Sets and their representations

• identify sets as well defined collections.
• represent sets in roster and set builder form.
• identify the symbols and and understand the difference between the two.
• Conversion from set builder form to roster form and vice versa.

Empty Set

• • Identify empty sets (null sets).

Singleton Set

• Identify singleton set and frame examples

Finite and infinite Sets

• identify finite and infinite sets; and their respective representations.

Equivalent and Equal Sets

• understand meaning of equal and equivalent sets.
• differentiate between equal and equivalent sets.
• determine whether the given pair of sets is equal or not.

Subsets

• identify the subsets of a given set and its symbol ( )
• understand that every set has two trivial subsets - null set and the set itself.
• understand the difference between a subset and proper subset.

Power Set

• identify power set as set of subsets.

Universal Set

• identify universal set and its symbol ( ).

Complement of a Set

• find the complement of a subset of a given set, within a given universe.

Intervals as Subsets of R

• closed interval, open interval, right half open interval, left half open interval.

Venn Diagrams

• represent sets using venn diagrams.

Union and Intersection of Sets

• find the intersection of sets and union of sets.
• show the intersection and union of sets using Venn diagrams.
• identify disjoint sets and its representation using venn diagram.

Difference of Sets

• find the difference of sets and their representation using venn diagram.

Laws of Operations on Sets

• apply the following laws of algebra on sets:
  • Laws of union of sets (commutative law, associative law, idempotent law, identity law)
  • laws of intersection of sets
  • distributive laws
• De Morgan's law

Properties of Complement Sets

• apply properties of complement sets

Practical Problems on union and Intersection of Sets

• solve practical problems on union and intersection of sets.
• apply results and solve problems on number of elements of sets using properties like.

Ordered Pairs

• identify the equality of two ordered pairs.
• identify an ordered pair.

Cartesian Product of Sets

• identify a cartesian product of two non empty sets.
• identify the two sets given their cartesian product.
• find the union and intersection on cartesian products.
• find ordered triplets (R R R).
• identify the number of elements in the cartesian product of two finite sets.
• identify Cartesian product of set of all real numbers with itself.

Definition of Relation

• understand relation of two sets as a subset of their cartesian product.

Arrow Diagram

• pictorial representation of a relation between two sets.

Domain, Codomain and Range of a Relation

• identify domain, co-domain and range of a relation.

Function as a Special Kind of Relation from one Set to another

• identify function as a special kind of relation from one set to another.
• determine when a relation is a function.
• describe and write functional relationships for given problem situations.
• understand that f c R c A X A.

Pictorial representation of a Function

• represent functions using graphs.
• to understand that every graph does not represent a function.

Domain, Codomain and Range of a Function

• identify domain, co-domain and range of a function.
• finding domain and range of a given function.
• identify even and odd functions.
• find specific function values
• find the algebra of functions covering:
  (f g)(x) = f(x) g(x) = g(x) f(x).

Real valued functions and their graphs

• recognise the following real valued functions
  • constant function
  • identity function
  • linear function
  • quadratic function
  • polynomial function
  • rational function
  • modulus function
  • signum function
  • greatest integer function

Positive and negative angles

• identify positive and negative angles.

Measuring angles in radians and in degrees and conversion from one measure to another

• measure angles in both degrees and in radians, and convert between these measures.

Definition of trigonometric functions with the help of unit circle

• define trigonometric functions with the help of unit circle.

Sign of trigonometric functions

• identify the change of signs of trigonometric functions in different quadrants.
• develop and apply the value of trigonometric functions at 0, /6,/4, /3, /2 radians and their multiples*.
• use the reciprocal and co-function relationships to find the values of the secant, cosecant and cotangent 0, /6, /4, /3, /2 radians value of trigonometric functions at n , where n is a positive integer

Domain and range of trigonometric functions

• identify the domain and range of trigonometric functions.

Trigonometric functions as periodic functions, their amplitude, argument, period and graph

• identify trigonometric functions as periodic functions with sine and cosine functions having a period of 2 , tangent and cotangent functions having a period of , secant and cosecant functions having a period of 2 .
• construct the graphs of trigonometric functions and describe their behaviour, including periodicity, amplitude, zeros and symmetry.

Trigonometric functions of sum and difference of two angles

• express sin (x ± y) and cos (x ± y) in terms of sin x, sin y, cos x and cos y.

Express sum and difference of T-Functions as the product of T-ratios

Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x

• deduce identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x, and apply them to simplify trigonometric equations.

General solution of trigonometric equations of the type sin = sin , cos = cos and tan = tan

• find the general solution of the trigonometric equations of the type sin = sin , cos = cos and tan = tan .

Proof and simple applications of sine and cosine rules

• prove the law of sines and law of cosine.
• solve for an unknown side or angle, using the law of sines or the law of cosine.
• apply law of sines and law of cosine in various problems.
• determine the area of a triangle or parallelogram, given the measure of two sides and the included angle.

