## Units and Topics

# | Unit/Topic | Marks |
---|---|---|

100 | Sets and Functions | - |

200 | Algebra | - |

300 | Coordinate Geometry | - |

400 | Calculus | - |

500 | Mathematical Reasoning | - |

600 | Statistics and Probability | - |

Total | - |

## Syllabus

- Concept of Angle
- Part of Angle - Initial Side,Terminal Side,Vertex
- Types of Angle - Positive and Negative Angles
- Measuring Angles in Radian
- Measuring Angles in Degrees
- initial side, terminal side, vertex,positive angle, negative angle
- Degree measure
- Radian measure
- Relation between radian and real numbers
- Relation between degree and radian

- Introduction of Trigonometric Functions
- Trigonometric Functions with the Help of Unit Circle

- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Domain and Range of Trignometric Functions and Their Graphs

- Trigonometric Functions of Sum and Difference of Two Angles
- Identities Related to Sin 2x, Cos2x, Tan 2x, Sin3x, Cos3x and Tan3x.
- Deducing the Identities
- Deducing the identities like the following:-

`tan(x+-y)=(tanx+-tany)/(1+-tanxtany)", "cot(x+-y)=(cotxcoty+-1)/(coty+-cotx)`

`sinalpha+-sinbeta=2"sin"1/2(alpha+-beta)"cos"1/2(alpha+-beta)`

`cosalpha+cosbeta=2"cos"1/2(alpha+beta)"cos"1/2(alpha-beta)`

`cosalpha-cosbeta=-2"sin"1/2(alpha+beta)"sin"1/2(alpha-beta)`

- Trigonometric Equations
- Trigonometric Functions
- Truth of the Identity
sin

^{2}x+cos^{2}x=1, for All X.

- Truth of the Identity
- Negative Function Or Trigonometric Functions of Negative Angles
- 90 Degree Plusminus X Function
- Conversion from One Measure to Another
- 180 Degree Plusminus X Function
- 2X Function
- 3X Function
- Graphs of Trigonometric Functions
- Transformation Formulae
- Values of Trigonometric Functions at Multiples and Submultiples of an Angle
- Sine and Cosine Formulae and Their Applications

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. Domain and range of trignometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications.Deducing the identities like the following:

`tan(x+-y)=(tanx+-tany)/(1+-tanxtany),`

`sinalpha+-sinbeta=2sin(1/2)(alpha+-beta)cos(1/2)(alpha+-beta)`

`cosalpha+cosbeta=2cos(1/2)(alpha+beta)cos(1/2)(alpha-beta)`

`cosalpha-cosbeta=-2sin(1/2)(alpha+beta)sin(1/2)(alpha-beta)`

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tanJx. General solution of trigonometric equations of the type siny = sina, cosy= cosa and tany = tana.

- Cartesian Product of Sets
- Number of Elements in the Cartesian Product of Two Finite Sets
- Cartesian Product of set of the Reals with Itself

- Relation
- Definition of Relation
- Domain
- Co-domain and Range of a Relation

- Concept of Functions
- Function, Domain, Co-domain, Range
- Types of function

1. One-one or One to one or Injective function

2. Onto or Surjective function - Representation of Function
- Graph of a function
- Value of funcation
- Some Basic Functions - Constant Function, Identity function, Power Functions, Polynomial Function, Radical Function, Rational Function, Exponential Function, Logarithmic Function, Trigonometric function

- Exponential Function
Domain and range of this function

- Logarithmic Functions
Concept of Logarithmic Functions

Domain and range of this function

- Some Functions and Their Graphs
- Identity function - Domain and range of this function
- Constant function - Domain and range of this function
- Polynomial function -Domain and range of this function
- Rational functions - Domain and range of this function
- The Modulus function - Domain and range of this function
- Signum function - Domain and range of this function
- Greatest integer function

- Algebra of Real Functions
- Sum, Difference,Product and Quotient of Function
- Addition of two real functions
- Subtraction of a real function from another
- Multiplication by a scalar
- Multiplication of two real functions
- Quotient of two real functions

- Sets
- Concept of Relations
- Pictorial Representation of a Function
Domain, Co-domain and Range of a function

- Pictorial Representation of a Function
- Graph of Function
- Brief Review of Cartesian System of Rectanglar Co-ordinates

Ordered pairs, Cartesian product of sets.Number of elements in the cartesian product of two finite sets.

