#### Topics

##### Relations and Functions

##### Relations and Functions

##### Inverse Trigonometric Functions

##### Algebra

##### Matrices

- Introduction of Operations on Matrices
- Inverse of a Matrix by Elementary Transformation
- Multiplication of Two Matrices
- Negative of Matrix
- Properties of Matrix Addition
- Transpose of a Matrix
- Subtraction of Matrices
- Addition of Matrices
- Symmetric and Skew Symmetric Matrices
- Types of Matrices
- Proof of the Uniqueness of Inverse
- Invertible Matrices
- Elementary Transformations
- Multiplication of Matrices
- Properties of Multiplication of Matrices
- Equality of Matrices
- Order of a Matrix
- Matrices Notation
- Introduction of Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Scalar Multiplication of a Matrix
- Properties of Transpose of the Matrices

##### Calculus

##### Vectors and Three-dimensional Geometry

##### Determinants

- Applications of Determinants and Matrices
- Elementary Transformations
- Inverse of a Square Matrix by the Adjoint Method
- Properties of Determinants
- Determinant of a Square Matrix
- Determinants of Matrix of Order One and Two
- Introduction of Determinant
- Area of a Triangle
- Minors and Co-factors
- Determinant of a Matrix of Order 3 × 3
- Rule A=KB

##### Linear Programming

##### Continuity and Differentiability

- Derivative - Exponential and Log
- Concept of Differentiability
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Algebra of Continuous Functions
- Continuous Function of Point
- Mean Value Theorem
- Second Order Derivative
- Derivatives of Functions in Parametric Forms
- Logarithmic Differentiation
- Exponential and Logarithmic Functions
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Composite Functions - Chain Rule
- Concept of Continuity

##### Probability

##### Applications of Derivatives

- Maximum and Minimum Values of a Function in a Closed Interval
- Maxima and Minima
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
- Increasing and Decreasing Functions
- Rate of Change of Bodies or Quantities
- Introduction to Applications of Derivatives

##### Sets

##### Integrals

- Definite Integrals Problems
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integrals
- Integrals of Some Particular Functions
- Indefinite Integral by Inspection
- Some Properties of Indefinite Integral
- Integration Using Trigonometric Identities
- Introduction of Integrals
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Fundamental Theorem of Calculus
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
- Methods of Integration: Integration by Parts
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Substitution
- Integration as an Inverse Process of Differentiation

##### Applications of the Integrals

##### Differential Equations

- Linear Differential Equations
- Solutions of Linear Differential Equation
- Homogeneous Differential Equations
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- General and Particular Solutions of a Differential Equation
- Order and Degree of a Differential Equation
- Basic Concepts of Differential Equation
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves

##### Vectors

- Direction Cosines
- Properties of Vector Addition
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Multiplication of a Vector by a Scalar
- Addition of Vectors
- Introduction of Vector
- Magnitude and Direction of a Vector
- Basic Concepts of Vector Algebra
- Vectors and Their Types
- Components of Vector
- Section Formula
- Vector Joining Two Points
- Vectors Examples and Solutions
- Projection of a Vector on a Line
- Introduction of Product of Two Vectors

##### Three - Dimensional Geometry

- Three - Dimensional Geometry Examples and Solutions
- Introduction of Three Dimensional Geometry
- Equation of a Plane Passing Through Three Non Collinear Points
- Relation Between Direction Ratio and Direction Cosines
- Intercept Form of the Equation of a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Angle Between Two Lines
- Vector and Cartesian Equation of a Plane
- Shortest Distance Between Two Lines
- Equation of a Line in Space
- Direction Cosines and Direction Ratios of a Line
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Plane Passing Through the Intersection of Two Given Planes

##### Linear Programming

##### Probability

- Variance of a Random Variable
- Probability Examples and Solutions
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Introduction of Probability
- Properties of Conditional Probability

- Property 1 - The value of the determinant remains unchanged if its rows are turned into columns and columns are turned into rows.
- Property 2 - If any two rows (or columns) of a determinant are interchanged then the value of the determinant changes only in sign.
- Property 3 - If any two rows ( or columns) of a determinant are identical then the value of the determinant is zero.
- Property 4 - If each element of a row (or column) of a determinant is multiplied by a constant k then the value of the new determinant is k times the value of the original determinant.
- Property 5 - If each element of a row (or column) is expressed as the sum of two numbers then the determinant can be expressed as the sum of two determinants
- Property 6 - If a constant multiple of all elements of any row (or column) is added to the corresponding elements of any other row (or column ) then the value of the new determinant so obtained is the same as that of the original determinant.
- Property 7 - (Triangle property) - If all the elements of a determinant above or below the diagonal are zero then the value of the determinant is equal to the product of its diagonal elements.

## Notes

In this section, some properties of determinants which simplifies its evaluation by obtaining maximum number of zeros in a row or a column. These properties are true for determinants of any order.

**Property 1:** The value of the determinant remains unchanged if its rows and columns are interchanged.

Verification Let `triangle = |(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|`

Expanding along first row, we get

`triangle = a_1|(b_2,b_3),(c_2,c_3)| - a_2 |(b_1,b_3),(c_1,c_3)| + a_3 |(b_1,b_2),(c_1,c_2)| `

= `a_1 (b_2 c_3 – b_3 c_2) – a_2 (b_1 c_3 – b_3 c_1) + a_3 (b_1 c_2 – b_2 c_1)`

By interchanging the rows and columns of ∆, we get the determinant

`∆_1 = |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

Expanding `∆_1` along first column, we get

`∆_1 = a_1 (b_2 c_3 – c_2 b_3) – a_2 (b_1 c_3 – b_3 c_1) + a_3 (b_1 c_2 – b_2 c_1)`

Hence ∆ = `∆_1`**Remark:** It follows from above property that if A is a square matrix, then det (A) = det (A′), where A′ = transpose of A.

