Topics
Sets, Relations, and Functions
 Introduction to Sets, Relations and Functions
 Sets and Their Representations
 Types of Sets
 Empty Set (Null or Void Set)
 Finite and Infinite Sets
 Equal Sets
 Subsets
 Power Set
 Universal Set
 Venn Diagrams
 Operations on Sets
 Union of Sets
 Intersection of Sets
 Disjoint Sets
 Difference of Sets
 Complement of a Set
 Cardinal Number of Sets
 Practical Problems on Union and Intersection of Two Sets
 Cartesian Product of Sets
 Concept of Relation
 Types of Relations
 Concept of Functions
 Types of Functions
 Algebra of Real Functions
 Algebraic Operations on Functions
Complex Numbers and Quadratic Equations
 Introduction of Complex Number
 Concept of Complex Numbers
 Complex Numbers as Ordered Pairs of Reals
 Representation of Complex Numbers
 Argand Plane and Polar Representation
 Algebraic Operations of Complex Numbers
 Properties of Conjugate, Modulus and Argument (or Amplitude) of Complex Numbers
 Square Root of a Complex Number
 Triangle Inequality
 Integral Powers of Iota
 Rotational Theorem of Complex Number
 Cube Root of Unity
 Geometry of Complex Numbers
 Demoiver's Theorem
 Powers of Complex Numbers
 Quadratic Equations
 Sum and Product of Root
 Nature of Roots of a Quadratic Equation
 Relation Between Coefficients and Roots of a Quadratic Equation
 Formation of a Quadratic Equation with Given Roots
 Application of Quadratic Equation
 Condition for Common Roots
 Maximum and Minimum Value of Quadratic Equation
 Quadratic Expression in Two Variables
 Solution of Quadratic Inequalities
Matrices and Determinants
 Introduction to Matrices and Determinants
 Matrices
 Algebraic Operations on Matrices
 Addition of Matrices
 Multiplication of a Matrix by a Scalar
 Properties of Matrix Addition
 Properties of Scalar Multiplication of a Matrix
 Multiplication of Matrices
 Properties of Multiplication of Matrices
 Types of Matrices
 Determinants
 Order of a Matrix
 Properties of Determinants
 Evaluation of Determinants
 Area of a Triangle Using Determinants
 Adjoint of a Matrix
 Inverse of Matrix
 Inverse of a Matrix by Elementary Transformation
 Inverse of a Square Matrix by the Adjoint Method
 Test of Consistency
 Applications of Determinants and Matrices
 Subtraction of Matrices
 Transpose of a Matrix
 Symmetric and Skew Symmetric Matrices
 Multiplication of Two Determinants
 Minors and Cofactors
 Some Special Cases of Matrix
 Rank of a Matrix
Permutations and Combinations
 Introduction of Permutations and Combinations
 Fundamental Principles of Counting
 Permutations
 Combination
 Meaning of P (n,r) and C (n,r)
 Simple Applications of Permutations and Combinations
 Factorials
 Division and Distribution of Objects
 Dearrangement Theorem
 Sum of Numbers
 Important Result About Point
Mathematical Inductions
 Mathematical Induction
 Principle of Mathematical Induction
 Motivation
 Simple Applications of Mathematical Induction
Binomial Theorem and Its Simple Applications
 Introduction of Binomial Theorem
 Binomial Theorem for Positive Integral Indices
 General and Middle Terms
 Properties of Binomial Coefficient with Simple Application
 Expansion of Binomial
 Coefficient of Any Power of 'X'
 Greatest Term
 Independent Term
 Particular Term from End in Binomial Expansion
 Greatest Binomial Coefficients
 Number of Terms in the Expansion of (x + y + z)n
 Multinomial Theorem
 Infinite Series
 Binomial Theorem for Any Index (Without Proof)
Sequence and Series
 Sequence and Series
 Introduction of Sequence and Series
 Relation Between Arithmetic Mean (A.M.), Geometric Mean (G.M.), Harmonic Mean (H.M.)
 Arithmeticogeometric Sequence
 Arithmetic Progression (A.P.)
 Geometric Progression (G. P.)
 Harmonic Progression (H. P.)