Cartesian product of the set of reals with itself (upto R x R x R). Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotient of functions.

- Sets and Their Representations
1) Roster or Tabular method or List method

2) Set-Builder or Rule Method

3) Venn Diagram - The Empty Set
- Empty Set
- null set or the void set

- Finite and Infinite Sets
- Equal Sets
- Subsets
- Subsets of a Set of Real Numbers Especially Intervals - With Notation
- Singleton Set
- Super Set
- Subsets of set of real numbers
- Intervals as subsets of R

- Power Set
- Universal Set
- Venn Diagrams
- Operations on Sets
- Union Set
- Some Properties of the Operation of Union

- Intersection of Sets
- Some Properties of Operation of Intersection

- Difference of Sets
- “ A minus B” Or "A – B"

- Union Set
- Complement of a Set
- Properties of Complement Sets

- Practical Problems on Union and Intersection of Two Sets
- Proper and Improper Subset
- Open and Close Intervals
- Operation on Set - Disjoint Sets
- Element Count Set

Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set of real numbers especially intervals (with notation). Power set. Universal Set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement.

- Introduction of Binomial Theorem
- History of Binomial Theorem

- Binomial Theorem for Positive Integral Indices
- Statement and Proof of the Binomial Theorem for Positive Integral Indices
- Proof of Binomial Therom by Induction
- Special Case in Binomial Therom
- Pascal's Triangle
- Binomial theorem for any positive integer n
- Some special cases-(In the expansion of (a + b)
^{n})

- General and Middle Terms
- Proof of Binomial Therom by Pattern
- Proof of Binomial Therom by Combination
- Rth Term from End
- 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.

- Concept of Sequences
- Finite sequence
- Infinite sequence
- Progression

- Concept of Series
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- N
^{th}Term of Geometric Progression (G.P.) - General Term of a Geometric Progression (G.P.)
- Sum of First N Terms of a Geometric Progression (G.P.)
- Sum of infinite terms of a G.P.
- Geometric Mean (G.M.)

- N
- Relationship Between A.M. and G.M.
- Relation Between Arithematic Mean (A.M.) and Geometric Mean (G.M.)

- Sum to N Terms of Special Series
- 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 first n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Formulae for the following special sums.

- Inequalities - Introduction
- Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation
- Graphical Solution of Linear Inequalities in Two Variables
Linear Inequalities - Graphical Representation of Linear Inequalities in Two Variables

- Solution of System of Linear Inequalities in Two Variables

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical representation of linear inequalities in two variables. Graphical method of finding a solution of system of linear inequalities in two variables.

- Concept of Complex Numbers
- Imaginary number
- Complex Number

- Algebra of Complex Numbers - Equality
Equality of Complex number

- Need for Complex Numbers
Need for complex numbers, especially √−1, to be motivated by inability to solve some of the quardratic equations

- Algebra of Complex Numbers
- Equality of two Complex Numbers
- Conjugate of a Complex Number
- Properties of `barz`
- Addition of complex numbers - Properties of addition, Scalar Multiplication
- Subtraction of complex numbers - Properties of Subtraction
- Multiplication of complex numbers - Properties of Multiplication
- Powers of i in the complex number
- Division of complex number - Properties of Division

- The Modulus and the Conjugate of a Complex Number
- Modulus of Complex Number
- Conjugate of Complex Number

- Argand Plane and Polar Representation
- Representation of Complex Number - Argand Plane Representation
- Representation of Complex Number - Polar Representation of Complex Numbers

- Quadratic Equations

Need for complex numbers, especially ../-1, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients) in the complex number system. Square root of a complex number.

