#### Topics

##### Sets and Functions

##### 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
- 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

##### Relations and Functions

- Cartesian Product of Sets
- 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

##### Sets

- Sets and Their Representations
- The Empty Set
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Intrdouction of Operations on Sets
- Union Set
- 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
- Operation on Set - Disjoint Sets
- Element Count Set

##### Algebra

##### Binomial Theorem

##### Sequence and Series

##### Linear Inequalities

##### Complex Numbers and Quadratic Equations

##### 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

##### Coordinate Geometry

##### Straight Lines

##### Introduction to Three-dimensional Geometry

##### 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

##### Calculus

##### 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

##### Mathematical Reasoning

##### Mathematical Reasoning

##### Statistics and Probability

##### 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

#### notes

A function f is said to be a polynomial function of degree n f(x) = `a_0+ a_1x+a_2x^2+ . . + a_nx^n` , where a_1s are real numbers such that `a _n` ≠ 0 for some natural number n.

`lim_(x->a)` x = a .

Hence

`lim_(x -> a)x^2 = lim_(x->a) (x.x)` = `lim_(x->a) x . lim_(x->a) x = a.a =a^2`

An easy exercise in induction on n tells us that

`lim_(x-> a) x^n = a^n`

Now, let f(x) = `a_0 + a_1x + a_2x^2 + ...+a_nx^n` be a polynomial function.

Suppose of each of `a_0 , a_1x , a_2x^2 , ...., a_nx^n ` as a function , we have

`lim_(x ->a) f(x) = lim_(x -> a) [a_0 + a_1x + a_2 x^2 + ...+a_nx^n]`

= `lim_(x -> a) a_0 + lim_(x -> a) a_1x + lim_(x -> a) a_2x^2 + ... + a_nx^n`

= `a_0 + a_1 lim_(x ->a) x + a_2 lim_(x ->a) x^2 + ... + a_n lim_(x ->a) x^n.`

= `a_0 + a_1a + a_2a^2 + ... + a_na^n`

= f(a)

A function f is said to be a rational function, if f(x) = `g(x)/(h(x))` , where g(x) and h(x) are polynomials such that h(x) ≠ 0.

Then `lim_(x ->a) f(x) = lim_(x ->a)g(x)/(h(x)) =(lim_(x -> a) g(x))/(lim_(x ->a) h(x)) = g(a)/(h(a))`.

**Case 1 -** h(a) = 0 and g(a) = k

`g(a)/(h(a)) = k/0 = ∞ `

Limit does not exist (undefined)

Example -

`lim_(x->2) (x^3 - 2 )/(x - 2) = (2^3 - 2)/(2-2) = (8-2)/0 = 6/0 = ∞`

**Case 2 - **

h(a) = 0 and g(a) = 0

`g(a)/(h(a)) = 0/0`

Example - `lim_(x->2) (x^2 - 4)/(x-2) = (2^2 - 4)/(2 - 2) = (4 - 4)/(2 - 2) = 0/0`

**Case 3 -** h(a) = k and g(a) = 0

`g(a)/(h(a)) = 0/k = 0`

Example - `lim_(x->2) (x - 2)/(2x + 2) = (2 - 2)/(2 . 2 + 2) = 0/(4 + 2) = 0/6 = 0`