Curriculum Mathematics Senior Secondary NIOS
SYLLABUS
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 
Compulsory Modules 
Minimum 
Marks 
No. 

Study Hours 

1. 
Complex Numbers & Quadratic Equations 
15 
10 
2. 
Determinants & Matrices 
15 
10 
3. 
Permutations & Combinations 
20 
08 
4. 
Sequences & Series 
20 
08 
5. 
Trigonometry 
30 
10 
6. 
Coordinate Geometry 
30 
10 
7. 
Differential Calculus 
45 
17 
8. 
Integral Calculus 
45 
17 

Optional Modules 



(The learner have to choose any one module) 


9. 
Statistics & Probability 



OR 


10. 
Vectors & Analytical Solid Geometry 
20 each 
10 each 

OR 


11. 
Linear Programming 



TOTAL 
240 
100 
CURRICULUM OF SENIOR SECONDARY MATHEMATICS
COMPULSORY MODULES
Module 1: Complex Numbers and Quadratic Equations
Study Time: 15 hrs. Max. Marks: 10
Prerequisites: 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 z_{1} = z _{2} ⇔ z_{1} = z_{2}
(ii) z_{1} + z _{2} ≤ z_{1} + z_{2}
(iii) 

z_{1} 

= 



z_{1} 




( z 
2 
≠ 0) 







z _{2} 



z_{2} 












































• 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
Prerequisites : 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:
 The properties of determinants to include the following:
 If any two rows or columns of a determinant are interchanged, then
the sign of the determinant is changed.
2. If each element of a row (or column) of a determinant is multiplied by a constant, the value of the determinant gets multiplied by.
3. 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
Prerequisites : 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 ^{n}P_{r}
 Expression for ^{n}P_{r}
? Combinations
 Meaning of ^{n}C_{r}
 Expression for ^{n}C_{r}
 Properties of ^{n}C_{r} namely

n _{C} 


n _{P} 




(i) 

= 

r 




r 
n! 





















(ii) 
n _{C} 
r 
= ^{n}C 
n 
−r 








(iii) 
^{n} C _{r} _{−}_{1} + ^{n} C _{r} = ^{n} ^{+}^{1}C_{r} 

• 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
Study Time: 20 hrs. Max. Marks: 8
Prerequisites : 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)
Extended Learning
• Arithmetic Mean, Geometric Mean
• Harmonic Progression, ArithmeticoGeometric Progression and their relationships
• Logarithms on any base
Module 5 : Trigonometry
Study Time: 30 hrs. Max. Marks: 10
Prerequisites : 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 Tratios
 Periodicity
 Tratios 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
Prerequisites: 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
 Slopeintercept form
 Two point form
 Pointslope 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 type l + 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 in standard form
 Condition for y = mx + c to be a tangent to these conics
 Point of tangency
• General second degree equation in two variables
Condition for it to represent :
 A pair of straight lines
 A circle
 Different conic sections
MODULES 7: Differential Calculus
Study Time: 45 hrs. Max. Marks: 17
Prerequisites: Trigonometry and Exponential
and Logarithmic series
Content and Extent of Coverage
• Limit and Coverage
 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 CLACULUS
Study Time: 45 hrs. Max. Marks: 17
Prerequisite : 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

b 
a 

(i) 
_{∫} f ( x ) dx = −_{∫} f ( x )dx 


a 
b 


b 
c 
b 
(ii) 
_{∫} f ( x ) dx = _{∫} f ( x ) dx + _{∫} f ( x )dx 


a 
a 
c 

2 a 
a 
a 
(iii) 
_{∫} f ( x ) dx = _{∫} f ( x ) dx + _{∫} f (2 a − x )dx 


0 
0 
0 

b 
b 

(iv) _{∫} f ( x ) dx = _{∫} f ( a + b − x )dx 


a 
a 


a 
a 

(v) 
_{∫} f ( x ) dx = _{∫} f ( a − x )dx 


0 
0 


2 a 
a 

(vi) 
_{∫} f ( x ) dx = 2 _{∫} f ( x )dx if f (2 a − x ) = f ( x) 


0 
0 



= 0 if f (2 a − x ) = − f ( x) 


a 
a 

(vii) 
∫ 
f ( x ) dx = 2 _{∫} f ( x )dx if ? is an even 


− a 
0 



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, antiderivative to be specified.
The following types of integrals may be taken up giving appropriate details.



dx 





dx 


dx 






dx 




dx 






, 









, 


, 





, 





, 





























∫ _{x}2 
± a^{2} 

∫ _{x}2 _{±} _{a}2 

∫ _{a}2 _{−} _{x}2 


∫ 


a^{2} −x^{2} 
∫_{ax}2 
+ bx +c 





dx 






( px + q)dx 






(px +q)dx 











, 







, 









, 












∫ _{ax}2 



∫ 









^{∫} ax^{2} + bx + c 
+ bx +c 

ax^{2} + bx +c 



_{∫}x^{2} ± a^{2} dx , _{∫}a^{2} −x^{2} dx , _{∫}e^{ax} sin bx dx ,
_{∫} ( px + q)ax^{2} + bx + c dx ,_{∫} sin^{−}^{1}x dx ,
∫ ^{sin}^{n} ^{x} ^{cos} ^{m} ^{x dx} ^{,}∫ _{a} _{+} _{b}^{dx}_{sinx} ^{,}∫_{a} _{+}_{b}^{dx}_{cos} _{x}
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
Prerequisites : 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  Ramdom experiments
 Sample space, events
 Types of events, viz. mutually exclusive events 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 cases favourable to an event and the total number of cases.
 Venn diagrams to be used as frequently as possible to give a pictorial representation of the concepts
 Use of addition theorem when product of event 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 hrs. : 20 Max. Marks: 10
Prerequisites: 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 their 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 the 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 the 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 : Centreradius 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
Prerequisites : 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
• Productmix problem
• Duality
• Simplex method.