changed.

If each element of a row (or column) of a determinant is multiplied by a constant, the value of the determinant gets multiplied by the same constant.

If k times a row (or column) is added to another row (or column) the value of the determinant remains unchanged.

The number of equations and variables to be restricted to three only.

Extended Learning

Cramer’s Rule for four or more equations
Determinant as a function
Matrix as a function
Matrices over complex numbers
Hermitian and Skew Hermitian
Rank of a Matrix
Inverse by elementary row transformations
Solution of 4 or more than 4 linear equations in 4 more than 4 variables

Module 3: Permutations, Combinations and Binomial Theorem

Study Time: 20 hrs. Max. Marks: 8

Pre-requisites : Number Systems

Content and Extent of Coverage

Mathematical Induction
Principle of mathematical induction
Application of the principle in solving problems

Permutations
Fundamental Principle of Counting
Meaning of ⁿPᵣ
Expression for ⁿPᵣ

Combinations
Meaning of ⁿCᵣ
Expression for ⁿCᵣ

Properties of ⁿCᵣ namely

(i)	ⁿCᵣ = ⁿPᵣ
(ii)	ⁿCᵣ = ⁿCₙ₋ᵣ
(iii)	ⁿCᵣ₋₁ + ⁿCᵣ = ⁿ⁺¹Cᵣ

Binomial Theorem
Binomial theorem for a positive index with proof.

Extended Learning

Circular permutations
Pascal’s triangle
Binomial theorem for negative index and rational indices (without proof)

Module 4: Sequences and Series

Selling of Goods and Services

Study Time: 20 hrs. Max. Marks: 8

Pre-requisites : Permutation, Combination and concept of a function, Exponential functions, Logarithmic functions and their properties, and graphs.

Content and Extent of Coverage

Arithmetic Progression
Concept of a sequence – A.P as a sequence
General term of an A.P
Sum upto ‘n’ terms of an A.P.

Geometric Progression
G.P as a sequence
General term of a G.P
Sum upto ‘n’ terms of a G.P.
Sum upto infinite terms of a G.P.

Series
Concept of a series
Some important series, etc. using method of differences and mathematical induction

Exponential and Logarithmic Series
Representation of e^x and log(1 + x) as series.
Properties of e^x and log(1 + x)

Content and Extent of Coverage

Arithmetic Mean, Geometric Mean
Harmonic Progression, Arithmetico-Geometric Progression and their relationships
Logarithms on any base

Module 5 : Trigonometry

Study Time: 30 hrs. Max. Marks: 10

Pre-requisites : Trigonometric ratios of an acute angle.

Content and Extent of Coverage

Functions
Concept of a function
Domain, codomain and range of a function
Graphs of functions
Odd and even functions
Some important functions

Composition of Functions
Composition of two or more functions
Inverse of a Function

Trigonometric Ratios
Radian measure of angles
Trigonometric ratios as functions
Graphs of T-ratios
Periodicity
T-ratios of allied angles
Inverse Trigonometric ratios

Addition and Multiplication Formulae
Addition and subtraction formulae for trigonometric functions
Sines, Cosines and Tangents of multiples and submultiples
Solution of simple trigonometric equations

Extended Learning

Properties of triangles
Solution of triangles
Properties of inverse functions
Trigonometric equations and their solutions
General solution of Trigonometric equations

Module 6 : Coordinate Geometry

Study Time: 30 hrs. Max. Marks: 10

Pre-requisites: Number systems and plotting of points on a graph.

Content and Extent of Coverage

Introduction (Basic Concepts)
Distance Formula
Section Formula
Area of a Triangle

Straight Line
Equation of a straight line in slope-intercept form
Two point form
Point-slope form
Parametric form
Intercepts form

General equation of first degree and its relationship with straight line

Parallel and Perpendicular Lines
Angle between two lines
Parallel lines
Perpendicular lines
Distance of a point from a line
Distance between two parallel lines
Family of lines

Circle
Equation of a circle whose radius and centre are given
Equation of a circle in terms of extremities of its diameter
General equation of a circle
Equations of tangents and normals
Parametric representation of a circle

Conic Sections
Acquaintance with equation of parabola and ellipse in standard form
Eccentricity, directrix and focus

NOTE

Problems on lines to include questions of the typel + λ l ‘ = 0
Conic sections to be introduced through examples of loci and not as a section of a cone

Extended Learning

Locus – Advanced examples of loci
System of Circles
Equation of a family of circles passing through the intersection of two circles
Condition for orthogonality of circles
Radical axis of two circles

Sections of a Cone (Conic Sections)
Derivation of equations of parabola, ellipse and hyperbola
Condition for y = mx + c to be a tangent
Point of tangency

General second degree equation in two variables
Condition to represent a pair of straight lines
A circle
Different conic sections

MODULE 7 : Differential Calculus

Study Time: 45 hrs. Max. Marks: 17

Pre-requisites: Trigonometry and Exponential and Logarithmic series

Content and Extent of Coverage

Limit and Continuity
Notion of limit (left hand and right hand limits)
Continuity of functions at a point
Continuity of functions in an interval

Differentiation
Derivatives from the first principle
Derivative as instantaneous rate of change
Geometrical meaning of derivative
Derivative of sum, difference, product and quotient of functions and chain rule
Derivatives of algebraic, trigonometric, exponential and logarithmic functions

Monotonicity of Functions
Monotonicity and sign of the derivative
Second derivative of a function
Maxima and Minima