 Insertion of Arithmetic
 Inserting Two or More Geometric Means Between Any Two Numbers
 Sum to N Terms of Special Series
 ArithmeticoGeometric Progression
 Some Special Sequences
Limit, Continuity, and Differentiability
 Concept of Limits
 RealValued Functions
 Limits by Factorisation, Substitution and Rationalisation
 Algebra of Limits
 Limits of Polynomials and Rational Functions
 Limits of Logarithmic Functions
 Limits of Exponential Functions
 Limits of Trigonometric Functions
 Inverse Functions
 Graphs of Simple Functions
 Concept of Continuity
 Concept of Differentiability
 Differentiation of the Sum, Difference, Product, and Quotient of Two Functions
 Derivatives of Composite Functions  Chain Rule
 Derivatives of Trigonometric Functions
 Derivatives of Inverse Trigonometric Functions
 Derivative of Logarithmic Functions
 Derivatives of Exponential Functions
 Derivative of Composite Functions
 Derivatives of Implicit Functions
 Derivatives of Functions in Parametric Forms
 Second Order Derivative
 Mean Value Theorem
 Simple Problems on Applications of Derivatives
 Rate of Change of Bodies or Quantities
 Increasing and Decreasing Functions
 Maxima and Minima
 Tangents and Normals
 Limits Using Lhospital's Rule
 Evaluation of Limits
 Infinite Series
 Successive Differentiation
 nth Derivative of Standard Functions
 Algebra of Derivative (Leibnitz or Product Rule)
 Rolle's Theorem
 Lagrange's Mean Value Theorem (LMVT)
 Approximations
Integral Calculas
 Integration
 Integration as an Inverse Process of Differentiation
 Fundamental Integrals Involving Algebraic Functions
 Fundamental Integrals Involving Trigonometric Functions
 Fundamental Integrals Involving Exponential Functions
 Fundamental Integrals Involving Logarithms Functions
 Methods of Integration: Integration by Substitution
 Methods of Integration: Integration by Parts
 Methods of Integration: Integration Using Partial Fractions
 Integration Using Trigonometric Identities
 Integrals of Some Particular Functions
 Definite Integral as the Limit of a Sum
 Fundamental Theorem of Calculus
 Properties of Definite Integrals
 Evaluation of Definite Integrals
 Area of the Region Bounded by a Curve and a Line
 Area Between Two Curves
 Area Under Simple Curves
 Integration of Some Special Irrational Algebraic Functions
 Evaluation of Definite Integrals by Substitution
 Summation of Series by Integration
Diffrential Equations
 Introduction to Ordinary Differential Equations
 Formation of Ordinary Differential Equations
 Order and Degree of a Differential Equation
 Formation of Differential Equations
 General and Particular Solutions of a Differential Equation
 Solutions of Linear Differential Equation
 Methods of Solving First Order, First Degree Differential Equations
 Differential Equations with Variables Separable Method
 Homogeneous Differential Equations
 Linear Differential Equations
 Linear Differential Equation of First Order
 Solution by Inspection Method
Coordinate Geometry
 Brief Review of Cartesian System of Rectanglar Coordinates
 Distance Formula
 Section Formula
 Locus
 Equation of Locus
 Translation of Axes
 Slope of a Line
 Parallel and Perpendicular Lines
 Intercepts of a Line on the Coordinate Axis
 Various Forms of the Equation of a Line
 Intersection of Two Lines
 Angle Between Two Lines
 Conditions for Concurrence of Three Lines
 Distance of a Point from a Line
 Straight Lines
 Equations of Internal and External by Sectors of Angles Between Two Lines Coordinate of the Centroid, Orthocentre, and Circumcentre of a Triangle
 Equations of Internal and External by Sectors of Angles Between Two Lines Coordinate of the Centroid, Orthocentre, and Circumcentre of a Triangle
 Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
 Equations of a Circle in Standard Form
 Equations of a Circle in General Form
 Equation of a Circle When the Endpoints of a Diameter Are Given
 Point of Intersection of a Line and a Circle
 General Equation of Tangents
 Conic Sections
 Parabola
 Ellipse
 Hyperbola
 Standard Equations of Parabola
 Standard Equations of an Ellipse
 Standard Equation of Hyperbola
 Condition for Y = mx + c to Be a Tangent and Point(s) of Tangency
 Results of Triangle
 Equation of Locus
 Slope of a Straight Line
 Slope of a Line Joining Two Points
 Various Forms of Equation of a Line
 Shortest Distance Between Two Lines
 Bisector of the Angle Between the Two Lines
 Perpendicular Distance of a Point from a Line
 Foot of the Perpendicular
 Position of a Point with Respect to a Line
 Pedal Points
 Pair of Straight Lines
 Circle
 Standard Equation of a Circle
Three Dimensional Geometry
 Three  Dimensional Geometry
 Coordinates of a Point in Space
 Distance Between Two Points
 Section Formula
 Direction Ratios
 Direction Cosines and Direction Ratios of a Line
 The Angle Between Two Intersecting Lines
 Skew Lines
 Shortest Distance Between Two Lines
 Equations of Line in Different Forms
 Equations of a Plane in Different Forms
 Intersection of the Line and Plane
 Coplanarity of Two Lines
 Angle Between Two Lines
 Projection of a Point on a Line
 Projection of a Line Segment Joining Two Points
 Equation of a Straight Line in Cartesian and Vector Form
 Condition of Parallelism and Perpendicularity of Two Lines
 Perpendicular Distance of a Point from a Line
 Distance Between Skew Lines and Parallel Lines
 Different Forms of Equation of a Plane
 Equation of a Plane
 Equation of Plane Passing Through the Intersection of Two Given Planes
 Angle Between Two Planes
 Angle Between Line and a Plane
 Distance Between Two Parallel Planes
 Position of Point and Line wrt a Plane
 Projection of a Line on a Plane
Vector Algebra
 Introduction to Vector Algebra
 Vectors and Scalars
 Addition of Vectors
 Components of Vector
 Scalar Product and Vector Product
 Scalar Triple Product of Vectors
 Vector Triple Product
 Algebra of Vectors
 Section Formula
 Linear Dependent and Independent Vectors
 Position Vector of a Point in a Space
 Modulus of a Vector
 Collinearity of Three Points
 Coplanarity of Three Vectors and Four Points
 Vector Inequality
 Product of Two Vectors
 Scalar (Or Dot) Product of Two Vectors
 Vector (Or Cross) Product of Two Vectors
 Projection of a Vector Along Any Other Vector
 Area of a Parallelogram
 Area of a Triangle
Statistics and Probability
 Measures of Discretion
 Arithmetic Mean  Raw Data
 Mean of Grouped Data
 Mean of Ungrouped Data
 Concept of Median
 Median of Grouped Data
 Median of Ungrouped Data
 Concept of Mode
 Mode of Grouped Data
 Mode of Ungrouped Data
 Standard Deviation
 Variance
 Mean Deviation
 Geometric Mean
 Harmonic Mean (H.M.)
