Algebra Class 10

Equations and Inequalities:
- Linear equations: ax + by = c, where a, b, and c are constants
- Quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
- Solving equations using substitution, elimination, and graphical methods
- Inequalities: linear and quadratic, solving using graphical and algebraic methods
Functions:
- Domain and range
- Composition of functions
- Inverse functions
- Graphical representation of functions
Matrices:
- Definition and notation
- Operations: addition, subtraction, multiplication, and inversion
- Determinants: calculation and properties
- Applications: solving systems of linear equations and finding eigenvalues

Limits:
- Definition and properties
- Rules of limits: sum, product, and chain rule
- Squeeze theorem
Derivatives:
- Definition and geometric interpretation
- Rules of differentiation: power rule, product rule, and quotient rule
- Derivatives of common functions: trigonometric, exponential, and logarithmic
- Higher-order derivatives
Applications of Derivatives:
- Finding maxima and minima
- Motion along a line, curve, and surface
- Optimization problems

Angles and Triangles:
- Degree and radian measure
- Trigonometric ratios: sine, cosine, and tangent
- Identities: Pythagorean and sum and difference formulas
Graphs of Trigonometric Functions:
- Sine, cosine, and tangent waves
- Amplitude, period, and phase shift
Identities and Equations:
- Solving trigonometric equations
- Proving trigonometric identities

Vectors:
- Definition and notation
- Operations: addition, subtraction, and scalar multiplication
- Magnitude and direction
Vector Products:
- Dot product: definition and properties
- Cross product: definition and properties
- Applications: finding the angle between vectors and finding the area of a parallelogram
Vector Equations:
- Solving vector equations
- Applications: finding the equation of a line and a plane

Equations and Inequalities
- Linear equations have the form ax + by = c, where a, b, and c are constants
- Quadratic equations have the form ax^2 + bx + c = 0, where a, b, and c are constants
- Equations can be solved using substitution, elimination, and graphical methods
- Inequalities can be linear or quadratic and are solved using graphical and algebraic methods
Functions
- Domain is the set of input values, while range is the set of output values
- Composition of functions is denoted as (f ∘ g)(x) = f(g(x))
- Inverse functions are denoted as f^(-1) and satisfy f(f^(-1)(x)) = x
- Graphical representation of functions helps in understanding their behavior
Matrices
- Matrices are denoted as [a_ij] and have m rows and n columns
- Operations on matrices include addition, subtraction, multiplication, and inversion
- Determinants are used to find the solvability of systems of linear equations and eigenvalues
- Applications of matrices include solving systems of linear equations and finding eigenvalues

Limits
- Limits are used to define the behavior of a function as the input approaches a certain value
- Rules of limits include sum, product, and chain rule
- Squeeze theorem is used to find the limit of a function by bounding it between two other functions
Derivatives
- Derivatives are used to measure the rate of change of a function
- Geometric interpretation of derivatives is the slope of the tangent line
- Rules of differentiation include power rule, product rule, and quotient rule
- Higher-order derivatives are used to find the rate of change of the rate of change
Applications of Derivatives
- Finding maxima and minima is used to optimize functions
- Motion along a line, curve, and surface is used to model real-world problems
- Optimization problems are used to find the best solution among many possible solutions

Angles and Triangles
- Degree measure is used for angles, while radian measure is used for trigonometric functions
- Trigonometric ratios are sine, cosine, and tangent, and are used to relate the angles of a triangle
- Identities include Pythagorean and sum and difference formulas
Graphs of Trigonometric Functions
- Sine, cosine, and tangent waves have amplitude, period, and phase shift
- Graphs of trigonometric functions are used to model periodic phenomena
Identities and Equations
- Solving trigonometric equations is used to find the values of unknown angles
- Proving trigonometric identities is used to establish relationships between different trigonometric functions

Dot product is used to find the angle between two vectors
Cross product is used to find the area of a parallelogram
Applications of vector products include finding the angle between vectors and finding the area of a parallelogram