#### Topics

##### Mathematical Logic

- Statements - Introduction in Logic
- Sentences and Statement in Logic
- Truth Value of Statement in Logic
- Open Sentences in Logic
- Compound Statement in Logic
- Quantifier and Quantified Statements in Logic
- Logical Connectives
- Truth Tables of Compound Statements
- Examples Related to Real Life and Mathematics
- Statement Patterns and Logical Equivalence
- Algebra of Statements
- Difference Between Converse, Contrapositive, Contradiction
- Application of Logic to Switching Circuits, Switching Table.

##### Mathematical Logic

- Truth Value of Statement in Logic
- Logical Connective, Simple and Compound Statements
- Truth Tables of Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Quantifier and Quantified Statements in Logic
- Duality
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits, Switching Table.

##### Matrics

##### Trigonometric Functions

##### Pair of Straight Lines

##### Vectors

- Representation of Vector
- Vectors and Their Types
- Algebra of Vectors
- Coplanar Vectors
- Vector in Two Dimensions (2-D)
- Three Dimensional (3-D) Coordinate System
- Components of Vector
- Position Vector of a Point P(X, Y, Z) in Space
- Component Form of a Position Vector
- Vector Joining Two Points
- Section formula
- Dot/Scalar Product of Vectors
- Cross/Vector Product of Vectors
- Scalar Triple Product of Vectors
- Vector Triple Product
- Addition of Vectors

##### Line and Plane

##### Linear Programming

##### Matrices

- Elementary Operation (Transformation) of a Matrix
- Inverse by Elementary Transformation
- Elementary Transformation of a Matrix Revision of Cofactor and Minor
- Inverse of a Matrix Existance
- Adjoint Method
- Addition of Matrices
- Solving System of Linear Equations in Two Or Three Variables Using Reduction of a Matrix Or Reduction Method
- Solution of System of Linear Equations by – Inversion Method

##### Differentiation

##### Applications of Derivatives

##### Indefinite Integration

##### Definite Integration

##### Application of Definite Integration

##### Differential Equations

##### Probability Distributions

##### Binomial Distribution

##### Trigonometric Functions

- Trigonometric equations
- General Solution of Trigonometric Equation of the Type
- Solution of a Triangle
- Hero’s Formula in Trigonometric Functions
- Napier Analogues in Trigonometric Functions
- Basic Concepts of Trigonometric Functions
- Inverse Trigonometric Functions - Principal Value Branch
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions

##### Pair of Straight Lines

- Pair of Lines Passing Through Origin - Combined Equation
- Pair of Lines Passing Through Origin - Homogenous Equation
- Theorem - the Joint Equation of a Pair of Lines Passing Through Origin and Its Converse
- Acute Angle Between the Lines
- Condition for Parallel Lines
- Condition for Perpendicular Lines
- Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines
- Point of Intersection of Two Lines

##### Circle

- Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle
- Tangent of a Circle - Equation of a Tangent at a Point to General Circle
- Condition of tangency
- Tangents to a Circle from a Point Outside the Circle
- Director circle
- Length of Tangent Segments to Circle
- Normal to a Circle - Equation of Normal at a Point

##### Conics

##### Vectors

- Vectors Revision
- Collinearity and Coplanarity of Vectors
- Linear Combination of Vectors
- Condition of collinearity of two vectors
- Conditions of Coplanarity of Three Vectors
- Section formula
- Midpoint Formula for Vector
- Centroid Formula for Vector
- Basic Concepts of Vector Algebra
- Scalar Triple Product of Vectors
- Geometrical Interpretation of Scalar Triple Product
- Application of Vectors to Geometry
- Medians of a Triangle Are Concurrent
- Altitudes of a Triangle Are Concurrent
- Angle Bisectors of a Triangle Are Concurrent
- Diagonals of a Parallelogram Bisect Each Other and Converse
- Median of Trapezium is Parallel to the Parallel Sides and Its Length is Half the Sum of Parallel Sides
- Angle Subtended on a Semicircle is Right Angle

##### Three Dimensional Geometry

##### Line

##### Plane

- Equation of Plane in Normal Form
- Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
- Equation of Plane Passing Through the Given Point and Parallel to Two Given Vectors
- Equation of a Plane Passing Through Three Non Collinear Points
- Equation of Plane Passing Through the Intersection of Two Given Planes
- Vector and Cartesian Equation of a Plane
- Angle Between Two Planes
- Angle Between Line and a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane

##### Linear Programming Problems

##### Continuity

- Introduction of Continuity
- Continuity of a Function at a Point
- Defination of Continuity of a Function at a Point
- Discontinuity of a Function
- Types of Discontinuity
- Concept of Continuity
- Algebra of Continuous Functions
- Continuity in Interval - Definition
- Exponential and Logarithmic Functions
- Continuity of Some Standard Functions - Polynomial Function
- Continuity of Some Standard Functions - Rational Function
- Continuity of Some Standard Functions - Trigonometric Function
- Continuity - Problems

##### Differentiation

- Revision of Derivative
- Relationship Between Continuity and Differentiability
- Every Differentiable Function is Continuous but Converse is Not True
- Derivatives of Composite Functions - Chain Rule
- Derivative of Inverse Function
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Implicit Functions
- Exponential and Logarithmic Functions
- Derivatives of Functions in Parametric Forms
- Derivative of Functions in Product of Function Form
- Derivative of Functions in Quotient of Functions Form
- Higher Order Derivative
- Second Order Derivative

