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
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
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Definition: Standard Forms of Linear Inequalities
The equation ax + by = c is called the associated equation of the inequality.
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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 |
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Definition: Solution Set of a System
The common region satisfying all the given inequalities is called the solution set.
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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.
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Definition: Constraints
The conditions represented by a system of linear inequalities, which the variables of a Linear Programming Problem must satisfy, are called constraints.
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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.
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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.
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Definition: Objective Function
The linear function whose maximum or minimum value is to be determined is called the objective function.
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Definition: Optimize
To optimise means to maximise or minimise.
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Definition: Non-Negativity Restrictions
The constraints which imply that the decision variables of an LPP are non-negative are called non-negativity restrictions.
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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.
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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.
-
A feasible region is always convex.
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Definition: Infeasible Solution
A solution which does not satisfy all the constraints and non-negativity restrictions is called an infeasible solution.
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Definition: General Linear Programming
A Linear Programming Problem involves maximising or minimising a linear function subject to linear constraints and non-negativity restrictions.
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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.
