Topics
Mathematical Logic
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified 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
- Properties of Matrix Multiplication
- Application of Matrices
- Applications of Determinants and Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space
- Overview of Pair of Straight Lines
Vectors
Line and Plane
- Vector and Cartesian Equations of a Line
- Distance of a Point from a Line
- Distance Between Skew Lines and Parallel Lines
- Equation of a Plane
- Angle Between Planes
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Overview of Line and Plane
Linear Programming
Differentiation
- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
- 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 the Curve, Axis and Line
- Area Between 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
- Application of Differential Equations
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables and Its Probability Distributions
- Types of 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
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and Binomial Distribution
- Overview of Binomial Distribution
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Standard Forms of Linear Inequalities
The equation ax + by = c is called the associated equation of the inequality.
Maharashtra State Board: Class 12
CISCE: Class 12
Key Points: Region representation
| Condition | Region represented |
|---|---|
| ( x > 0 ) | Right of the y-axis |
| ( x < 0 ) | Left of the y-axis |
| ( y > 0 ) | Above x-axis |
| ( y < 0 ) | Below x-axis |
| ( x ≥ 0 ) | Includes y-axis |
| ( y ≥ 0 ) | Includes x-axis |
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Solution Set of a System
The common region satisfying all the given inequalities is called the solution set.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Linear Programming Problem
A Linear Programming Problem (LPP) involves a linear function of variables subject to a set of linear constraints, where the objective is to maximise or minimise the function.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Constraints
The conditions represented by a system of linear inequalities, which the variables of a Linear Programming Problem must satisfy, are called constraints.
Maharashtra State Board: Class 12
CISCE: Class 12
Theorem: Fundamental Theorem of Linear Programming
Statement:
If a linear objective function has a maximum or minimum value over a feasible region, then the maximum or minimum occurs at one of the corner points of the feasible region.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Linear Programming
Linear Programming is a method of solving problems in which a linear objective function is maximised or minimised subject to conditions expressed as a system of linear inequalities.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Objective Function
The linear function whose maximum or minimum value is to be determined is called the objective function.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Optimize
To optimise means to maximise or minimise.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Non-Negativity Restrictions
The constraints which imply that the decision variables of an LPP are non-negative are called non-negativity restrictions.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Convex Region
A region is said to be convex if the line segment joining any two points in the region lies entirely within the region.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Feasible Region & Feasible (Optimal) Solution
The common region satisfying all the constraints of an LPP is called the feasible region, and every point in this region is called a feasible solution.
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A feasible region is always convex.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: Infeasible Solution
A solution which does not satisfy all the constraints and non-negativity restrictions is called an infeasible solution.
Maharashtra State Board: Class 12
CISCE: Class 12
Definition: General Linear Programming
A Linear Programming Problem involves maximising or minimising a linear function subject to linear constraints and non-negativity restrictions.
Maharashtra State Board: Class 12
CISCE: Class 12
Key Points: Corner Point Method
This method is based on the fundamental extreme point theorem.
- Step 1: Draw the feasible region and find all corner points (vertices).
- Step 2: Evaluate the objective function Z = ax + by at each corner point.
Feasible Region is Bounded
Maximum value of Z = the largest value at a corner
Minimum value of Z = the smallest value at a corner
If the Feasible Region is Unbounded
If a maximum (or minimum) exists, it must occur at a corner point.
But maximum or minimum may not exist.
Extreme point theorem:
An optimum solution to a linear programming problem, if it exists, occurs at one of the corners (or extreme) points of the feasible region.
