#### 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

- 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

## Formula

Area of triangle = `(1/2) × "base" × "height" = 1/2 × b × h`.

## Notes

**Area of a triangle:**

To get the area of the triangle, we first choose one of the sides to be the base (b). Then we draw a perpendicular line segment from a vertex of the triangle to the base. This is the height (h) of the triangle. The area of a triangle is equal to half the product of the base and the height.

Area of triangle = `(1/2) × "base" × "height" = 1/2 × b × h`

**All the congruent triangles are equal in the area but the triangles equal in the area need not be congruent.**

**Areas of different types of triangles:**

Consider an acute and an obtuse triangle.

Area of each triangle = `(1/2) × "base" × "height" = 1/2 × b × h`

## Example

Find the area of the following triangle:

Area of triangle = `1/2bh`

= `1/2` × QR × PS

= `1/2` × 4cm × 2cm

= 4 cm^{2}

## Example

Find the area of the following triangle:

Area of triangle = `1/2bh`

= `1/2` × MN × LO

= `1/2` × 3cm × 2cm

= 3 cm^{2}

## Example

Find BC, if the area of the triangle ABC is 36 cm^{2} and the height AD is 3 cm.

Height = 3 cm, Area = 36 cm^{2}

Area of the triangle ABC =`1/2bh`

36 = `1/2` × b × 3

b = `(36 xx 2)/3` = 24 cm

So, BC = 24 cm.

## Example

In ∆PQR, PR = 8 cm, QR = 4 cm and PL = 5 cm.

Find:

(i) the area of the ∆PQR

(ii) QM.

(i) QR = base = 4 cm, PL = height = 5 cm

Area of the triangle PQR =`1/2bh`

= `1/2 xx 4 xx 5`

= 10 cm^{2}

(ii) PR = base = 8 cm, QM = height = ?, Area = 10 cm^{2}

Area of triangle = `1/2 xx b xx h`

10 = `1/2 xx 8 xx h`

h = `10/4 = 5/2 = 2.5`.

QM = 2.5 cm