 Measures of Central Tendency  Quartile
 Quartile Deviation (Semi  Inter Quartile Range)
 Coefficient of Variation
 Probability of an Event
 Addition Theorem of Probability
 Multiplication Theorem on Probability
 Bayes’ Theorem
 Random Variables and Its Probability Distributions
 Bernoulli Trials and Binomial Distribution
 Random Experiments
 Sample Space
 Event
 Mutually Exclusive Events
 Exhaustive Events
 Equally Likely Outcomes
 Odds in Favour and Against
 Boole's Inequality
 Demorgan's Law
 Independent Events
 Conditional Probability
 Probability Distribution
 Poisson Distribution
Trigonometry
 Trigonometric Identities
 Trigonometric Equations
 Trigonometric Functions
 Inverse Trigonometric Functions
 Properties of Inverse Trigonometric Functions
 Heights and Distances
 Circular System
 Trigonometric Ratios
 Domain and Range of Trigonometric Functions
 Trigonometric Ratios of Allied Angles
 Conditional Trigonometric Identities
 Greatest and Least Value of Trigonometric Expressions
 Solution of Trigonometric Equations (Solution in the Specified Range)
 Domain and Range of Inverse Trigonometric Functions
 Principal Value of Inverse Trigonometric Functions
 Intervals for Inverse Trigonometric Functions
 Infinite Series of Inverse Trigonometric Functions
Mathematical Reasoning
 Mathematical Reasoning
 Introduction of Validating Statements
 Mathematically Acceptable Statements
 Truth Value of Statement
 Tautology, Contradiction, and Contingency
 Logical Connective
 Truth Tables
 Logical Equivalance
 Duality
 Converse, Inverse and Contrapositive of the Conditional Staternent
 Negative of a Compound Statement
 Algebra of Statements
Linear Inequality
 Linear Inequality
 Solution of Linear Inequality
 System of Linear Inequalities
 Inequalities of Various Functions
Properties of Triangles
 Properties of Triangle
 Solutions of Triangle
 Inscribed Circle
 Regular Polygon
 Heights and Distances
Formula
Area of parallelogram = base x height
Notes
Area of the parallelogram:
To get the area of the parallelogram, we first draw a perpendicular line segment from one corner of the parallelogram to the opposite side. This is the height (h) of the parallelogram. The area of a parallelogram is equal to the product of its length and height.
Area of a parallelogram = base x height
You may notice that `lh` is also the area of a rectangle with dimensions l and h. The diagram below will explain why. If we cut out the triangle ABC and add it to the other side (triangle DEF), you will have a rectangle with dimensions l and h that has the same area as the original parallelogram.
Area of parallelogram ABCD = (base x height)

Any side of a parallelogram can be chosen as the base of the parallelogram.

The perpendicular dropped on that side from the opposite vertex is known as height (altitude).
Example
Find the height ‘x’ if the area of the parallelogram is 24 cm^{2} and the base is 4 cm.
Area of parallelogram = b × h
Therefore, 24 = 4 × `x`
`24/4 = x`
x = 6 cm
So, the height of the parallelogram is 6 cm.
Example
The two sides of the parallelogram ABCD are 6 cm and 4 cm. The height corresponding to the base CD is 3 cm.
Find the
(i) area of the parallelogram.
(ii) the height corresponding to the base AD.
(i) Area of parallelogram = b × h
= 6 cm × 3 cm = 18 cm^{2 }
(ii) base (b) = 4 cm, height = x (say), Area = 18 cm^{2}
Area of parallelogram = b × `x`
18 = 4 × `x`
`18/4 = x`
Therefore, x = 4.5 cm
Thus, the height corresponding to base AD is 4.5 cm.