Need for complex number,especially iota to be motivated by inability to solve some of quadratic equation

• understand the need of Imaginary Quantities.
• understand the concept of iota and its application.

Standard form of complex number

• define a complex number (z = a+ib) and identify its real and imaginary parts
• concept of purely real and purely imaginary complex number
• get familiar with equality of complex numbers
• understand the addition and subtraction of complex numbers and its properties.

Modulus and conjugate of complex number

• identify the conjugate of a complex number and familiarized with its properties
• identify the modulus of a complex number and familiarized with its properties.

Multiplication and division of complex numbers

• understand the multiplication of complex numbers and its properties.
• understand the division of complex numbers and its properties.
• identify the multiplicative inverse or reciprocal of a complex number.

Polar representation Of complex number Polar Representation of Complex number

• understand the polar or trigonometrical form of a complex number.
• find the modulus of a complex number.
• find the argument of a complex number.

Argand Plane

• geometrical representation of a complex number.
• understand different properties of complex numbers and its representation on argand plane.
• solve different mathematical problems using argand plane.

Statement of Fundamental theorem of algebra

• get familiar with fundamental theorem of algebra.

Square root of a complex number

• find the square root of a complex number.

Solution of quadratic equations in the complex number system

• solve the quadratic equations in the complex number system.

Cube root of unity and its properties

• familiar with cube roots of unity and their properties.

De Moivre's theorem

• prove and apply De-Moivre's theorem.

Linear Inequations

• understand linear inequalities.

Algebraic solutions of linear inequations in one variable

• find algebraic solutions of linear inequalities in one variable.
• represent the solution of linear inequalities in one variable on a number line.
• simultaneous solution of two linear inequalities algebraically as well as on number line.

Algebraic solutions of linear in equations in two variables

• find algebraic solutions of linear inequalities in two variables.

Graphical solution of linear inequations in two variables

• Solution of linear inequality in two variables and the graph of its solution set.
• Solution of system of linear inequalities in two variables and the graph of its solution set.

Inequations solving modulus functions

• inequalities involving modulus function.
• understand wavy curve method for 2nd degree and higher degree polynomials expressed in the form (x+a)(x+b) ...... (the number of such terms corresponding to the degree of the polynomial).

Fundamental principles of counting

• know the fundamental addition principles of counting and apply it to find out number of ways particular event can occur.
• know the fundamental multiplication principles of counting and apply it to find out number of ways particular event can occur.

Factorial n (n!)

• know the meaning of factorial and its symbol.
• know how to compute factorial.
• know how to represent product of consecutive numbers in factorial.
• know how to represent product of consecutive numbers in factorial.

Permutation

Combinations

Derivation of properties of combination

Types of permutations

• linear permutations.
• circular permutation.
• restricted permutation.
• permutations when particular thing is to be included everytime.
• permutations when particular thing is never to be included.
• permutation of objects are not all different Permutation with repetition.

Simple applications

• solve the simple practical problems on permutation.
• solve the simple practical problems on combination.

Pascal's Triangle

• get familiar with the Pascal's triangle.
• observe different patterns of numbers followed in pascals triangle.

History, statement and proof of the binomial theorems for positive integral indices

• know the binomial theorems for positive integral indices and their proof.
• expand an expression using binomial theorem.

General and middle term in binomial expansion

• get familiar with the general term in binomial expansion.
• get familiar with middle term in binomial expansion when number of terms are even/odd.
• get familiar with pth term from the end.

Application of binomial theorem

• compute simple application problems using binomial theorems.

Arithmetic Progression, Geometric Progression

• identify an arithmetic or geometric sequence.
• find the formula for the nth term of an arithmetic sequence.
• find the formula for the nth term of a geometric sequence.
• prove a given sequence from an arithmetic progression or a geometric progression.
• determine a specified term of an arithmetic sequence.
• determine a specified term of a geometric sequence.
• generate or construct sequences from given recursive relationships.

Arithmetic Mean

• find the arithmetic mean.
• insert 'n' arithmetic means between 2 given numbers.

Geometric Mean

• find the geometric mean.
• insert 'n' geometric means between 2 given numbers.

Sum to n terms of an A.P.

• find the sum of finite terms of an arithmetic progression.

Sum to n terms of a G.P.

• find the sum of finite terms of a geometric progression.

Infinite G.P. and its sum

• find the sum of an infinite geometric progression.

Relation between A.M. and G.M.

• identify and apply the relation between arithmetic mean and geometric mean.

Brief recall of two dimensional geometry from earlier classes

• distance between two points.
• area of triangle whose vertices are given.
• co-ordinates of a point divides the join of two given coordinates in the particular ratio.
• co-ordinates of midpoint of a line segment joining two coordinates.
• co-ordinates of centroid and incenter of a triangle.

