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
Notes
Coplanarity of Two Lines be
`vec r = vec a_1 + lambda vec b _1` ...(1)
and `vec r = vec a _2 + mu vec b _2` ...(2)
The line (1) passes through the point, say A, with position vector `vec a_1` and is parallel to `vec b _1.` The line (2) passes through the point , say B with position vector `vec a_2` and is parallel to `vec b_2.`
Thus , `vec (AB)= vec a _2 - vec a_1`
The given lines are coplanar if and only if `vec (AB)` is perpendicular to `vec b_1 xx vec b_2`.
i.e. `vec (AB) . (vec b _1 xx vec b_2) = 0` or
`(vec a_2 - vec a_1) . (vec b_1 xx vec b_2) = 0 `
Cartesian form
Let `(x_1, y_1, z_1)` and `(x_2, y_2, z_2)` be the coordinates of the points A and B respectively.
Let `a_1, b_1, c_1` and `a_2, b_2, c_2` be the direction ratios of `vec b _1` and `vec b _2` , respectively. Then
`vec (AB) = (x_2 - x_1) hat i + (y_2 - y_1) hat j + (z_2 - z_1) hat k`
`vec b_1 = a_1 hat i + b_1 hat j + c_1 hat k` and
`vec b_2 = a_2 hat i + b_2 hat j + c_2 hat k`
The given lines are coplanar if and only if `vec (AB) . (vec b_1 xx vec b_2) = 0 .` In the cartesian form , it can be expressed as
`|(x_2 - x_1 , y_2 - y_1 , z_2 - z_1), (a_1 , a_1 , a_1), (a_2 , b_2 , c_2)|` = 0 ...(4)