##### Applications of Derivative

##### Integration

- Methods of Integration - Integration by Substitution
- Methods of Integration - Integration Using Partial Fractions
- Methods of Integration - Integration by Parts
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Calculus
- Properties of Definite Integrals
- Evaluation of Definite Integrals by Substitution
- Integration by Non-repeated Quadratic Factors

##### Applications of Definite Integral

##### Differential Equation

- Basic Concepts of Differential Equation
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Formation of Differential Equation by Eliminating Arbitary Constant
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Linear Differential Equation
- Applications of Differential Equation

##### Statistics

##### Probability Distribution

- Conditional Probability
- Random Variables and Its Probability Distributions
- Discrete and Continuous Random Variable
- Probability Mass Function (P.M.F.)
- Probability Distribution of a Discrete Random Variable
- Cumulative Probability Distribution of a Discrete Random Variable
- Expected Value, Variance and Standard Deviation of a Discrete Random Variable
- Probability Density Function (P.D.F.)
- Distribution Function of a Continuous Random Variable

##### Bernoulli Trials and Binomial Distribution

#### description

- Distance between two skew lines
- Distance between parallel lines

#### notes

If two lines in space intersect at a point, then the shortest distance between them is zero. Also, if two lines in space are parallel, then the shortest distance between them will be the perpendicular distance, i.e. the length of the perpendicular drawn from a point on one line onto the other line. Further, in a space, there are lines which are neither intersecting nor parallel. In fact, such pair of lines are non coplanar and are called skew lines.Fig.

The line GE that goes diagonally across the ceiling and the line DB passes through one corner of the ceiling directly above A and goes diagonally down the wall. These lines are skew because they are not parallel and also never meet.

By the shortest distance between two lines we mean the join of a point in one line with one point on the other line so that the length of the segment so obtained is the smallest.

For skew lines, the line of the shortest distance will be perpendicular to both the lines.

**Distance between two skew lines:**

We now determine the shortest distance between two skew lines in the following way: Let `l_1` and `l_2` be two skew lines with equations in fig.

`vec r = vec a _1 + lambda vec b_1` ...(1)

and `vec r = vec a _2 + mu vec b _2` ...(2)

Take any point S on `l_1`with position vector `vec a_1` and T on `l_2`, with position vector `vec a_2`. Then the magnitude of the shortest distance vector will be equal to that of the projection of ST along the direction of the line of shortest distance.

If `vec (PQ)` is the shortest distance vector between `l_1` and `l_2` , then it being perpendicular to both `vec b_1` and `vec b_2` , the unit vector `hat n` along `vec (PQ)` would therefore be

`hat n = (vec b_1 xx vec b_2)/ |vec b_1 xx vec b_2|` ...(3)

Then `vec (PQ) = d . hat n `

where, d is the magnitude of the shortest distance vector. Let θ be the angle between `vec (ST)` and `vec (PQ).` Then

`PQ = ST |cos θ| `

But cos θ `= |(vec (PQ) . vec (ST))/(|vec (PQ)| |vec (ST)|)| `

`= |(d . hat n (vec a_2 - vec a_1)) / (d ST)|` (since `vec (ST) = vec a_2 - vec a_1`)

`= |((vec b _1 xx vec b_2) . (vec a_2 - vec a_1)) /( ST |vec b_1 xx vec b_2|)|` [From (3)]

Hence, the required shortest distance is

d = PQ =ST |cos θ|

or d=` |((vec b_1 xx vec b_2) . (vec a_2 xx vec a_1))/(|vec b_1 xx vec b_2|)|`

The shortest distance between the lines

`l_1 : (x - x_1)/a_1 = (y - y_1)/b_1 = (z - z_1)/c_1`

and `l_2 : (x - x_2)/a_2 = (y - y_2)/b_2 = (z - z_2)/c_2`

is `||(x_2-x_1 , y_2 - y_1 , z_2 - z_1),(a_1 , b_1 , c_1), (a_2 , b_2 , c_2)|/ sqrt ((b_1c_2 - b_2c_1)^2 + (c_1a_2 - c_2a_1)^2 + (a_1b_2 - a_2b_1)^2) |`

Video link : https://youtu.be/BXzj9mJvTKQ

**Distance between parallel lines: **

If two lines `l_1` and `l_2` are parallel, then they are coplanar. Let the lines be given by

`vec r = vec a_1 + lambda vec b` ...(1)

and `vec r = vec a_2 + mu vec b` ...(2)

where , `vec a_1` is the position vector of a point S on `l_1` and `vec a_2` is the position vector of a point T on `l_2` in following fig.

As `l_1, l_2` are coplanar, if the foot of the perpendicular from T on the line `l_1` is P, then the distance between the lines `l_1` and `l_2` = |TP|.

Let θ be the angle between the vectors `vec (ST)` and `vec b`. Then

`vec b xx vec (ST) = ( |vec b| |vec (ST)| sin θ) hat n ` ...(3)

where `hat n` is the unit vector perpendicular to the plane of the lines `l_1` and `l_2` But `vec (ST) = vec a_2 -vec a_1`

Therefore, from (3), we get

`vec b xx (vec a_2 - vec a_1) = |vec b| PT hat n` (since PT =ST sin θ )

i.e. `|vec b xx (vec a_2 - vec a_1)| = |vec b| PT . 1` (as `|hat n|` = 1)

Hence, the distance between the given parallel lines is

`d = |vec (PT)| = |(vec b xx (vec a_2 - vec a _1))/|vec b||`