- Fundamental Principle of Counting
- Concept of Permutations
- Concept of Combinations
- Introduction of Permutations and Combinations
- Permutation Formula to Rescue and Type of Permutation
- Without repetition
- With repetition

- Smaller Set from Bigger Set
- Derivation of Formulae and Their Connections
Derivation of formulae for

^{n}P_{r }and^{n}C_{r}and their connections - Introduction of Permutations and Combinations
- Factorial N (N!) Permutations and Combinations

Fundamental principle of counting. Factorial n (n!) Permutations and combinations, derivation of

formulae for ^{n}p_{r} and ^{n}c_{r},. and their connections, simple applications.

- Motivation
- Motivating the Application of the Method by Looking at Natural Numbers as the Least Inductive Subset of Real Numbers

- Principle of Mathematical Induction

Process 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.

- Slope of a Line
- Slope of a Line Or Gradient of a Line.
- Parallelism of Line
- Perpendicularity of Line in Term of Slope
- Collinearity of Points
- Slope of a line when coordinates of any two points on the line are given
- Conditions for parallelism and perpendicularity of lines in terms of their slopes
- Angle between two lines
- Collinearity of three points

- Various Forms of the Equation of a Line
- General Equation of a Line
- Different forms of Ax + By + C = 0 - Slope-intercept form, Intercept form, Normal form

- Distance of a Point from a Line
- Introduction of Distance of a Point from a Line
- Distance between two parallel lines

- Straight Lines
- Shifting of Origin

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slopeintercept form, two-point form, intercept form and normal form. General equation of a line.Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line.

- Three - Dimensional Geometry
- Coordinate Axes and Coordinate planes
Coordinate Axes and Coordinate Planes in Three Dimensions

- Distance Between Two Points
- Distance Between Two Points in 3-D Space

- Coordinate Axes and Coordinate planes
- Three Dimessional Space

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula.

- Sections of a Cone
- Concept of Circle
- Introduction of Parabola
- Parabola
- Introduction of Ellipse
- Ellipse
- Latus Rectum
- Latus Rectum in Ellipse

- Latus Rectum
- Introduction of Hyperbola
- Hyperbola
- Circle

Sections of a cone: circle, ellipse, parabola, hyperbola, a point, a straight line and a 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.

- Intuitive Idea of Derivatives
- Introduction of Limits
- Introduction to Calculus
- Differential Calculus
- Integral calculus

- Concept of Limits
- Limits of Trigonometric Functions
- Introduction of Derivatives
- Derivative
- Theorem for Any Positive Integer n
- Graphical Interpretation of Derivative
- Derive Derivation of x^n

`x^n`

Derivative introduced as rate of change both as that of distance function and geometrically.

lntuitive idea of limit. Limits of polynomials and rational functions, trigonometric, exponential and logarithmic functions. 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.

- Mathematically Acceptable Statements
- New Statements from Old
- Special Words Or Phrases
- Implications
- Introduction of Validating Statements
- Validating the Statements Involving the Connecting Words
- statement “p and q”, Statements with “Or”, Statements with “If-then”, Statements with “if and only if ”

- Validation by Contradiction
- Mathematical Reasoning
- Consolidating the Understanding
"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

- Consolidating the Understanding

- 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.

- Measures of Dispersion
- Concept of Range
- Measures of Dispersion - Range

- Mean Deviation
- Introduction of Variance and Standard Deviation
- Variance and Standard Deviation
- Introduction of Analysis of Frequency Distributions
- Analysis of Frequency Distributions
- Statistics
- Central Tendency - Mean
- Central Tendency - Median
- Concept of Mode
- Standard Deviation - by Short Cut Method

Measures of dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.

- Random Experiments
- Introduction of Event
- Event
- Types of Events
- Simple or elementary event
- Occurrence and non-occurrence of event
- Sure Event
- Impossible Event
- Complimentary Event

- Exhaustive Events
- Types of Event - Exhaustive Events

- Mutually Exclusive Events
- Types of Event - Mutually Exclusive Events

- Types of Events
- Axiomatic Approach to Probability
- Probability

Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, 'not', 'and' and 'or' events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories studied in earlier classes. Probability of an event, probability of 'not', 'and' and 'or' events.