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
- nCr , nCn =1, nC0 = 1, nCr = nCn–r, nCx = nCy, then x + y = n or x = y, n+1Cr = nCr-1 + nCr
- When all things are different
- When all things are not different.
- Mixed problems on permutation and combinations.
Theorem
Theorem: `"^n P_r`= `"^n C_r` r!, 0 < r ≤ n.
Proof: Corresponding to each combination of `"^nC_r`, we have r ! permutations, because r objects in every combination can be rearranged in r ! ways.
Hence, the total number of permutations of n different things taken r at a time is `"^nCr` × r!. On the other hand, it is P n r . Thus
`"^n P_r` =`"^n C_r` * r!, 0 < r ≤ n.
Notes
1) From above n!/(n-r)!= `"^n C_r` * r!, i.e., `"^n C_r`= n!/[r!(n-r)!]
In particular, if r= n, `"^n C_n`= n!/(n!0!)= 1
2) We define `"^nC_0` = 1, i.e., the number of combinations of n different things taken nothing at all is considered to be 1. Counting combinations is merely counting the number of ways in which some or all objects at a time are selected. Selecting nothing at all is the same as leaving behind all the objects and we know that there is only one way of doing so. This way we define `"^nC_0` = 1.
3) As `(n!)/[0!(n-0)!]`= 1= `"^nC_0`, the formula `"^n C_r`= `(n!)/[r!(n-r)!]` is applicable for r=0 also. Hence
`"^n C_r`= `(n!)/[r!(n-r)!], 0 < r ≤ n`.
4) `"^n C_n-r`= `(n!)/ [(n-r)! (n-(n-r))!]= (n!)/[(n-r)!r!]= ``"^n C_r`,
i.e., selecting r objects out of n objects is same as rejecting (n – r) objects.
5) `"^nC_a` = `"^nC_b` ⇒ a = b or a = n – b, i.e., n = a + b
Theorem
Theorem: `"^nC_r` + `"^nC_r-1`= `"^(n+1)C_r`
Proof: We have `"^nC_r` + `"^nC_r-1= (n!)/[r!(n-r)!] + (n!)/[(r-1)!(n-r+1)!]`
= `(n!)/ [r*(r-1)!(n-r)!] + (n!)/[(r-1)!(n-r+1)(n-r)!]`
= `(n!)/[(r-1)!(n-r)!] [(1/r) + 1/(n-r+1)]`
= `(n!)/[(r-1)!(n-r)!] * (n-r+1+r)/[r(n-r+1)]`
= `(n+1)!/[r!(n+1-r)!`
= `"^(n+1) C_r`
