The syllabus of NCERT Maths Class 10 comprises of the following chapters. We have prepared Video Leactures, relevant for the student of CBSE Class 10, who may be studying in CBSE as per the NCERT Maths Class 10 syllabus.
Syllabus of Maths class 5
Shapes & Spatial Understanding
• Gets the feel of perspective while drawing a 3-D object in 2-D.
• Gets the feel of an angle through observation and paper folding.
• Identifies right angles in the environment.
• Classifies angles into right, acute and obtuse angles.
• Represents right angle, acute angle and obtuse angle by drawing and tracing.
• Explores intuitively rotations and reflections of familiar 2-D shapes.
• Explores intuitively symmetry in familiar 3-D shapes.
• Makes the shapes of cubes, cylinders and cones using nets especially designed for this purpose.
Numbers and Operations
• Finds place value in numbers beyond 1000.
• Appreciates the role of place value in addition, subtraction and multiplication algorithms.
• Uses informal and standard division algorithms.
• Explains the meaning of factors and multiples.
3. MENTAL ARITHMETIC
• Estimates sums, differences, products and quotients and verifies using approximation.
4. FRACTIONAL NUMBERS
• Finds the fractional part of a collection.
• Compares fractions.
• Identifies equivalent fractions.
• Estimates the degree of closeness of a fraction to known fractions
• Uses decimal fractions in the context of units of length and money.
• Expresses a given fraction in decimal notation and vice versa.
• Applies the four operations in solving problems involving money.
• Determines area and perimeter of simple geometrical figures.
• Applies the four operations in solving problems involving length, weight and volume.
• Relates commonly used larger and smaller units of length, weight and volume and converts one to the other.
• Applies simple fractions to quantities.
• Converts fractional larger unit into complete smaller units.
• Appreciates volume of a solid body: intuitively and also by informal measurement.
• Uses addition and subtraction in finding time intervals in simple cases.
8. DATA HANDLING
• Collects two-dimensional quantitative data. represents the data in the form of a table.
• Draws a bar graph or a pictograph to present a data.
• Identifies patterns in square numbers, triangular numbers.
• Relates sequences of odd numbers between consecutive square numbers.
• Makes border strip and tiling patterns.
Syllabus of Maths class 4
Shapes & Spatial Understanding
• Draws a circle free hand and with compass.
• Identifies centre, radius and diameter of a circle.
• Uses Tangrams to create different shapes.
• Tiles geometrical shapes: using one or two shapes.
• Chooses a tile among a given number of tiles that can tile a given region both intuitively and experimentally.
• Explores intuitively the area and perimeter of simple shapes.
• Makes 4-faced, 5-faced and 6- faced cubes from given nets especially designed for the same.
• Explores intuitively the reflections through inkblots, paper cutting and paper folding.
• Reads and draws 3-D objects, making use of the familiarity with the conventions used in this.
• Draws intuitively the plan, elevation and side view of simple objects.
Numbers and Operations
• Writes multiplication facts.
• Writes tables upto 10 × 10.
• Multiplies two and three digit numbers using lattice algorithm and the standard (column) algorithm.
• Divides a given number by another number in various ways such as:
– by drawing dots.
– by grouping.
– by using multiplication facts.
– by repeated subtraction.
• Applies the four operations to life situations.
• Frames word problems.
• Estimates sums, differences and products of given numbers.
3. MENTAL ARITHMETIC
• Adds and subtracts multiples of 10 and 100, mentally.
• Completes multiplication facts by adding partial products, mentally (e.g. 7 × 6 = 5 × 6 + 2 × 6).
4. FRACTIONAL NUMBERS
• Identifies half, one fourth and three- fourths of a whole.
• Identifies the symbols, 1/2, 1/4, 3/4
• Explains the meaning of 1/2. 1/4, 3/4
• Appreciates equivalence of 2/4 and 1/2; and of 2/2, 3/3, 4/4 and 1.
• Converts Rupees to Paise.
• Adds and subtracts amounts using column addition and subtraction with regrouping.
