Topics
Sets and Relations
- Introduction of Set
- Representation of a Set
- Intervals
- Types of Sets
- Operations on Sets
- Relations of Sets
- Types of Relations
Functions
- Concept of Functions
- Types of Functions
- Representation of Function
- Graph of a Function
- Fundamental Functions
- Algebra of Functions
- Composite Function
- Inverse Functions
- Some Special Functions
Complex Numbers 33
- Introduction of Complex Number
- Imaginary Number
- Concept of Complex Numbers
- Conjugate of a Complex Number
- Algebraic Operations of Complex Numbers
- Square Root of a Complex Number
- Solution of a Quadratic Equation in Complex Number System
- Cube Root of Unity
Sequences and Series
- Concept of Sequences
- Geometric Progression (G.P.)
- General Term Or the nth Term of a G.P.
- Sum of the First n Terms of a G.P.
- Sum of Infinite Terms of a G. P.
- Recurring Decimals
- Harmonic Progression (H. P.)
- Types of Means
- Special Series (Sigma Notation)
Locus and Straight Line
- Locus
- Equation of Locus
- Line
- Equations of Lines in Different Forms
- General Form Of Equation Of Line
Determinants
- Determinants
- Properties of Determinants
- Application of Determinants
- Determinant method
- Consistency of Three Linear Equations in Two Variables
- Area of a Triangle Using Determinants
- Collinearity of Three Points
Limits
- Definition of Limit of a Function
- Algebra of Limits
- Evaluation of Limits
- Direct Method
- Factorization Method
- Rationalization Method
- Limits of Exponential and Logarithmic Functions
Continuity
- Continuous and Discontinuous Functions
- Continuity of a Function at a Point
- Definition of Continuity
- Continuity from the Right and from the Left
- Properties of Continuous Functions
- Continuity in the Domain of the Function
- Examples of Continuous Functions Whereever They Are Defined
Differentiation
- The Meaning of Rate of Change
- Definition of Derivative and Differentiability
- Derivative by the Method of First Principle
- Rules of Differentiation (Without Proof)
- Applications of Derivatives
Partition Values
- Concept of Median
- Partition Values
- Quartiles
- Deciles
- Percentiles
- Relations Among Quartiles, Deciles and Percentiles
- Graphical Location of Partition Values
Measures of Dispersion
- Measures of Dispersion
- Range of Data
- Quartile Deviation (Semi - Inter Quartile Range)
- Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Skewness
- Skewness
- Asymmetric Distribution (Positive Skewness)
- Asymmetric (Negative Skewness)
- Measures of Skewness
- Karl Pearson’S Coefficient of Skewness (Pearsonian Coefficient of Skewness)
- Features of Pearsonian Coefficient
- Bowley’s Coefficient of Skewness
Bivariate Frequency Distribution and Chi Square Statistic
- Bivariate Frequency Distribution
- Classification and Tabulation of Bivariate Data
- Marginal Frequency Distributions
- Conditional Frequency Distributions
- Categorical Variables
- Contingency Table
- Chi-Square Statistic ( χ2 )
Correlation
- Correlation
- Concept of Covariance
- Properties of Covariance
- Concept of Correlation Coefficient
- Scatter Diagram
- Interpretation of Value of Correlation Coefficient
Permutations and Combinations
- Introduction of Permutations and Combinations
- Fundamental Principles of Counting
- Concept of Addition Principle
- Concept of Multiplication Principle
- Concept of Factorial Function
- Permutations
- Permutations When All Objects Are Distinct
- Permutations When Repetitions Are Allowed
- Permutations When All Objects Are Not Distinct
- Circular Permutations
- Properties of Permutations
- Combination
- Properties of Combinations
Probability
- Introduction of Probability
- Types of Events
- Algebra of Events
- Elementary Properties of Probability
- Addition Theorem of Probability
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
Linear Inequations
- Linear Inequality
- Solution of Linear Inequality
- Graphical Representation of Solution of Linear Inequality in One Variable
- Graphical Solution of Linear Inequality of Two Variable
- Solution of System of Linear Inequalities in Two Variables
Commercial Mathematics
- Percentage
- Profit and Loss
- Simple and Compound Interest (Entrance Exam)
- Depreciation
- Partnership
- Goods and Service Tax (GST)
- Shares and Dividends
- Simple or elementary event
- Occurrence and non-occurrence of event
- Sure Event
- Impossible Event
- Complimentary Event
Notes
Events can be classified into various types on the basis of the elements they have.
i) Impossible and Sure Events:
The empty set φ and the sample space S describe events. In fact φ is called an impossible event and S, i.e., the whole sample space is called the sure event.
To understand these let us consider the experiment of rolling a die. The associated sample space is
S = {1, 2, 3, 4, 5, 6}
Let E be the event “ the number appears on the die is a multiple of 7”. Can you write the subset associated with the event E? Clearly no outcome satisfies the condition given in the event, i.e., no element of the sample space ensures the occurrence of the event E. Thus, we say that the empty set only correspond to the event E. In other words we can say that it is impossible to have a multiple of 7 on the upper face of the die. Thus, the event E = φ is an impossible event.
ii) Simple Event:
If an event E has only one sample point of a sample space, it is called a simple (or elementary) event.
In a sample space containing n distinct elements, there are exactly n simple simple events.
For example in the experiment of tossing two coins, a sample space is S={HH, HT, TH, TT}
There are four simple events corresponding to this sample space. These are `E_1= {HH}, E_2={HT}, E_3= { TH} and E_4={TT}.`
iii) Compound Event:
If an event has more than one sample point, it is called a Compound event.
For example, in the experiment of “tossing a coin thrice” the events
E: ‘Exactly one head appeared’
F: ‘Atleast one head appeared’
G: ‘Atmost one head appeared’ etc.
are all compound events. The subsets of S associated with these events are E={HTT,THT,TTH}
F={HTT,THT, TTH, HHT, HTH, THH, HHH}
G= {TTT, THT, HTT, TTH}
Each of the above subsets contain more than one sample point, hence they are all compound events.