Shifting of origin

• comprehend the change in equation on shifting the point of origin.

Slope of a line

• find the slope of a line when angle of inclination is given.
• identify the slope of a line in terms of co-ordinates of any two points on it.
• familiar with condition of parallel lines and perpendicular lines in terms of slope.
• use slopes of lines to investigate geometric relationships, including parallel lines, perpendicular lines.

Angle between two lines

• Angle between two lines.

Various forms of equation of a line: parallel to axis, point slop form, slopintercept form, two point form, intercept form, and normal form

• equation of lines parallel to the co-ordinate axis.
• form the equation of line when co-ordinates of point through which line passes and slope is given (point-slope form).
• form the equation of line when co-ordinates of two points through which line passes are given (two point form).
• familiar with intercepts of a line on the axes.
• form the equation of line making slope m and making an intercept c on y/x axis (slope intercept form).
• form the equation of line when a line cuts off intercepts a & b respectively on x and y axis (intercept form).
• form the equation of line when the length of the perpendicular on it and angle of that perpendicular is given (normal form of line).
• use different forms of a line to find out missing parameters of a line in symmetric form.

General equation of a line

• identify general equation and transform it in different standard forms.

Equation of family of lines passing through the point of intersection of two lines

• find the point of intersection of two lines.
• understand the concept of family of lines passing through the intersection of lines ll and l2 in terms of l1 + k l2 = 0.
• give the equation of lines passing through the point of intersection of two lines under given conditions.

Distance of a point from a line

• compute the distance of a point from a line.

Distance between parallel lines

• compute the distance between parallel lines.

Introduction to section of a cone

• identify the circle, parabola, ellipse and hyperbola as cross sections of a double napped cone by a plane.

Circle (Standard form)

• identify the equation of a circle in standard form having the Centre (h, k) and radius r.
• equation of a circle having centre at origin and radius r.
• equation of a circle when the end points of a diameter are given.

Circle (general form)

Parabola (standard form)

• identify the standard parabola (right handed, left handed, upward and downward parabola).
• find the axis, vertex, focus, directrix and the latus rectum of the standard parabola.

Parabola (general form)

• identify the general equation of a parabola.
• reduction of general form of parabola to the standard form.
• find the axis, vertex, focus, directory and the latus rectum from the general equation of the parabola.
• find the equation of parabola under given condition.

Ellipse (standard form) horizontal & vertical ellipse

• identify the vertical and horizontal ellipse.
• find the vertices, major and minor axis, foci, directrix, centre, eccentricity and latus rectum of the vertical and horizontal ellipse.

Ellipse (general form)

• identify the general form of an ellipse (vertical & horizontal).
• reduction of general form of ellipse to the standard form.
• find the vertices, major and minor axis, foci, directrix, centre, eccentricity and latus rectum from the general from of ellipse.
• find the equation of an ellipse under given conditions.

Hyperbole (standard form)

• identify the hyperbola in standard form (also conjugate hyperbola).
• find the centre, vertices, foci, directrix, transverse and conjugate axes, eccentricity and length of latus rectum.

Hyperbole (general form)

• identify the general form of hyperbole.
• reduction of general form of hyperbola to standard form.
• find the centre, vertices, foci directrix, transverse and conjugate axes, eccentricity & latus rectum from the general equation of hyperbola.
• find the equation of hyperbole under given condition.

Application of conic section

• apply the concepts of parabola, ellipse and hyperbola in the given problems.

Co-ordinate axes and co-ordinate planes in three dimensions

• identify co-ordinate axes in three dimensions.
• identify co-ordinate planes in three dimensions.
• find co-ordinates of a point in space.

Distance between two points and section formula

• find distance between two points.
• apply section formula.

Some results on line in space

• direction cosines of a line.
• direction ratios of a line.
• angle between two lines.

Random experiment: outcomes, sample spaces (set representation)

• learn the concept of random experiment, outcomes of random experiment and sample spaces.
• list the sample spaces of a random experiment.

Events: occurrence of events, 'or', 'and', & 'not' events

• understand the term event as a subset of sample space.
• write events/sample space for a given experiment.
• recognise 'or', 'and' & 'not' events.

Exhaustive events, mutually exclusive events Axiomatic (set theoretic) probability

• identify impossible events and sure events.
• Identify simple and compound events.
• identify mutually exclusive events.
• identify exhaustive events.
• get familiar with independent events, equally likely events, and complementary events*.

Probability of an event

• find the probability of occurrence of an event.

Odds of an event

• Odds in favour of an event.
• Odds against an event.

Probability of occurrence of a complementary events

• Find the probability of complement of an event using the relation P(E) = 1 – P( E).