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
Trigonometric Functions
Pair of Straight Lines
Vectors
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- 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
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous 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
- Bernoulli Trial
- Mean and Variance of Binomial Distribution
- Probability using Binomial Distribution
- Overview of Binomial Distribution
Estimated time: 3 minutes
- Equation of a line passing through a given point and parallel to given vector
- Equation of a line passing through given two points
Maharashtra State Board: Class 12
Formula: Angle between Two Lines
Vector:
Angle between two lines: \[\cos\theta=\left|\frac{\mathbf{b}_{1}\cdot\mathbf{b}_{2}}{|\mathbf{b}_{1}||\mathbf{b}_{2}|}\right|\]
If two lines are perpendicular: b₁ · b₂ = 0
If two lines are parallel: b₁ = λb₂
Cartesian:
\[\cos\theta=\frac{|a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]
If two lines are perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
If two lines are parallel: \[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]
Maharashtra State Board: Class 12
Key Points: Vector and Cartesian Equations of a Line
| Case | Vector Form | Cartesian Form (Symmetric Form) |
|---|---|---|
| 1. Through a point + parallel to vector | r = a + λb | x = x₁ + lλ y = y₁ + mλ z = z₁ + nλ |
| 2. Through two points | r = a + λ(b − a) | x − x₁ / (x₂ − x₁) = y − y₁ / (y₂ − y₁) = z − z₁ / (z₂ − z₁) |