• Uses operations to find totals, change, multiple costs and unit cost.
• Estimates roughly the totals and total cost.
• Relates metre with centimetre;
• Converts metre into centimetres and vice versa.
• Solves problems involving length and distances.
• Estimates length of an object and distance between two given locations.
• Weighs objects using a balance and standard units.
• Determines sums and differences of weights.
• Estimates the weight of an object and verifies using a balance.
• Measures volumes of given liquid using containers marked with standard units.
• Determines sums and differences of volumes.
• Estimates the volume of a liquid contained in a vessel and verifies by measuring.
• Computes the number of weeks in a year.
• Correlates the number of days in a year with the number of days in each month.
• Justifies the reason for the need of a leap year.
• Reads clock time to the nearest hours and minutes.
• Expresses time, using the terms, ‘a.m.’ and ‘p.m.’
• Estimates the duration of familiar events.
• Finds approximate time elapsed by (to the nearest hour) forward counting.
• Computes the number of days between two dates.
8. DATA HANDLING
• Collects data and represents in the form of bar graphs;
• Draws Inferences by discussing with the teacher.
• Identifies patterns in multiplication and division: multiples of 9,
• Casts out nines from a given number to check if it is a multiple of nine.
• Multiplies and divides by 10s, 100s.
• Identifies geometrical patterns based on symmetry.
Syllabus of Maths class 3
Shapes & Spatial Understanding
• Creates shapes through paper folding, paper cutting.
• Identifies 2-D shapes
• Describes the various 2-D shapes by counting their sides, corners and diagonals.
• Makes shapes on the dot-grid using straight lines and curves.
• Creates shapes using tangram pieces.
• Matches the properties of two 2-D shapes by observing their sides and corners (vertices).
• Tiles a given region using a tile of a given shape.
• Distinguishes between shapes that tile and that do not tile.
• Intuitive idea of a map. Reads simple maps (not necessarily scaled)
• Draws some 3D-objects.
Number Sequence Upto 1000
• Reads and writes 3-digit numbers.
• Expands a number w.r.t. place values.
• Counts in different ways – starting from any number.
• Compares numbers.
• Forms greatest and smallest numbers using given digits.
3. ADDITION AND SUBTRACTION
• Adds and subtracts numbers by writing them vertically in the following two cases:
– without regrouping.
– with regrouping.
• Uses the place value in standard algorithm of addition and subtraction.
• Solves addition and subtraction problems in different situations presented through pictures and stories.
• Frames problems for addition and subtraction facts.
• Estimates the sum of, and difference between, two given numbers.
• Explains the meaning of multiplication (as repeated addition).
• Identifies the sign of multiplication.
• Constructs the multiplication tables of 2, 3, 4, 5 and 10
• Uses multiplication facts in situations.
• Multiplies two digit numbers using standard algorithm and Lattice multiplication algorithm.
• Explains the meaning of division from context of equal grouping and sharing.
• Relates division with multiplication.
• Completes division facts:
– by grouping
– by using multiplication tables.
6. MENTAL ARITHMETIC
• Adds and subtracts single digit numbers and two digit numbers mentally.
• Doubles two digit numbers mentally (result not exceeding two digits).
• Converts Rupee. to Paise using play money.
• Adds and subtracts amounts using column addition, and subtraction without regrouping.
• Makes rate charts and bills.
• Appreciates the need for a standard unit.
• Measures length using appropriate standard units of length by choosing between centimetres. and metres.
• Estimates the length of given object in standard units and verifies by measuring.
• Uses a ruler
• Relates centimetre. and metre.
• Weighs objects using non standard Units.
• Appreciates the conservation of weight.
• Measures and compares the capacity of different containers in terms of non-standard units.
• Appreciates the conservation of volume.
• Reads a calendar to find a particular day and date.
• Reads the time correct to the hour.
• Sequences the events chronologically.
9. DATA HANDLING
• Records data using tally marks.
• Collects data and represents in terms of pictograph choosing appropriate scale and unit for display through pictographs.
• Draws conclusions from the data by discussing with the teacher.
