#### Topics

##### Sets

- Sets and Their Representations
- Empty Set (Null or Void Set)
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Intrdouction of Operations on Sets
- Union of Sets
- Intersection of Sets
- Difference of Sets
- Complement of a Set
- Practical Problems on Union and Intersection of Two Sets
- Proper and Improper Subset
- Open and Close Intervals
- Disjoint Sets
- Element Count Set

##### Mathematical Reasoning

- Mathematically Acceptable Statements
- New Statements from Old
- Special Words Or Phrases
- Contrapositive and Converse
- Introduction of Validating Statements
- Validation by Contradiction
- Difference Between Contradiction, Converse and Contrapositive
- Consolidating the Understanding

##### Sets and Functions

##### Relations and Functions

- Cartesian Product of Sets
- Concept of Relation
- Concept of Functions
- Some Functions and Their Graphs
- Algebra of Real Functions
- Ordered Pairs
- Equality of Ordered Pairs
- Pictorial Diagrams
- Graph of Function
- Pictorial Representation of a Function
- Exponential Function
- Logarithmic Functions
- Brief Review of Cartesian System of Rectanglar Co-ordinates

##### Algebra

##### Trigonometric Functions

- Concept of Angle
- Introduction of Trigonometric Functions
- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Trigonometric Functions of Sum and Difference of Two Angles
- Trigonometric Equations
- Trigonometric Functions
- 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
- Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
- Graphs of Trigonometric Functions
- Transformation Formulae
- Values of Trigonometric Functions at Multiples and Submultiples of an Angle
- Sine and Cosine Formulae and Their Applications

##### Coordinate Geometry

##### Calculus

##### Complex Numbers and Quadratic Equations

- Concept of Complex Numbers
- Algebraic Operations of Complex Numbers
- The Modulus and the Conjugate of a Complex Number
- Argand Plane and Polar Representation
- Quadratic Equations
- Algebra of Complex Numbers - Equality
- Algebraic Properties of Complex Numbers
- Need for Complex Numbers
- Square Root of a Complex Number

##### Mathematical Reasoning

##### Linear Inequalities

##### Statistics and Probability

##### Permutations and Combinations

- Fundamental Principles of Counting
- Permutations
- Combination
- Introduction of Permutations and Combinations
- Permutation Formula to Rescue and Type of Permutation
- Smaller Set from Bigger Set
- Derivation of Formulae and Their Connections
- Simple Applications of Permutations and Combinations
- Factorial N (N!) Permutations and Combinations

##### Principle of Mathematical Induction

##### Binomial Theorem

- Introduction of Binomial Theorem
- Binomial Theorem for Positive Integral Indices
- General and Middle Terms
- Proof of Binomial Therom by Pattern
- Proof of Binomial Therom by Combination
- Rth Term from End
- Simple Applications of Binomial Theorem

##### Sequence and Series

##### Straight Lines

- Slope of a Line
- Various Forms of the Equation of a Line
- General Equation of a Line
- Distance of a Point from a Line
- Brief Recall of Two Dimensional Geometry from Earlier Classes
- Shifting of Origin
- Equation of Family of Lines Passing Through the Point of Intersection of Two Lines

##### Conic Sections

- Sections of a Cone
- Concept of Circle
- Introduction of Parabola
- Standard Equations of Parabola
- Latus Rectum
- Introduction of Ellipse
- Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse
- Special Cases of an Ellipse
- Eccentricity
- Standard Equations of an Ellipse
- Latus Rectum
- Introduction of Hyperbola
- Eccentricity
- Standard Equation of Hyperbola
- Latus Rectum
- Standard Equation of a Circle

##### Introduction to Three-dimensional Geometry

##### Limits and Derivatives

- Intuitive Idea of Derivatives
- Introduction of Limits
- Introduction to Calculus
- Algebra of Limits
- Limits of Polynomials and Rational Functions
- Limits of Trigonometric Functions
- Introduction of Derivatives
- Algebra of Derivative of Functions
- Derivative of Polynomials and Trigonometric Functions
- Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
- Limits of Logarithmic Functions
- Limits of Exponential Functions
- Derivative of Slope of Tangent of the Curve
- Theorem for Any Positive Integer n
- Graphical Interpretation of Derivative
- Derive Derivation of x^n

