Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Adjoint & Inverse of Matrix
- Application of Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle>Polar Co-Ordinates
- Solving a Triangle>Solving a Triangle
- Basics of Inverse Trigonometric Functions
- Graphs of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Properties of Inverse Trigonometric Functions > Self-adjusting Property
- Overview of Trigonometric Functions
- Properties of Inverse Trigonometric Functions > Reciprocal Property
- Properties of Inverse Trigonometric Functions > Complementary Property
- Properties of Inverse Trigonometric Functions > Addition & Subtraction Formula for Inverse Tangent
- Properties of Inverse Trigonometric Functions > Double-angle Property
- Properties of Inverse Trigonometric Functions > Triple-angle Property
- Properties of Inverse Trigonometric Functions > Addition–Subtraction Formula for Inverse Sine & Cosine
- Properties of Inverse Trigonometric Functions > Negative Argument Property
Pair of Straight Lines
Vectors
- Overview of Vectors
- Basic Concepts of Vector Algebra
- Types of Vectors in Algebra
- Algebra of Vectors > Scalar Multiplication
- Algebra of Vectors > Addition & Subtraction of Two Vectors
- Collinearity and Coplanarity of Vectors
- Vectors in Coordinate Geometry
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors > Scalar (Dot) Product
- Product of Two Vectors > Vector (Cross) Product
- Direction Ratios, Direction Cosine & Direction Angles in Vector
- Scalar Triple Product
- Vector Triple Product
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivatives of Functions in Parametric Forms
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Applications of Differential Equation
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
Estimated time: 13 minutes
Maharashtra State Board: Class 12
Key Points: Combined Equation of a Pair Lines
| General Equation | Combined equation of a pair of lines through the origin | Combined equation of a pair of lines not passing through the origin |
|---|---|---|
| ax² + 2hxy + by² = 0 | ax² + 2hxy + by² + 2gx + 2fy + c = 0 | |
| Necessary Conditions for Real Lines | h² − ab ≥ 0 | \[\begin{vmatrix} \mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{h} & \mathrm{b} & \mathrm{f} \\ \mathrm{g} & \mathrm{f} & \mathrm{c} \end{vmatrix}=0,\] h² − ab ≥ 0 |
| Point of intersection | (0, 0) | \[\left(\frac{\mathrm{hf-bg}}{\mathrm{ab-h^2}},\frac{\mathrm{gh-af}}{\mathrm{ab-h^2}}\right)\] |
| Angle between the lines | \[\tan\theta=\left|\frac{2\sqrt{\mathrm{h}^2-\mathrm{ab}}}{\mathrm{a}+\mathrm{b}}\right|\] | \[\tan\theta=\left|\frac{2\sqrt{\mathrm{h}^{2}-\mathrm{ab}}}{\mathrm{a}+\mathrm{b}}\right|\] |
| For parallel (coincident) lines | h² − ab = 0 | h² − ab = 0, bg² = af², \[\frac{\mathrm{a}}{\mathrm{h}}=\frac{\mathrm{h}}{\mathrm{b}}=\frac{\mathrm{g}}{\mathrm{f}}\] |
| For perpendicular lines | a + b = 0 | a + b = 0 |
Maharashtra State Board: Class 12
Formula: Sum and Product of Slopes
\[m_1+m_2=-\frac{2h}{b}\]
\[m_1m_2=\frac{a}{b}\]