• Identifies simple symmetrical shapes and patterns.
• Makes patterns and designs from straight lines and other geometrical shapes.
• Identifies patterns in the numerals for odd and even numbers and in adding odd and even numbers.
• Partitions a number in different ways.
• Identifies patterns in his surroundings
• Identifies patterns in multiplication with, and dividing by 10s.
Syllabus of Maths class 2
• Reads and writes numerals for numbers up to ninetynine.
• Expands a number with respect to place values.
• Counts and regroups objects into tens and ones.
• Uses the concept of place value in the comparison of numbers.
• Counts in various ways:
– Starting from any number.
– Group counting etc.
• Arranges numbers upto hundred in ascending and descending order.
• Forms the greatest and the smallest two digit numbers with and without repetition of given digits.
• Indicates and identifies the position of an object in a line.
2. ADDITION AND SUBTRACTION
• Adds and subtracts two digit numbers by drawing representations of tens and ones without and with regrouping.
• Adds zero to a number and subtracts zero from a number.
• Observes the commutative property of addition through patterns.
• Solves addition, subtraction problems presented through pictures and verbal description.
• Describes orally the situations that correspond to the given addition and subtraction facts.
• Estimates the result of addition and subtraction and compares the result with another given number.
3. PREPARATION FOR MULTIPLICATION AND DIVISION
• Discussion of situations involving repeated addition and situations involving equal sharing.
• Activities of making equal groups.
4. MENTAL ARITHMETIC
• Adds and subtracts single digit numbers mentally.
• Adds and subtracts multiples of ten mentally.
• Measures lengths & distances along short & long paths
using uniform (non-standard) units, extends to longer
• Compares two or more objects by their weight.
• Appreciates the need for a simple balance.
• Compares weights of given objects using simple balance.
• Compares and orders containers in terms of internal volume(capacity).
• Orders given containers as per their capacities on the basis of perception & verifies by pouring out etc.
• Identifies currency – notes and coins.
• Puts together amounts of money not exceeding Rs 50/-.
• Adds and subtracts small amounts of money mentally.
• Transacts an amount using 3-4 notes.
• Gets familiar with the days of the week and months of the year.
• Gets a feel for sequence of seasons (varying locally).
• Sequences the events occurring over longer periods in terms of dates/days.
Shapes & Spatial Understanding 3-D and 2-D Shapes
• Observes objects in the environment and gets a qualitative feel for their geometrical attributes.
• Identifies the basic 3-D shapes such as cuboid, cylinder, cone, sphere by their names.
• Traces the 2-D outlines of 3-D objects.
• Observes and identifies these 2-D shapes.
• Identifies 2-D shapes viz., rectangle, square, triangle, circle by their names.
• Describes intuitively the properties of these 2-D shapes.
• Identifies and makes straight lines by folding, straight edged objects, stretched strings and draws free hand and with a ruler.
• Draws horizontal, vertical and slant lines (free hand).
• Distinguishes between straight and curved lines.
• Identifies objects by observing their shadows.
• Observes and extends patterns in sequence of shapes and numbers.
• Searches for patterns in different ways of splitting a number.
• Creates block patterns by stamping thumbprints, leaf prints, vegetable prints, etc.
• Creates patterns of regular shapes by stamping.
10. DATA HANDLING
• Collects data through measurement.
• Represents the data followed by discussion (e.g. heights of children).
• Collects and presents the data on birthdays.
• Draws inferences from the data at the appropriate level.
Syllabus of Maths class 1
Developing a Sense of Numberness, Counting and Operations of Numbers 1 – 9 and Zero
• Observes object and makes collections of objects.
• Arranges the collection of objects in order by
– Matching and
– One to one correspondence
• Counts the number of objects in a collection.
• Makes collection of objects corresponding to a specific number.
• Recognises and speaks numbers from 1 to 9.
• Uses numbers from 1 to 9 in counting and comparison. (Real objects and repeated events like clapping to be used for counting)
• Reads and writes numerals from 1 to 9.
• Adds and subtracts using real objects and pictures.