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

**Property 2:** If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

Verification Let ∆ =`|(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|`

Expanding along first row, we get

∆ =` a_1 (b_2 c_3 – b_3 c_2) – a_2 (b_1 c_3 – b_3 c_1) + a_3 (b_1 c_2 – b_2 c_1)`

Interchanging first and third rows, the new determinant obtained is given by

`∆_1 = |(c_1,c_2,c_3),(b_1,b_2,b_3),(a_1,a_2,a_3)|`

Expanding along third row, we get

`∆_1 = a_1 (c_2 b_3 – b_2 c_3) – a_2 (c_1 b_3 – c_3 b_1) + a_3 (b_2 c_1 – b_1 c_2)

= – [a_1 (b_2 c_3 – b_3 c_2) – a_2 (b_1 c_3 – b_3 c_1) + a_3 (b_1 c_2 – b_2 c_1)] `

Clearly `∆_1` = – ∆

Similarly, we can verify the result by interchanging any two columns.

**Property 3:** If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.

Proof: If we interchange the identical rows (or columns) of the determinant ∆, then ∆ does not change. However, by Property 2, it follows that ∆ has changed its sign

Therefore ∆ = – ∆

or ∆ = 0

**Property 4:** If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

Verification Let ∆ = `|(a_1,b_1,c_1 ),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

and `∆_1` be the determinant obtained by multiplying the elements of the first row by k. Then

`∆_1 = |(ka_1,kb_1,kc_1 ),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

Expanding along first row, we get

`∆_1 = k a_1 (b_2 c_3 – b_3 c_2) – k b_1 (a_2 c_3 – c_2 a_3) + k c_1 (a_2 b_3 – b_2 a_3)`

= `k [a_1 (b_2 c_3 – b_3 c_2) – b_1 (a_2 c_3 – c_2 a_3) + c_1 (a_2 b_3 – b_2 a_3)]`

= `k ∆`

Hence ` |(ka_1,kb_1,kc_1 ),(a_2,b_2,c_2),(a_3,b_3,c_3)|` = k `|(a_1,b_1,c_1 ),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

**Remarks:**

(i) By this property, we can take out any common factor from any one row or any one column of a given determinant.

(ii) If corresponding elements of any two rows (or columns) of a determinant are proportional (in the same ratio), then its value is zero. For example

∆ = `|(a_1,a_2,a_3 ),(b_1,b_2,b_3),(ka_1,ka_2,ka_3)|` =0 (rows `R_1` and `R_2` are proportional)

Video link : https://youtu.be/KIwfdtyCjV4

**Property 5:** If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

For example, `|(a_1+lambda_1 , a_2 + lambda_2 , a_3 + lambda_3),(b_1,b_2 ,b_3),(c_1,c_2,c_3)|` = `|(a_1,a_2,a_3),(b

_1,b_2,b_3), (c_1,c_2,c_3)|` + `|(lambda_1,lambda_2,lambda_3),( b_1,b_2,b_3), (c_1,c_2,c_3)|`

Verification L.H.S.

= `|(a_1+lambda_1 , a_2 + lambda_2 , a_3 + lambda_3),(b_1,b_2 ,b_3),(c_1,c_2,c_3)|`

Expanding the determinants along the first row, we get

∆ = `(a_1 + λ_1) (b_2 c_3 – c_2 b_3) – (a_2 + λ_2) (b_1 c_3 – b_3 c_1) + (a_3 + λ_3) (b_1 c_2 – b_2 c_1)`

`= a_1 (b_2 c_3 – c_2 b_3) – a_2 (b_1 c_3 – b_3 c_1) + a_3 (b_1 c_2 – b_2 c_1) + λ_1 (b_2 c_3 – c_2 b_3) – λ_2 (b_1 c_3 – b_3 c_1) + λ_3 (b_1 c_2 – b_2 c_1) ` (by arranging terms)

`= |(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)| + |(lambda_1,lambda_2,lambda_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|` = R.H.S.

Video link : https://youtu.be/09-fQzshves

**Property 6:** If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation `R_i → R_i + kR_j or C_i → C_i + kC_j. `

Verification Let

∆ =` |(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|` and

`∆_1 = |((a_1 +kc_1),(a_2+kc_2),(a_3 + kc_3)),(b_1,b_2,b_3),(c_1,c_2,c_3)|`,

where `∆_1` is obtained by the operation `R_1 → R_1 + kR_3` . Here, we have multiplied the elements of the third row `(R_3)` by a constant k and added them to the corresponding elements of the first row `(R_1)`.

Symbolically, we write this operation as `R_1 → R_1 + k R_3`.

Now, again `∆_1 = |(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)| + |(kc_1,kc_2,kc_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|` (Using Property 5)

= ∆ + 0 (since `R_1` and `R_3` are proportional)

Hence ∆ = `∆_1`

**Remarks: **

(i) If ∆1 is the determinant obtained by applying `R_i → kR_i or C_i → kC_i` to the determinant ∆, then `∆_1` = k∆.

(ii) If more than one operation like `R_i → R_i + kR_j` is done in one step, care should be taken to see that a row that is affected in one operation should not be used in another operation. A similar remark applies to column operations.

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