NOTE

The concept of monotonic function will be introduced at the appropriate stage

Extended Learning

Differentials and errors
Approximation
Rolle’s theorem
Lagrange’s mean value theorem
Derivatives of higher orders
Points of inflexion
Concavity and convexity of functions

MODULE 8 : Integral Calculus

Study Time: 45 hrs. Max. Marks: 17

Pre-requisite : Differential Calculus

Content and Extent of Coverage

Introduction to Integral Calculus
Integration as inverse of differentiation
Properties of integrals

Techniques of Integration
Integration by Substitution
Integration by parts
Integration using partial fractions

Definite Integrals
Idea of definite integral as limit of a sum
Geometrical interpretation of definite integrals in simple cases
Properties of definite integrals

(i)	∫ab f(x) dx = − ∫ba f(x) dx
(ii)	∫ac f(x) dx = ∫ab f(x) dx + ∫bc f(x) dx
(iii)	∫02a f(x) dx = ∫0a f(x) dx + ∫0a f(2a − x) dx
(iv)	∫ab f(x) dx = ∫ab f(a + b − x) dx
(v)	∫0a f(x) dx = ∫0a f(a − x) dx
(vi)	∫02a f(x) dx = 2 ∫0a f(x) dx if f(2a − x) = f(x) = 0 if f(2a − x) = −f(x)
(vii)	∫−aa f(x) dx = 2 ∫0a f(x) dx if f is an even function of x = 0 if f is an odd function of x

Fundamental theorem of Integral Calculus (statement only)
Application of definite integrals in finding area under a curve

Differential Equations
Notion of differential equation, its order and degree
Solution of first order, first degree differential equations

NOTE

The fact that integral is called primitive, anti-derivative to be specified

The following types of integrals may be taken up giving appropriate details.

1.	∫ x² ± a² dx
2.	∫ (a² − x²) dx
3.	∫ (ax² + bx + c) dx
4.	∫ (px + q) dx
5.	∫ (px + q)/(ax² + bx + c) dx
6.	∫ eax sin bx dx
7.	∫ sin⁻¹x dx
8.	∫ sinⁿx cosᵐx dx
9.	∫ (a + b) sin x dx
10.	∫ (a + b) cos x dx

Extended Learning

Application of definite integrals in finding the area under a curve
Formation of a differential equation
Higher order differential equations reducible to variable separable cases

OPTIONAL MODULES

(The learner have to choose any one out three modules)

Module 9 : Statistics and Probability

Study Time: 20 hrs. Max. Marks: 10

Pre-requisites : Mean, median and mode of ungrouped and grouped data.

Content and Extent of Coverage

Measures of Dispersion
Range
Mean Deviation
Variance and Standard Deviation

Random Experiments and Events
Random experiments
Sample space and events
Types of events – mutually exclusive and equally likely events

Probability
Concept of probability
Use of permutation and combination in probability
Probability as a function
Conditional probability and independent events
Random variable as a function on sample space

Probability Distribution
Introduction to probability distribution
Binomial distribution
Expected value of a random variable
Mean and variance of a binomial distribution

NOTE

Probability to be explained as the ratio of number of favourable cases to the total number of cases.
Venn diagrams to be used as frequently as possible for pictorial representation.
Use of addition theorem when product of events is easily identifiable.

Extended Learning

Correlation and Regression
Curve fitting (fitting a line)
Mean and variance of Poisson distribution
Bivariate probability distributions

Module 10 : Vectors & Analytical Solid Geometry

Study Time: 20 hrs. Max. Marks: 10

Pre-requisites : Knowledge of Two-Dimensional Geometry, Coordinate Geometry and Trigonometry.

Content and Extent of Coverage

Vectors
Scalars and vectors
Vectors as directed line segments
Magnitude and direction of a vector
Null vector and unit vector
Equality of vectors
Position vector of a point

Algebra of Vectors
Addition and subtraction of vectors and their properties
Multiplication of a vector by a scalar and its properties

Resolution of a Vector
Resolution of a vector in two dimensions
Resolution of a vector in three dimensions
Section formula

Coordinates of a Point
Coordinates of a point in space
Distance between two points
Coordinates of a division point
Direction cosines and projection
Condition of parallelism and perpendicularity of two lines

The Plane
General equation of a plane
Equation of a plane passing through three points
Equation of a plane in normal and intercept form
Angle between two planes
Plane bisecting angles between two planes
Homogeneous equations of second degree representing two planes
Projection and area of a triangle
Volume of tetrahedron

The Straight Line
Equation of a line in symmetrical form
Deduction of general equation into symmetrical form
Perpendicular distance of a point from a straight line
Angle between a line and a plane
Condition of coplanarity of two lines

The Sphere
Equation of a sphere (centre-radius form)
Equation of a sphere through four non-coplanar points
Diameter form of the equation of a sphere
Plane section of a sphere and sphere through a given circle
Intersection of a sphere and a line

Extended Learning

Skew lines
Intersection of three planes
Pole and polar plane in a sphere
Equation of a cylinder and its properties
Equation of a cone and its properties

Module 11 : Linear Programming

Study Time: 20 hrs. Max. Marks: 10

Pre-requisites : Matrices

Content and Extent of Coverage

Introduction
Introduction through a real life problem
Solution by graphical method
General terms used in linear programming (inequation, objective function, convex polygon, feasible solution, optimal solution, etc.)
Constraints in a linear programming problem
Feasible and optimal solutions
Simplex method

Applications
Dual problem
Assignment problem
Transportation problem

Extended Learning

Product-mix problem
Duality
Simplex method