• Adds and subtracts the numbers using symbols ‘+’ and ‘-’.
• Approaches zero through the subtraction pattern (such as 3 – 1 = 2, 3 – 2 = 1, 3 – 3 = 0).
2. NUMBERS FROM (10 – 20)
• Forms Number sequence from 10 to 20.
• Counts objects using these numbers.
• Groups objects into a group of 10s and single objects.
• Develops the vocabulary of group of ‘tens’ and ‘ones’.
• Shows the group of tens and ones by drawing.
• Counts the number of tens and ones in a given number.
• Writes the numerals for eleven to nineteen.
• Writes numerals for ten and twenty.
• Compares numbers upto 20.
3. ADDITION AND SUBTRACTION (UPTO 20)
• Adds and subtracts numbers upto 20.
4. NUMBERS FROM 21 – 99
• Writes numerals for Twenty-one to Ninety-nine. Groups objects into tens and ones.
• Draws representation for groups of ten and ones.
• Groups a number orally into tens and ones.
5. MENTAL ARITHMETIC
• Adds two single digit numbers mentally.
• Describes sequences of simple patterns found in shapes in the surroundings and in numbers, e.g. stamping activity using fingers and thumb.
• Completes a given sequence of simple patterns found in shapes in the surroundings and in numbers.
Shapes & Spatial Understanding
• Develops and uses vocabulary of spatial relationship (Top, Bottom, On, Under, Inside, Outside, Above, Below, Near, Far, Before, After)
SOLIDS AROUND US
• Collects objects from the surroundings having different sizes and shapes like pebbles, boxes, balls, cones, pipes, etc.
• Sorts, Classifies and describes the objects on the basis of shapes, and other observable properties.
• Observes and describes the way shapes affect
• Sorts 2 – D shapes such as flat objects made of card etc.
• Distinguishes between near, far, thin, thick, longer/taller, shorter, high, low.
• Seriates objects by comparing their length.
• Measures short lengths in terms of non-uniform units (in the context of games e.g. ‘Gilli Danda’ and ‘marblegames’).
• Estimates distance and length, and verifies using nonuniform units (e.g. hand span etc.)
• Compares between heavy and light objects.
• Distinguishes between events occurring in time using terms -earlier and later.
• Gets the qualitative feel of long & short duration, of school days v/s holidays.
• Narrates the sequence of events in a day.
• Identifies common currency notes and coins.
• Puts together small amounts of money.
11. DATA HANDLING
• Collects, represents and interprets simple data such as measuring the arm length or circumference of the head using a paper strip.
(i) Rational Numbers:
- Properties of rational numbers. (including identities). Using general form of expression to describe properties
- Consolidation of operations on rational numbers.
- Representation of rational numbers on the number line
- Between any two rational numbers there lies another rational number (Making children see that if we take two rational numbers then unlike for whole numbers, in this case you can keep finding more and more numbers that lie between them.)
- Word problem (higher logic, two operations, including ideas like area)
- Integers as exponents.
- Laws of exponents with integral powers
(iii)Squares, Square roots, Cubes, Cube roots.
- Square and Square roots
- Square roots using factor method and division method for numbers containing (a) no more than total 4 digits and (b) no more than 2 decimal places
- Cubes and cubes roots (only factor method for numbers containing at most 3 digits)
- Estimating square roots and cube roots. Learning the process of moving nearer to the required number.
(iv) Playing with numbers
- Writing and understanding a 2 and 3 digit number in generalized form (100a + 10b + c , where a, b, c can be only digit 0-9) and engaging with various puzzles concerning this. (Like finding the missing numerals represented by alphabets in sums involving any of the four operations.) Children to solve and create problems and puzzles.
- Number puzzles and games
- Deducing the divisibility test rules of 2, 3, 5, 9, 10 for a two or three-digit number expressed in the general form.
