Topics
Applications of Matrices and Determinants
- Concept of Matrices
- Operations on Matrices> Matrix Multiplication
- Symmetric and Skew Symmetric Matrices
- Applications of Matrices: Solving System of Linear Equations
- Applications of Matrices: Consistency of System of Linear Equations by Rank Method
Complex Numbers
- Concept of Complex Numbers
- Basic Algebraic Properties of Complex Numbers
- Conjugate of a Complex Number
- Modulus of a Complex Number
- Geometry and Locus of Complex Numbers
- Polar and Euler Form of a Complex Number
- de Moivre’s Theorem and Its Applications
Theory of Equations
- Introduction to Theory of Equations
- Basics of Polynomial Equations
- Vieta’s Formulae and Formation of Polynomial Equations
- Nature of Roots and Nature of Coefficients of Polynomial Equations
- Roots of Higher Degree Polynomial Equations
- Polynomials with Additional Information
- Polynomial Equations with No Additional Information
- Descartes Rule
Inverse Trigonometric Functions
- Basics of Inverse Trigonometric Functions
- Some Fundamental Concepts
- Sine Function and Inverse Sine Function
- The Cosine Function and Inverse Cosine Function
- The Tangent Function and the Inverse Tangent Function
- The Cosecant Function and the Inverse Cosecant Function
- The Secant Function and Inverse Secant Function
- The Cotangent Function and the Inverse Cotangent Function
- Properties of Inverse Trigonometric Functions
Two Dimensional Analytical Geometry-II
- Two Dimensional Analytical Geometry-II
- Advanced Concept of Circle
- Fundamentals of Conic Sections
- Parametric Form of Conics
- Tangents and Normals to Conics
- Real Life Applications of Conics
Applications of Vector Algebra
- Basic Concepts of Vector Algebra
- Geometric Introduction to Vectors
- Scalar Product(Dot Product)
- Scalar Triple Product
- Vector Triple Product
- Jacobi’S Identity and Lagrange’S Identity
- Application of Vectors to 3-dimensional Geometry
- Different Forms of Equation of a Plane
- Image of a Point in a Plane
- Meeting Point of a Line and a Plane
Applications of Differential Calculus
- Differential Calculus
- Meaning of Derivatives
- Mean Value Theorem
- Series Expansions
- Indeterminate Forms
- Applications of First Derivative
- Applications of Second Derivative
- Applications in Optimization
- Symmetry and Asymptotes
- Sketching of Curves
Differentials and Partial Derivatives
- Introduction to Differentials and Partial Derivatives
- Linear Approximation and Differentials
- Functions of Several Variables
- Limit and Continuity of Functions of Two Variables
- Partial Derivatives
- Linear Approximation and Differential of a Function of Several Variables
Applications of Integration
- Applications of Integrations
- Definite Integral as the Limit of a Sum
- Integral Calculus
- Improper Integrals
- Reduction Formulae
- Gamma Integral
- Evaluation of a Bounded Plane Area by Integration
- Volume of a Solid Obtained by Revolving Area About an Axis
Ordinary Differential Equations
- Introduction to Ordinary Differential Equations
- Order and Degree of a Differential Equation
- Classification of Differential Equations
- Formation of Differential Equations
- Solution of Ordinary Differential Equations
- Solution of First Order and First Degree Differential Equations
- First Order Linear Differential Equations
- Applications of First Order Ordinary Differential Equations
Probability Distributions
- Concept of Probability
- Random Variable
- Random Variables
- Continuous Distributions
- Mathematical Expectation
- Theoretical Distributions: Some Special Discrete Distributions
Discrete Mathematics
- Introduction to Discrete Mathematics
- Binary Operations
- Mathematical Logic
Estimated time: 4 minutes
Maharashtra State Board: Class 12
Definition: Modulus of a Complex Number
The modulus (or absolute value) of a complex number, z = a + ib, is defined as the non-negative real number
√(a² + b²). It is denoted by |z| i.e. |z| = √(a² + b²)
Maharashtra State Board: Class 12
Key Points: Modulus of a Complex Number
- |z| = √(a² + b²)
- |z| = 0 ⇔ z = 0
- −|z| ≤ Re(z) ≤ |z|; −|z| ≤ Im(z) ≤ |z|
- |z₁z₂| = |z₁| |z₂|
- \[\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}\], z₂ ≠ 0
- |zⁿ| = |z|ⁿ
- |z₁ + z₂|² = |z₁|² + |z₂|² + 2Re(z₁ z̄₂)
- |z₁ − z₂|² = |z₁|² + |z₂|² − 2Re(z₁ z̄₂)
- |z₁ + z₂|² + |z₁ − z₂|² = 2(|z₁|² + |z₂|²)
- |z₁ + z₂| ≤ |z₁| + |z₂|
- |z₁ − z₂| ≥ ||z₁| − |z₂||
- z·z̄ = |z|²
- z₁z̄₂ + z̄₁z₂ = 2|z₁||z₂| cos(θ₁ − θ₂), where θ₁ = arg(z₁) and θ₂ = arg(z₂)
