The syllabus for Class 10 Mathematics typically covers a range of topics that build on the concepts learned in previous years and prepare students for more advanced studies. Here's a complete summary of the major topics usually included in the Class 10 Mathematics syllabus:

### 1. **Real Numbers**
- **Euclid's Division Lemma**: Understanding the concept of divisibility and applying it to find the Highest Common Factor (HCF).
- **Fundamental Theorem of Arithmetic**: Prime factorization of integers and the concept of HCF and LCM.
- **Irrational Numbers**: Proof of irrationality of numbers like √2, √3, etc.
- **Decimal Expansions**: Representation of rational numbers as terminating or non-terminating recurring decimals.

### 2. **Polynomials**
- **Polynomials and Their Degrees**: Understanding the concept of polynomials, degrees, and coefficients.
- **Zeros of a Polynomial**: Relationship between zeros and coefficients of quadratic polynomials.
- **Division Algorithm**: Division of polynomials and finding remainders.

### 3. **Pair of Linear Equations in Two Variables**
- **Graphical Method**: Solving linear equations by plotting on a graph.
- **Algebraic Methods**: Substitution, elimination, and cross-multiplication methods to solve equations.
- **Equations Reducible to Linear Form**: Converting non-linear equations to linear form and solving.

### 4. **Quadratic Equations**
- **Standard Form of Quadratic Equations**: Understanding ax² + bx + c = 0.
- **Solutions of Quadratic Equations**: Methods such as factorization and the quadratic formula.
- **Nature of Roots**: Discriminant and its significance in determining the nature of roots.

### 5. **Arithmetic Progressions (AP)**
- **General Form of AP**: Understanding the common difference and nth term.
- **Sum of First n Terms**: Deriving and applying the formula for the sum of an arithmetic progression.

### 6. **Triangles**
- **Similar Triangles**: Criteria for similarity of triangles (AA, SAS, SSS).
- **Theorems on Similarity**: Basic proportionality theorem (Thales theorem), and its applications.
- **Pythagoras Theorem**: Proof and applications in problems.

### 7. **Coordinate Geometry**
- **Distance Formula**: Calculating the distance between two points.
- **Section Formula**: Finding the coordinates of a point dividing a line segment in a given ratio.
- **Area of a Triangle**: Using coordinate points to determine the area of a triangle.

### 8. **Introduction to Trigonometry**
- **Trigonometric Ratios**: Definitions of sine, cosine, tangent, and their reciprocals.
- **Trigonometric Identities**: Fundamental identities and their applications.
- **Trigonometric Ratios of Complementary Angles**: Relationships between the ratios of complementary angles.

### 9. **Some Applications of Trigonometry**
- **Height and Distance**: Solving real-life problems involving angles of elevation and depression.

### 10. **Circles**
- **Tangent to a Circle**: Properties of tangents, including the perpendicularity of the radius to the tangent at the point of contact.
- **Number of Tangents from a Point**: Tangents drawn from an external point to a circle.

### 11. **Constructions**
- **Division of a Line Segment**: Dividing a line segment in a given ratio.
- **Construction of Similar Triangles**: Constructing triangles similar to a given triangle.
- **Tangent to a Circle**: Constructing tangents from an external point to a circle.

### 12. **Areas Related to Circles**
- **Perimeter and Area**: Calculating the perimeter and area of circles, sectors, and segments.
- **Problems Involving Area of Combinations**: Solving problems related to the area of combinations of figures involving circles.

### 13. **Surface Areas and Volumes**
- **Surface Area**: Calculating the surface area of solids like cubes, cuboids, spheres, hemispheres, cones, and cylinders.
- **Volume**: Finding the volume of these solids.
- **Conversion of Solids**: Problems involving the conversion of one type of solid into another.

### 14. **Statistics**
- **Mean, Median, and Mode**: Measures of central tendency for grouped data.
- **Cumulative Frequency**: Understanding and interpreting cumulative frequency curves (ogives).

### 15. **Probability**
- **Classical Definition of Probability**: Basic concepts of probability, including the probability of a simple event.
- **Probability of Complementary Events**: Understanding and applying the concept of complementary events.

Summary
This syllabus provides a strong foundation in various mathematical concepts, from basic arithmetic and algebra to geometry, trigonometry, and probability. It helps in developing problem-solving skills and logical thinking, essential for higher studies and practical applications in daily life.