(i) Algebraic Expressions
- Multiplication and division of algebraic exp.(Coefficient should be integers)
- Some common errors (e.g. 2 + x ≠ 2x, 7x + y ≠ 7xy )
- Identities (a±b)² = a² ± 2ab + b²
a²-b² = (a – b) (a + b)
Factorisation (simple cases only) as examples the following types a(x + y), (x±y)², a²-b², (x + a).(x + b)
- Solving linear equations in one variable in contextual problems involving multiplication and division (word problems) (avoid complex coefficient in the equations)
Ratio and Proportion
- Slightly advanced problems involving applications on percentages, profit & loss, overhead expenses, Discount, tax.
- Difference between simple and compound interest (compounded yearly up to 3 years or half-yearly up to 3 steps only), Arriving at the formula for compound interest through patterns and using it for simple problems.
- Direct variation – Simple and direct word problems
- Inverse variation – Simple and direct word problems
- Time & work problems– Simple and direct word problems
(i) Understanding shapes:
- Properties of quadrilaterals – Sum of angles of a quadrilateral is equal to 3600 (By verification)
- Properties of parallelogram (By verification)
(i)Opposite sides of a parallelogram are equal,
(ii) Opposite angles of a parallelogram are equal,
(iii) Diagonals of a parallelogram bisect each other. [Why (iv), (v) and (vi) follow from (ii)]
(iv) Diagonals of a rectangle are equal and bisect each other.
(v) Diagonals of a rhombus bisect each other at right angles.
(vi) Diagonals of a square are equal and bisect each other at right angles.
(ii) Representing 3-D in 2-D
- Identify and Match pictures with objects [more complicated e.g. nested, joint 2-D and 3-D shapes (not more than 2)].
- Drawing 2-D representation of 3-D objects (Continued and extended)
- Counting vertices, edges & faces & verifying Euler’s relation for 3-D figures with flat faces (cubes, cuboids, tetrahedrons, prisms and pyramids)
Construction of Quadrilaterals:
- Given four sides and one diagonal
- Three sides and two diagonals
- Three sides and two included angles
- Two adjacent sides and three angles
(i)Area of a trapezium and a polygon.
(ii) Concept of volume, measurement of volume using a basic unit, volume of a cube, cuboid and cylinder
(iii) Volume and capacity (measurement of capacity)
(iv) Surface area of a cube, cuboid, cylinder.
(i) Reading bar-graphs, ungrouped data, arranging it into groups, representation of grouped data through bar-graphs, constructing and interpreting bar-graphs.
(ii)Simple Pie charts with reasonable data numbers
(iii) Consolidating and generalising the notion of chance in events like tossing coins, dice etc. Relating it to chance in life events. Visual representation of frequency outcomes of repeated throws of the same kind of coins or dice.
Throwing a large number of identical dice/coins together and aggregating the result of the throws to get large number of individual events. Observing the aggregating numbers over a large number of repeated events. Comparing with the data for a coin. Observing strings of throws, notion of randomness
Introduction to graphs
(i) Axes (Same units), Cartesian Plane
(ii) Plotting points for different kind of situations (perimeter vs length for squares, area as a function of side of a square, plotting of multiples of different numbers, simple interest vs number of years etc.)
(iii) Reading off from the graphs
- Reading of linear graphs
- Reading of distance vs time graph
I. Relations and Functions
IV. Vectors and Three-Dimensional Geometry
V. Linear Programming
1. Proofs in Mathematics
2. Mathematical Modelling
1.1 Relations and Functions
1.2 Inverse Trigonometric Functions
3.1 Continuity and Differentiability
3.2 Applications of Derivatives
3.4 Applications of the Integrals
3.5 Differential Equations
4.2 Three-dimensional Geometry
5.1 Linear Programming
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 branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.
Unit II: Albegra
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Noncommutativity 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).
Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors 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 function. Concept of exponential and logarithmic functions and their derivatives. 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 interpretations.
2. Applications of Derivatives
Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normals, 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).
Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts
Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ ellipses (in standard form only), area between 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, 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.
Unit IV: Vectors and Three-Dimensional Geometry
Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. 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. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors.
2. Three-dimensional Geometry
Direction cosines/ratios of a line joining two points. Cartesian 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
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, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).
Unit VI: Probability
Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem. Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution.