##### Statistics

- Measures of Dispersion
- Concept of Range
- Mean Deviation
- Introduction of Variance and Standard Deviation
- Standard Deviation
- Standard Deviation of a Discrete Frequency Distribution
- Standard Deviation of a Continuous Frequency Distribution
- Shortcut Method to Find Variance and Standard Deviation
- Introduction of Analysis of Frequency Distributions
- Comparison of Two Frequency Distributions with Same Mean
- Statistics Concept
- Central Tendency - Mean
- Central Tendency - Median
- Concept of Mode
- Measures of Dispersion - Quartile Deviation
- Standard Deviation - by Short Cut Method

##### Probability

- Random Experiments
- Introduction of Event
- Occurrence of an Event
- Types of Events
- Algebra of Events
- Exhaustive Events
- Mutually Exclusive Events
- Axiomatic Approach to Probability
- Probability of 'Not', 'And' and 'Or' Events

- 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

## Definition of Angle

What is an angle? In-Plane Geometry, a figure formed by two rays or lines that share a common endpoint is called an angle. The word “angle” is derived from the Latin word “angulus”, which means “corner”. The two rays are called the sides of an angle, and the common endpoint is called the vertex.

One complete revolution from the position of the initial side, as indicated in Fig

## Notes

**Positive Angle:** If the angle goes counterclockwise, it is called a positive angle.

**Negative Angle:** If the angle goes clockwise, it is called a negative angle.

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side, and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is positive, and if the direction of rotation is clockwise, then the angle is negative.

**Degree measure-** If a rotation from the initial side to the terminal side is `(1/360)"th"` of a revolution, the angle is said to have a measure of one degree, written as 1°.

A degree is divided into 60 minutes and a minute into 60 seconds. One-sixtieth of a degree is called a minute, written as 1′, and one-sixtieth of a minute is called a second, written as 1″.

Thus, 1° = 60′, 1′ = 60″ Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are shown in Fig

**Radian measure - **There is another unit for the measurement of an angle called the radian measure. The angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.

The figures show the angles whose measures are `1 radian, –1 radian, 1 1/2 radian and -1 1/2 radian.`

Thus, if in a circle of radius r, an arc of length l subtends an angle θ radian at the centre, we have

`θ= l/r` or `l= rθ.`

Here, θ will always be represented in terms of radian.

Relation between radian and real numbers:

Consider the unit circle with centre O. Let A be any point on the circle. Consider OA as the initial side of an angle. Then the length of an arc of the circle will give the radian measure of the angle at which the arc will subtend at the centre of the circle. Consider the line PAQ which is tangent to the circle at A. Let point A represent the real number zero, AP represents a positive real number, and AQ represents negative real numbers (Fig). Suppose we rope the line AP in the anticlockwise direction along the circle and AQ in the clockwise direction. In that case, every real number will correspond to a radian measure and conversely. Thus, radian measures and real numbers can be considered the same.

**Relation between degree and radian: **

Circumference of a circle =2πr

Number of arcs= 2π

One full rotation= 2π Radians

2π Radians= 360°

π= 180°

`π/2` Radians= 90°

1 Radian= `"360°"/"2π"` ∼ 57.2958°

The relation between degree measures and radian measure of some common angles are given in the following table:

**Notational Convention:**

Since angles are measured either in degrees or in radians, we adopt the convention that whenever we write angle θ°, we mean the angle whose degree measure is θ and whenever we write angle β, we mean the angle whose radian measure is β. Note that the word ‘radian’ is frequently omitted when an angle is expressed in radians.

Thus, `π= 180°` and `π/4= 45°` are written with the understanding that `π` and `π/4` are radian measures. Thus, we can say that

`"Radian measure" = π/180xx "Degree measure"`

`"Degree measure" = 180/π xx "Radian measure"`