- 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
- 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
Sets and Functions
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
Introduction to Three-dimensional Geometry
Relations and Functions
- Cartesian Product of Sets
- 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
Sequence and Series
- Sets and Their Representations
- The Empty Set
- Finite and Infinite Sets
- Equal Sets
- 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
- 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
- Standard Equations of an Ellipse
- Latus Rectum
- Introduction of Hyperbola
- Standard Equation of Hyperbola
- Latus Rectum
- Standard Equation of a Circle
Complex Numbers and Quadratic Equations
Permutations and Combinations
- Fundamental Principles of Counting
- 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
Statistics and Probability
Principle of Mathematical Induction
The connecting words which are found in compound statements like “And”, “Or”, etc. are often used in Mathematical Statements. These are called connectives.
1) The word “And”:
Let us look at a compound statement with “And”.
p: A point occupies a position and its location can be determined.
The statement can be broken into two component statements as
q : A point occupies a position.
r : Its location can be determined.
Here, we observe that both statements are true.
2) The word “Or”:
p: An ice cream or pepsi is available with a Thali in a restaurant.
This means that a person who does not want ice cream can have a pepsi along with Thali or one does not want pepsi can have an ice cream along with Thali. That is, who do not want a pepsi can have an ice cream. A person cannot have both ice cream and pepsi. This is called an exclusive “Or”.
Rule for the compound statement with ‘Or’:
For example, consider the following statement.
p: Two lines intersect at a point or they are parallel
The component statements are
q: Two lines intersect at a point.
r: Two lines are parallel.
Then, when q is true r is false and when r is true q is false.
Therefore, the compound statement p is true.
4) Quantifiers :
Quantifiers are phrases like, “There exists” and “For all”. Another phrase which appears in mathematical statements is “there exists”. For example, consider the statement. p: There exists a rectangle whose all sides are equal. This means that there is atleast one rectangle whose all sides are equal.
A word closely connected with “there exists” is “for every” (or for all). Consider a statement.
p: For every prime number p,`sqrt p` is an irrational number.
This means that if S denotes the set of all prime numbers, then for all the members p of the set S,`sqrt p` is an irrational number.