1. Proofs in Mathematics
Through a variety of examples related to mathematics and already familiar to the learner, bring out different kinds of proofs: direct, contrapositive, by contradiction, by counter-example.
2. Mathematical Modelling
Modelling real-life problems where many constraints may really need to be ignored (continuing from Class XI). However, now the models concerned would use techniques/results of matrices, calculus and linear programming.
I. Sets and Functions
III. Coordinate Geometry
V. Mathematical Reasoning
VI. Statistics and Probability
1. Infinite Series,
2. Mathematical Modelling
1.2 Relations and Functions
1.3 Trigonometric Functions
2.1 Principle of Mathematical Induction
2.2 Complex Numbers and Quadratic Equations
2.3 Linear Inequalities
2.4 Permutations and Combinations
2.5 Binomial Theorem
2.6 Sequence and Series
3.1 Straight Lines
3.2 Conic Sections
3.3 Introduction to Three-dimensional
4.1 Limits and Derivatives
5.1 Mathematical Reasoning
Unit I: Sets and Functions
Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of the set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set.
2. Relations and Functions
Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the reals with itself (upto RxRxR).
Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain and range of a function. Real valued function of the real variable, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer functions with their graphs. Sum, difference, product and quotients of functions.
3. Trigonometric Functions
Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin2x + cos2x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x. General solution of trigonometric equations of the type sin θ = sin α, cos θ = cos α and tan θ = tan α. Proofs and simple applications of sine and cosine formulae.
Unit II: Algebra
1. Principle of Mathematical Induction
Processes of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications.
2. Complex Numbers and Quadratic Equations
Need for complex numbers, especially , to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system.
3. Linear Inequalities
Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables – graphically.
4. Permutations and Combinations
Fundamental principle of counting. Factorial n. Permutations and combinations, derivation of formulae and their connections, simple applications.
5. Binomial Theorem
History, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle, general and middle term in binomial expansion, simple applications.
6. Sequence and Series
Sequence and Series. Arithmetic progression (A. P.), arithmetic mean (A.M.). Geometric progression (G.P.), general term of a G. P., sum of n terms of a G.P., geometric mean (G.M.), relation between A.M. and G.M. Sum to n terms of the special series:
Unit III: Coordinate Geometry
1. Straight Lines
Brief recall of 2D from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axes, point-slope form, slope-intercept form, two-point form, intercepts form and normal form. General equation of a line. Distance of a point from a line.
2. Conic Sections
Sections of a cone: Circles, ellipse, parabola, hyperbola, a point, a straight line and pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.
3. Introduction to Three-dimensional Geometry
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula.
Unit IV: Calculus
Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
Unit V: Mathematical Reasoning
Mathematically acceptable statements. Connecting words/phrases — consolidating the understanding of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”, “or”, “there exists” and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words – difference between contradiction, converse and contrapositive.
Unit VI: Statistics and Probability
Measure of dispersion; mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.
Random experiments: Outcomes, sample spaces (set representation). Events: Occurrence of events, ‘not’, ‘and’ & ‘or’ events, exhaustive events, mutually exclusive events. Axiomatic (set theoretic) probability, connections with the theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ & ‘or’ events.
1. Infinite Series
Binomial theorem for any index, infinite geometric series, exponential and logarithmic series.
2. Mathematical Modelling
Consolidating the understanding developed up to Class X. Focus on modelling problems related to real-life (like environment, travel, etc.) and connecting with other subjects of study where many constraints may really need to be ignored, formulating the model, looking for solutions, interpreting them in the problem situation and evaluating the model.
I. Number Systems
III. Coordinate Geometry
VI. Statistics and Probability
1. Proofs in Mathematics,
2. Introduction to Mathematical Modelling.
Unit I: Number Systems
Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating/non-terminating recurring decimals, on the number line through successive magnification. Rational numbers as recurring/terminating decimals.
Examples of nonrecurring/non terminating decimals such as √2, √3, √5 etc. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line, and conversely, every point on the number line represents a unique real number.
