#### Topics

##### Mathematical Logic

- Statements - Introduction in Logic
- Sentences and Statement in Logic
- Truth Value of Statement
- Open Sentences in Logic
- Compound Statement in Logic
- Quantifier and Quantified Statements in Logic
- Logical Connective, Simple and Compound Statements
- Logical Connective, Simple and 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

##### 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

##### 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
- Scalar Product of Vectors (Dot)
- Vector Product of Vectors (Cross)
- Scalar Triple Product of Vectors
- Vector Triple Product
- Addition of Vectors

##### Line and Plane

##### Linear Programming

##### Matrices

- Elementary Transformations
- 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
- Solutions of 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

#### definition

A random variable is a real valued function whose domain is the sample space of a random experiment.

For example, let us consider the experiment of tossing a coin two times in succession.

The sample space of the experiment is S = {HH, HT, TH, TT}.

If X denotes the number of heads obtained, then X is a random variable and for each outcome, its value is as given below :

X(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0.

More than one random variables can be defined on the same sample space. For example, let Y denote the number of heads minus the number of tails for each outcome of the above sample space S.

Then Y(HH) = 2, Y (HT) = 0, Y (TH) = 0, Y (TT) = – 2.

Thus, X and Y are two different random variables defined on the same sample space S.

#### text

In most of the random experiments , we were not on;ly interested in the particular outcome that occurs but rather in some number associated with that outcomes.

Example :

(i) In tossing two dice, we may be interested in the sum of the numbers on the two dice.

(ii) In tossing a coin 50 times, we may want the number of heads obtained.

(iii) In the experiment of taking out four articles (one after the other) at random from a lot of 20 articles in which 6 are defective, we want to know the number of defectives in the sample of four and not in the particular sequence of defective and nondefective articles.

In all of the above experiments, We have a rule which assigns to each outcome of the experiment a single real number.

This single real number may vary with different outcomes of the experiment.

Its value depends upon the outcome of a random experiments.

It is called random variable.

Random variable is denoted by X.

Video link : https://youtu.be/M-Wz5wRhaNQ

#### notes

**Probability distribution of a random variable:**

The experiment of selecting one family out of ten families `f_1, f_2 ,..., f_10` in such a manner that each family is equally likely to be selected. Let the families `f_1, f_2, ... , f_10` have 3, 4, 3, 2, 5, 4, 3, 6, 4, 5 members, respectively.

Let us select a family and note down the number of members in the family denoting X. Clearly, X is a random variable defined as below :

`X(f_1) = 3, X(f_2) = 4, X(f_3) = 3, X(f_4) = 2, X(f_5) = 5, X(f_6) = 4, X(f_7) = 3, X(f_8) = 6, X(f_9) = 4, X(f_10) = 5 `

Thus, X can take any value 2,3,4,5 or 6 depending upon which family is selected. Now, X will take the value 2 when the family `f_4` is selected. X can take the value 3 when any one of the families `f_1, f_3, f_7` is selected.

Similarly,

X = 4, when family `f_2`, `f_6` or `f_9` is selected,

X = 5, when family `f_5` or `f_10` is selected

and

X = 6, when family `f_8` is selected.

Since we had assumed that each family is equally likely to be selected, the probability that family `f_4` is selected is `1/10`.

Thus, the probability that X can take the value 2 is `1/10`. We write P(X = 2) = `1/10`

Also, the probability that any one of the families` f_1, f_3 "or" f_7` is selected is `P({f_1, f_3, f_7}) `= `3/10`

Thus, the probability that X can take the value 3 = `3/10`

We write P(X = 3) = `3/10`

Similarly, we obtain P(X = 4) = `P({f_2, f_6, f_9}) = 3/10`

P(X = 5) = `P({f_5, f_10}) = 2/10`

and P(X = 6) = `P({f_8}) = 1/10`

The values of the random variable along with the corresponding probabilities is called the probability distribution of the random variable X

**Defination :** The probability distribution of a random variable X is the system of numbers

X | `x_1` | `x_2` | ... | `x_n` |

P(X) | `p_1` | `p_2` | ... | `p_n` |

where , `p_i > 0 sum_(i=1)^n p_i = 1 , i = 1 , 2,..., n`

The real numbers `x_1, x_2,..., x_n` are the possible values of the random variable X and `p_i (i = 1,2,..., n)` is the probability of the random variable X taking the value `x_i` i.e., `P(X = x_i) = p_i`

Video link : https://youtu.be/HYKdhi0c-KM

#### Video Tutorials

#### Shaalaa.com | Probability Distribution

#### Related QuestionsVIEW ALL [38]

A random variable X has the following probability distribution

X |
2 | 3 | 4 |

P(x) |
0.3 | 0.4 | 0.3 |

Then the variance of this distribution is

**Solve the following problem :**

Following is the probability distribution of a r.v.X.

X |
– 3 | – 2 | –1 | 0 | 1 | 2 | 3 |

P(X = x) |
0.05 | 0.1 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |

Find the probability that X is positive.