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
- Equality of two Complex Numbers
- Conjugate of a Complex Number
- Properties of `barz`
- Addition of complex numbers - Properties of addition, Scalar Multiplication
- Subtraction of complex numbers - Properties of Subtraction
- Multiplication of complex numbers - Properties of Multiplication
- Powers of i in the complex number
- Division of complex number - Properties of Division
- The square roots of a negative real number
- Identities
Notes
1) Addition of two complex numbers :
Let `z_1` = a + ib and `z_2` = c + id be any two complex numbers. Then, the sum `z_1` + `z_2 `is defined as follows:
`z_1` + `z_2` = (a + c) + i (b + d), which is again a complex number.
For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8
The addition of complex numbers satisfy the following properties:
(i) The closure law The sum of two complex numbers is a complex number, i.e., `z_1` + `z_2` is a complex number for all complex numbers `z_1` and `z_2`.
(ii) The commutative law For any two complex numbers `z_1` and `z_2`, `z_1` + `z_2` =` z_2` + `z_1`
(iii) The associative law For any three complex numbers `z_1, z_2, z_3, (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)`.
(iv) The existence of additive identity There exists the complex number 0 + i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number z, z + 0 = z.
(v) The existence of additive inverse To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. We observe that z + (–z) = 0 (the additive identity).
2) Difference of two complex numbers:
Given any two complex numbers `z_1` and `z_2`, the difference `z_1 – z_2` is defined as follows:
z_1 – z_2 = z_1 + (– z_2).
For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i
and (2 – i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i
3) Multiplication of two complex numbers:
Let `z_1` = a + ib and `z_2` = c + id be any two complex numbers. Then, the product `z_1 z_2` is defined as follows:
`z_1 z_2` = (ac – bd) + i(ad + bc) For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28
The multiplication of complex numbers possesses the following properties, which we state without proofs.
(i) The closure law :The product of two complex numbers is a complex number, the product `z_1 z_2` is a complex number for all complex numbers `z_1 and z_2`.
(ii) The commutative law: For any two complex numbers `z_1` and `z_2, z_1 z_2 = z_2 z_1`.
(iii) The associative law: For any three complex numbers `z_1, z_2, z_3, (z_1 z_2) z_3 = z_1 (z_2 z_3)`.
(iv) The existence of multiplicative identity: There exists the complex number 1 + i 0 (denoted as 1), called the multiplicative identity such that z.1 = z, for every complex number z.
(v) The existence of multiplicative inverse: For every non-zero complex number z = a + ib or a + bi(a ≠ 0, b ≠ 0), we have the complex number `a/(a^2+b^2)` + i -`b/(a^2+b^2)` (denoted by `1/z or z^-1`), called the multiplicative inverse of z such that
z,`1/z`= 1 (the multiplicative identity).
(vi) The distributive law: For any three complex numbers `z_1, z_2, z_3`,
(a) `z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3`
(b) ` (z_1 + z_2) z_3 = z_1 z_3 + z_2 z_3`
4) Division of two complex numbers: Given any two complex numbers `z_1` and `z_2`,
where z_2 ≠ 0, the quotient `z_1/z_2` is defined by
`z_1/z_2= z_1 1/z_2`
For example, let `z_1 `= 6 + 3i and ` z_2` = 2 – i
Then `z_1/z_2`= `[(6+3i)xx 1/(2-i)]`
= `(6+3i) [2/ [2^2+ (-1)^2] + i -(-1)/[2^2+(-1)^2]]`
= `(6+3i) [(2+i)/5]`
= `1/5 [12-3+i(6+6)]`
= `1/5 (9+12i)`
5) Power of i:
we know that
`i^3= i^2i= (-1)i= -i,`
`i^4= (i^2)^2= (-1)^2= 1`
`i^5= (i^2)^2 i= (-1)^2 i= i,`
`i^6= (i^2)^3= (-1)^3= -1`, etc.
Also, we have `i^-1= (1/i) xx (i/i)= i/-1= -i,`
`i^-2= 1/i^2= 1/-1= 1,`
`i^-3= 1/i^3= (1/-i)xx (i/i)= i/1= i,`
`i^-4= 1/i^4= 1/1= 1`
In general, for any integer k, `i^(4k)=1, i^(4k+1)=i, i^(4k+2)= -1, i^(4k+3)= -i`
6) The square roots of a negative real number:
If a is a positive real number, `sqrt -a= sqrt a sqrt-1= sqrt a i,`
`sqrt a xx sqrt b= sqrt ab ` for all positive real number a and b. This result also holds true when either a > 0, b < 0 or a < 0, b > 0.
`sqrt a xx sqrt b ≠ sqrt ab` if both a and b are negative real numbers.
If any of a and b is zero, then, `sqrt a xx sqrt b= sqrt ab = 0`
7) Identities:
1) `(z_1 + z_2)^2= z_1^2+ z_2^2+ 2z_1z_2`
2)` (z_1 - z_2)^2= z_1^2- z_2^2+ 2z_1z_2`
3) `(z_1+ z_2)^3= z_1^3+ 3z_1^2z_2+ 3z_1z_2^2+ z_2^3`
4) `(z_1- z_2)^3= z_1^3- 3z_1^2z_2+ 3z_1z_2^2- z_2^3`
5) `z_1^2- z_2^2= (z_1+z_2) (z_1- z_2)`