Existence of √x for a given positive real number x (visual proof to be emphasized). Definition of nth root of a real number
Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws)
Unit II: Algebra
Definition of a polynomial in one variable, its coefficients, with examples and counter examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and multiples. Zeros/roots of a polynomial/equation. State and motivate the Remainder Theorem with examples and analogy to integers. Statement and proof of the Factor Theorem. Factorisation of ax² + bx + c, a ≠ 0 where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem
Linear Equations in Two Variables
Recall of linear equations in one variable. Introduction to the equation in two variables. Prove that a linear equation in two variables has infinitely many solutions, and justify their being written as ordered pairs of real numbers, plotting them and showing that they seem to lie on a line. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.
Unit III: Coordinate Geometry
The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane, graph of linear equations as examples; focus on linear equations of the type ax + by + c = 0 by writing it as y =mx + c and linking with the chapter on linear equations in two variables.
Unit IV: Geometry
1. Introduction to Euclid’s Geometry
History – Euclid and geometry in India. Euclid’s method of formalizing observed phenomenon into rigorous mathematics with definitions, common/obvious notions, axioms/postulates, and theorems. The five postulates of Euclid. Equivalent versions of the fifth postulate. Showing the relationship between axiom and theorem.
1. Given two distinct points, there exists one and only one line through them.
2. (Prove) Two distinct lines cannot have more than one point in common.
2. Lines and Angles
1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse.
2. (Prove) If two lines intersect, the vertically opposite angles are equal.
3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines.
4. (Motivate) Lines, which are parallel to a given line, are parallel.
5. (Prove) The sum of the angles of a triangle is 180°.
6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence).
2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).
3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).
4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle.
5. (Prove) The angles opposite to equal sides of a triangle are equal.
6. (Motivate) The sides opposite to equal angles of a triangle are equal.
7. (Motivate) Triangle inequalities and relation between ‘angle and facing side’; inequalities in a triangle.
1. (Prove) The diagonal divides a parallelogram into two congruent triangles.
2. (Motivate) In a parallelogram opposite sides are equal and conversely.
3. (Motivate) In a parallelogram opposite angles are equal and conversely.
4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.
5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.
6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and (motivate) its converse.
Review concept of area, recall area of a rectangle.
1. (Prove) Parallelograms on the same base and between the same parallels have the same area.
2. (Motivate) Triangles on the same base and between the same parallels are equal in area and its converse.
Through examples, arrive at definitions of circle related concepts, radius, circumference, diameter, chord, arc, subtended angle.
1. (Prove) Equal chords of a circle subtend equal angles at the centre and (motivate) its converse.
2. (Motivate) The perpendicular from the centre of a circle to a chord bisects the chord and conversely, the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
3. (Motivate) There is one and only one circle passing through three given non-collinear points.
4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the centre(s) and conversely.
5. (Prove) The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
6. (Motivate) Angles in the same segment of a circle are equal.
7. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.
8. (Motivate) The sum of the either pair of the opposite angles of a cyclic quadrilateral is 1800 and its converse.
1. Construction of bisectors of a line segment and angle, 60°, 90°, 45° angles etc, equilateral triangles.
2. Construction of a triangle given its base, sum/difference of the other two sides and one base angle.
3. Construction of a triangle of given perimeter and base angles.
Unit V: Mensuration
Area of a triangle using Heron’s formula (without proof) and its application in finding the area of a quadrilateral.
2. Surface Areas and Volumes
Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right circular cylinders/cones.
Unit VI: Statistics and Probability
Introduction to Statistics: Collection of data, presentation of data – tabular form, ungrouped/ grouped, bar graphs, histograms (with varying base lengths), frequency polygons, qualitative analysis of data to choose the correct form of presentation for the collected data. Mean, median, mode of ungrouped data.
History, Repeated experiments and observed frequency approach to probability. Focus is on empirical probability. (A large amount of time to be devoted to group and to individual activities to motivate the concept; the experiments to be drawn from real-life situations, and from examples used in the chapter on statistics).
1. Proofs in Mathematics
What a statement is; when is a statement mathematically valid. Explanation of axiom/ postulate through familiar examples. Difference between axiom, conjecture and theorem. The concept and nature of a ‘proof’ (emphasize deductive nature of the proof, the assumptions, the hypothesis, the logical argument) and writing a proof. Illustrate deductive proof with complete arguments using simple results from arithmetic, algebra and geometry (e.g., product of two odd numbers is odd etc.). Particular stress on verification not being proof. Illustrate with a few examples of verifications leading to wrong conclusions – include statements like “every odd number greater than 1 is a prime number”. What disproving means, use of counter examples.
2. Introduction to Mathematical Modelling
The concept of mathematical modelling, review of work done in earlier classes while looking at situational problems, aims of mathematical modelling, discussing the broad stages of modelling – real-life situations, setting up of hypothesis, determining an appropriate model, solving the mathematical problem equivalent, analyzing the conclusions and their real-life interpretation, validating the model. Examples to be drawn from ratio, proportion, percentages, etc.
1. Real Numbers
- Application of H.C.F. and L.C.M.
- Polynomial and its Types
- Value of Polynomial and Division Algorithm
- Geometrical Representation of Zeroes of a Polynomial
3. Pair of Linear Equations in Two Variables
- Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations in Two Variables
- Solution of Linear Equations in Two Variables
4. Quadratic Equations
- Quadratic Equations
- Solving Quadratic Equation
- Application of Quadratic Equations
- Nature of Roots
5. Arithmetic Progressions
- Arithmetic Progression
- Basic Proportionality Theorem and Equal Intercept Theorem
- Similarity of Triangles
- Pythagoras Theorem and its Application
7. Coordinate Geometry
- Distance Formula
- Section Formula
- Areas of Triangle and Quadrilateral
8. Introduction to Trigonometry
- Trigonometric Ratios
- Trigonometric Identities
9. Some Applications of Trigonometry
- Heights and Distances
- Tangents to the Circle
- Construction Related to Segment and Triangles
- Construction of Tangent to a Circle
12. Areas Related to Circles
- Perimeter and Area of Circle and Semicircle
- Area and Perimeter Related to Arcs of a Circle
13. Surface Areas and Volumes
- Frustum of Cone
- Combination of Solids
- Cumulative Frequency Curve
- Event and its Types
CHAPTER 1: Visualising Solid Shapes
1.1 Visualising Solid Shapes
CHAPTER 2: Symmetry
2.1 Rotational Symmetry
2.2 Lines of Symmetry
CHAPTER 3: Exponents and Powers
3.1 Laws of Exponents
CHAPTER 4: Algebraic Expressions
4.1 Operations on Algebraic Expressions
4.2 Monomials, Binomials, Trinomials and Polynomials
4.3 Expressions and Its Parts
CHAPTER 5: Perimeter and Area
5.2 Parallelogram and Triangles
5.3 Squares and Rectangles
CHAPTER 6: Practical Geometry
CHAPTER 7: Rational Numbers
7.1 Operations on Rational Numbers
7.2 Rational Numbers
CHAPTER 8: Comparing Quantities
8.1 Uses of Percentage
8.2 Comparison Using Percentage
CHAPTER 9: Congruence of Triangles
9.1 Congruence of Triangles
CHAPTER 10: The Triangle and its Properties
10.1 Right-angled Triangle
10.2 Equilateral and Isosceles Triangles
10.3Basics of Triangles
CHAPTER 11: Lines and Angles
11.1 Pair of Lines
11.2 Related Angles
11.3 Angles and its Types
CHAPTER 12: Simple Equations
12.1 Simple Equations
CHAPTER 13: Data Handling
13.1 Chance and Probability
13.2 Bar Graphs
13.3 Median and Mode
13.4 Arithmetic Mean
CHAPTER 14: Fractions and Decimals
14.1 Multiplication and Division of Decimal Numbers
14.2 Decimal Numbers
14.3 Division of Fractions
14.4 Multiplication of Fractions
CHAPTER 15: Integers
15.1 Operations on Integers
15.2 Basics of Integers